Fourier based statistics for irregular spaced spatial …suhasini/papers/fourier_based...Fourier based statistics for irregular spaced spatial data Suhasini Subba Rao Department of

Fourier based statistics for irregular spaced spatial data

Suhasini Subba Rao

Department of Statistics,

Texas A&M University,

College Station, U.S.A.

[email protected]

December 8, 2015

Abstract

A class of Fourier based statistics for irregular spaced spatial data is introduced, examples

include, the Whittle likelihood, a parametric estimator of the covariance function based on the

L2-contrast function and a simple nonparametric estimator of the spatial autocovariance which

is a non-negative function. The Fourier based statistic is a quadratic form of a discrete Fourier-

type transform of the spatial data. Evaluation of the statistic is computationally tractable,

requiring O(nb) operations, where b are the number Fourier frequencies used in the definition of

the statistic (which varies according to the application) and n is the sample size. The asymptotic

sampling properties of the statistic are derived using mixed spatial asymptotics, where the

number of locations grows at a faster rate than the size of the spatial domain and under the

assumption that the spatial random field is stationary and the irregular design of the locations

are independent, identically distributed random variables. The asymptotic analysis allows for

the case that the frequency grid over which the estimator is defined is unbounded. These results

are used to obtain the asymptotic sampling properties of certain estimators and also in variance

estimation.

Keywords and phrases: Fixed and increasing frequency domain asymptotics, mixed spatial

asymptotics, random locations, spectral density function, stationary spatial random fields.

1 Introduction

In recent years irregular spaced spatial data has become ubiquitous in several disciplines as varied

as the geosciences to econometrics. The analysis of such data poses several challenges which do

not arise in data which is sampled on a regular lattice. One important problem is the computa-

tional costs when dealing with large irregular sampled data sets. If spatial data are sampled on

a regular lattice then algorithms such as the Fast Fourier transform can be employed to reduce

the computational burden (see, for example, Chen, Hurvich, and Lu (2006)). Unfortunately, such

algorithms have little benefit if the spatial data are irregularly sampled. To address this issue,

within the spatial domain, several authors, including, Vecchia (1988), Cressie and Huang (1999),

1

Stein, Chi, and Welty (2004), have proposed estimation methods which are designed to reduce the

computational burden.

In contrast to the above references, Fuentes (2007) argues that working within the frequency

domain can simplify the problem. Fuentes assumes that the irregular spaced data can be embed-

ded on a grid and the missing mechanism is deterministic and ‘locally smooth’. Based on these

assumptions Fuentes proposes a tapered Whittle estimator to estimate the parameters of a spatial

covariance function. However, a possible drawback is that, if the locations are extremely irregular,

the local smooth assumption will not hold. Therefore, in order to work within the frequency domain,

methodology and inference devised specifically for irregular sampled data is required. Matsuda and

Yajima (2009) and Bandyopadhyay and Lahiri (2009) have pioneered this approach, by assuming

that the irregular locations are independent, identically distributed random variables (thus allow-

ing the data to be extremely irregular) and define the irregular sampled discrete Fourier transform

(DFT) as

Jn(!) =�d/2

n

nX

j=1

Z(sj) exp(is0j!), (1)

where sj 2 [��/2,�/2]d denotes the spatial locations observed in the space [��/2,�/2]d and

{Z(sj)} denotes the spatial random field at these locations. It’s worth mentioning a similar trans-

formation on irregular sampled data goes back to Masry (1978), who defines the discrete Fourier

transform of Poisson sampled continuous time series. Using the definition of the DFT given in (1)

Matsuda and Yajima (2009) define the Whittle likelihood for the spatial data asZ

⌦

✓log f✓(!) +

|Jn(!)|2

f✓(!)

◆d!, (2)

where ⌦ is a compact set in Rd, here we will assume that ⌦ = 2⇡[�C,C]d and f✓(!) is the

candidate parametric spectral density function. A clear advantage of this approach is that it avoids

the inversion of a large matrix. However, one still needs to evaluate the integral, which can be

computationally quite di�cult, especially if the aim is to minimise over the entire parameter space

of ✓.

For frequency domain methods to be computationally tractable and to be used as a viable

alternative to ‘spatial domain’ methods, the frequency domain needs to be gridified in such a way

that tangable estimators are defined on the grid. The problem is how to choose an appropriate

lattice on Rd. A possible solution can be found in Bandyopadhyay and Subba Rao (2015), Theorem

2.1, who show that on the lattice {!k = (2⇡k1� , . . . , 2⇡kd� );k = (k1

, . . . , kd) 2 Zd} the DFT {Jn(!k)}is ‘close to uncorrelated’ if the random field is second order stationary and the locations {sj} are

independent, identical, uniformly distributed random variables. Heuristically, this result suggests

that {Jn(!k)} contains all the information about the random field, such that one can perform the

analysis on the transformed data {Jn(!k)}. For example, rather than use (2) one can use the

discretized likelihood

1

�d

C�X

k1,...,kd=�C�

✓log f✓(!k) +

|Jn(!k)|2

f✓(!k)

◆, (3)

2

to estimate the parameters. Indeed Matsuda and Yajima (2009), Remark 2, mention that in practice

one should use the discretized likelihood to estimate the parameters, but, unfortunately, they did

not derive any results for the discretized likelihood.

Integrated periodogram statistics such as the discretized Whittle likelihood defined above, are

widely used in time series and their properties well understood. However, with the exception of

Bandyopadhyay, Lahiri, and Norman (2015), as far as we are aware, there does not exist any results

for integrated periodograms of irregularly sampled spatial data. Therefore one of the objectives of

this paper is to study the class of integrated statistics which have the form

Qa,�(g✓; r) =1

�d

aX

k1,...,kd=�a

g✓(!k)Jn(!k)Jn(!k+r) r 2 Zd, (4)

where ✓ is a finite dimension parameter, Jn(!k) is defined in (1) and g✓ : Rd ! R is a weight function

which depends on the application. Note that for r = 0, a = C� and g✓(!) = f✓(!)�1 we have the

discretized Whittle likelihood (up to a non-random constant). We have chosen to consider the case

r 6= 0 because in the case that the locations are uniformly sampled, Qa,�(g✓; 0) and {Qa,�(g✓; r); r 6=0} asymptotically have the same variance, which we exploit to estimate the variance of Qa,�(g✓; 0)

in Section 5. In terms of computation, evaluation of {Jn(!k);k = (k1

, . . . , kd), kj = �a, . . . , a}requires O(adn) operations (as far as are aware the FFT cannot be used to reduce the number of

operations for irregular spaced data). However, once {Jn(!k);k = (k1

, . . . , kd), kj = �a, . . . , a}has been evaluated the evaluation of Qa,�(g✓; r) only requires O(ad) operations.

To understand why such a class of statistics may be of interest in spatial statistics, consider the

case r = 0 and replace |Jn(!k)|2 in Qa,�(g✓; 0) with the spectral density, f : Rd ! R of the spatial

process {Z(s); s 2 Rd}. This replacement suggests Qa,�(g✓; 0) is estimating the functional I(g✓;a�)

where

I⇣g✓;

a

�

⌘=

1

(2⇡)d

Z

[�a/�,a/�]dg✓(!)f(!)d!. (5)

In Section 3 we show that this conjecture is true up to a multiplicative constant. Before giving

examples of functionals I�g✓;

a�

�of interest we first compare Qa,�(g✓;0) to the integrated peri-

odogram estimators which are often used in time series (see for example, Walker (1964), Hannan

(1971), Dunsmuir (1979), Dahlhaus and Janas (1996), Can, Mikosch, and Samorodnitsky (2010)

and Niebuhr and Kreiss (2014)). However, there are some fundamental di↵erences, which mean the

analysis of (4) is very di↵erent to those in classical time series. We observe that the user-chosen

frequency grid {!k;k = (k1

, . . . , kd), kj 2 [�a, a]} used to define Qa,�(g✓; r) influences the limits

of the functional I(g; a�). This is di↵erent to regular sampled locations where frequency grid is

defined over [0, 2⇡]d (outside this domain aliasing occurs). Returning to irregular sampled spatial

data, in certain situations, such as the Whittle likelihood described in (3) or the spectral density

estimator (described in Section 2), bounding the frequency grid to a = C� is necessary. In the

case of the Whittle likelihood this is important, because |f✓(!)| ! 0 as k!k ! 1 (for any norm

k · k), thus the discretized Whittle likelihood is only well defined over a bounded frequency grid.

3

The choice of C is tied to how fast the tails in the parametric class of spectral density functions

{f✓; ✓ 2 ⇥} (where ⇥ is a compactly supported parameter space) decay to zero. In contrast, in

most situations the aim is to estimate the functional I (g✓;1). For example, the spatial covariance

can be written as c(v) = I(eiv·;1) or a loss function defined over all frequencies. In this case, the

only plausible method for estimating I(g✓; r) without asymptotic bias is to allow the frequency grid

{!k;k = (k1

, . . . , kd),�a ki a} to be unbounded, in other words let a/� ! 1 as � ! 1. In

which case I(g✓;a�) ! I(g;1) as the domain � ! 1. To understand whether this is feasible, we

recall that if the spatial process were observed only on a grid then aliasing means that it is impossi-

ble to estimate I(g✓;1). On the other hand, unlike regularly spaced or near regularly spaced data,

‘truely’ irregular sampling means that the DFT can estimate high frequencies, without the curse

of aliasing (a phenomena which was noticed as early as Shapiro and Silverman (1960) and Beutler

(1970)). In this case, in the definition of Qa,�(g✓; r) there’s no need for the frequency grid to be

bounded, and a can be magnitudes larger than �. One contribution of this paper is to understand

how an unbounded frequency grid influences inference.

As alluded to in the previous paragraph, in this paper we derive the asymptotic sampling prop-

erties of Qa,�(g✓; r) under the asymptotic set-up that the spatial domain � ! 1. The main focus

will be the mixed asymptotic framework, introduced in Hall and Patil (1994) (see also Hall, Fisher,

and Ho↵man (1994) and used in, for example, Lahiri (2003), Matsuda and Yajima (2009), Bandy-

opadhyay and Lahiri (2009), Bandyopadhyay et al. (2015) and Bandyopadhyay and Subba Rao

(2015)). This is where � ! 1 (the size of the spatial domain grows) and the number of obser-

vations n ! 1 in such a way that �d/n ! 0. In other words, the sampling density gets more

dense as the spatial domain grows. The results di↵er slightly under the pure increasing domain

framework (see Bandyopadhyay et al. (2015) who compares the mixed and pure increasing domain

framework), where �d/n ! c (0 < c < 1) as n ! 1 and � ! 1. In Section 3.3 we outline some of

these di↵erences. We should be mention that under the fixed-domain asymptotic framework (where

� is kept fixed but the number of locations, n grows) considered in Stein (1994, 1999), Qa,�(g✓; r)

cannot consistently estimate I(g✓;a�). There is a non-trivial bias (which can be calculated using

the results in Section 3.1) and a variance that does not asymptotically converge to zero as n ! 1.

Obtaining the moments of Qa,�(g✓; r) in the case that a = O(�) is analogous to those for regular

sampled time series and spatial processes and follows from the DFT analysis given in Section 2.1.

However, when the frequency grid is unbounded, the number of terms within the summand of

Qa,�(g✓; r) is O(ad) which is increasing at a rate faster than the standardisation �d. This turns the

problem into one that is technically rather challanging; bounds on the cumulant of DFTs cannot be

applied to estimators defined on an increasing frequency grid. One of the aims in this paper is to

develop the machinery for the analysis of estimators defined with an unbounded frequency grid. We

show that under Gaussianity of the spatial process and some mild conditions on the spectral density

function, the choice of a does not play a significant role in the sampling properties of Qa,�(g✓; r).

In particular, we show that Qa,�(g✓; r) is a mean squared consistent estimator of I(g✓;1) (up

to a finite constant) for any a so long as a/� ! 1 as � ! 1. Furthermore, by constraining

the rate of growth of the frequency grid to a = O(�k) for some k such that 1 k < 1 (both

4

the bounded and unbounded frequency grid come under this canopy), the asymptotic sampling

properties (mean, variance and asymptotic normality) of Qa,�(g✓; r) can be derived. On the other

hand, we show that if the spatial random process is non-Gaussian then the additional constraint

that (�a)d/n2 ! c < 1 is required as �, a, n ! 1. This bound is quite strong and highlights stark

di↵erences between the statistical properties of a Gaussian random field to those of a non-Gaussian

random field. The results in this paper (at least in the case of Gaussianity) can be used to verify

condition C.4(ii) in Bandyopadhyay et al. (2015).

We now summarize the paper. In Section 2 we obtain the properties of the DFT Jn(!k)

and give examples of estimators which can be written as Qa,�(g✓; r). In Section 3 we derive the

asymptotic sampling properties of Qa,�(g✓; r). In Section 4 we apply the results derived in Section

3 to some of the statistics proposed in Section 2. The expressions for the variance Qa,�(g✓; 0) are

complex and di�cult to estimate in practice. In Section 5, we exploit the results in Section 3.4,

where we show that in the case that the locations are uniformly distributed {Qa,�(g✓; r)} forms a

‘near uncorrelated’ sequence over r which asymptotically has the same variance when r/� ! 0 as

� ! 1. Using these results we define an ‘orthogonal’ sample which can be used to obtain a simple

estimator of the variance of Qa,�(g✓; 0) which is computationally fast to evaluate and has useful

theoretical properties.

An outline of the proofs in the case of uniform sampling can be found in Appendix A. The

proofs for non-uniform sampling and technical proofs can be found in the supplementary material,

Subba Rao (2015b); many of these results build on the work of Kawata (1959) and may be of

independent interest.

2 Preliminary results and assumptions

We observe the spatial random field {Z(s); s 2 Rd} at the locations {sj}nj=1

. Throughout this

paper we will use the following assumptions on the spatial random field.

Assumption 2.1 (Spatial random field) (i) {Z(s); s 2 Rd} is a second order stationary ran-

dom field with mean zero and covariance function c(s1

� s2

) = cov(Z(s1

), Z(s2

)|s1

, s2

). We

define the spectral density function as f(!) =RRd c(s) exp(�is0!)ds.

(ii) {Z(s); s 2 Rd} is a stationary Gaussian random field.

In the following section we require the following definitions. For some finite 0 < C < 1 and

� > 0, let

��(s) =

(C |s| 2 [�1, 1]

C|s|�(1+�) |s| > 1. (6)

Let ��(s) =Qd

j=1

��(sj). To minimise notation we will often usePa

k=�a to denote the multiple

sumPa

k1=�a . . .Pa

kd=�a. Let k · k1 denote the `1

-norm of a vector and k · k2

denote the `2

-norm.

5

2.1 Properties of the DFTs

We first consider the behaviour of the DFTs under both uniform and non-uniform sampling of the

locations.

Below we summarize Theorem 2.1, Bandyopadhyay and Subba Rao (2015), which defines fre-

quencies where the Fourier transform is ‘close to uncorrelated’, this result requires the following

additional assumption on the distribution of the spatial locations.

Assumption 2.2 (Uniform sampling) The locations {sj} are independent uniformly distributed

random variables on [��/2,�/2]d.

Theorem 2.1 Let us suppose that {Z(s); s 2 Rd} is a stationary spatial random field whose covari-

ance function (defined in Assumption 2.1(i)) satisfies |c(s)| �1+�(s) for some � > 0. Furthermore,

the locations {sj} satisfy Assumption 2.2. Then we have

cov [Jn(!k1), Jn(!k2)] =

(f(!k) +O( 1� + �d

n ) k1

= k2

(= k)

O( 1

�d�b ) k1

� k2

6= 0

where b = b(k1

�k2

) denotes the number of zero elements in the vector k1

�k2

, !k = (2⇡k1� , . . . , 2⇡kd� ),

k 2 Zd and the bounds are uniform in k,k1

,k2

2 Zd.

PROOF See Theorem 2.1, Bandyopadhyay and Subba Rao (2015). ⇤

To understand what happens in the case that the locations are not uniformly sampled, we

adopt the assumptions of Hall and Patil (1994), Matsuda and Yajima (2009) and Bandyopadhyay

and Lahiri (2009) and assume that {sj} are iid random variables with density 1

�dh(·�), where

h : [�1/2,�1/2]d ! R. We use the following assumptions on the sampling density h.

Assumption 2.3 (Non-uniform sampling) The locations {sj} are independent distributed ran-

dom variables on [��/2,�/2]d, where the density of {sj} is 1

�dh(·�), and h(·) admits the Fourier

representation

h(u) =X

j2Zd

�j exp(i2⇡j0u),

whereP

j2Zd k�jk1 < 1 such that |�j | CQd

i=1

|ji|�(1+�)I(ji 6= 0) (for some � > 0). This assump-

tion is satisfied if the second derivative of h is bounded on the d-dimensional torus [�1/2, 1/2]d.

Note that if h is such that sups2[�1/2,1/2]d |@m1+...,mdh(s1,...,sd)

@sm11 ...@s

mdd

| < 1 (0 mi 2) but h is

not continuous on the d-dimensional torus [�1/2, 1/2]d then |�j | CQd

i=1

|ji|�1I(ji 6= 0) and

the above condition will not be satisfied. However, this assumption can be induced by tapering the

observations such that X(sj) is replaced with eX(sj), where eX(sj) = t(sj)X(sj), t(s) =Qd

i=1

t(si)

and t is a weight function which has a bounded second derivative, t(�1/2) = t(1/2) = 0 and

t0(1/2) = t0(�1/2) = 0. By using eX(sj) instead of X(sj), in all the derivations we can replace the

density h(s) with t(s)h(s). This means the results now rely on the Fourier coe�cients of t(s)h(s),

6

which decay at the rate |R[�1/2,1/2]d t(s)h(s) exp(i2⇡j

0s)ds| CQd

i=1

|ji|�2I(ji 6= 0), and thus the

above condition is satisfied. Note that Matsuda and Yajima (2009), Definition 2, use a similar

data-tapering scheme to induce a similar condition.

We now show that under this general sampling scheme the near ‘uncorrelated’ property of the

DFT given in Theorem 2.1 does not hold.

Theorem 2.2 Let us suppose that {Z(s); s 2 Rd} is a stationary spatial random field whose covari-

ance function (defined in Assumption 2.1(i)) satisfies |c(s)| �1+�(s) for some � > 0. Furthermore,

the locations {sj} satisfy Assumption 2.3. Then we have

cov [Jn(!k1), Jn(!k2)] = h�, �(k2�k1)

if (!k1) +c(0)�k2�k1�

d

n+O

✓1

�

◆, (7)

where the bounds are uniform in k1

,k2

2 Zd and h�, �ri =P

j2Zd �j�r�j .

PROOF See Appendix B. ⇤In the case of uniform sampling �j = 0 for all j 6= 0 and �

0

= 1, in this case cov [Jn(!k1), Jn(!k2)] =

[f(!k1)+ c(0)�d/n]I(k1

= k2

)+O(��1). Thus we observe that Theorem 2.2 includes Theorem 2.1

as a special case (up to the number of zeros in the vector k1

� k2

; if b(k1

� k2

) (d� 2) Theorem

2.1 gives a faster rate). On the other hand, we note that in the case the locations are not sampled

from a uniform distribution the DFTs are not (asymptotically) uncorrelated. However, they do

satisfy the property

cov [Jn(!k1), Jn(!k2)] = O

✓1 +

I(k1

6= k2

)

kk1

� k2

k1

� 1 +

�d

n

�+

1

�

◆.

In other words, the correlations between the DFTs decay the further apart the frequencies. A

similar result was derived in Bandyopadhyay and Lahiri (2009) who show that the correlation

between Jn(!1

) and Jn(!2

) are asymptotic uncorrelated if their frequencies are ‘asymptotically

distant’ such that �dk!1

� !2

k1

! 1.

Remark 2.1 In both Theorems 2.1 and 2.2 we use the assumption that |c(s)| �1+�(sj). This

assumption is satisfied by a wide range of covariance functions. Examples include:

• The Wendland covariance, since its covariance is bounded and has a compact support.

• The Matern covariance, which for ⌫ > 0 is defined as c⌫(ksk2) = 1

�(⌫)2⌫�1 (p2vksk

2

)⌫K⌫(p2⌫ksk

2

)

(K⌫ is the modified Bessel function of the second kind). To see why, we note that if ⌫ > 0

then c⌫(s) is a bounded function. Furthermore, for large ksk2

we note that c⌫(ksk2) ⇠C⌫ksk⌫�0.5

2

exp(p2vksk

2

) as ksk2

! 1 (where C⌫ is a finite constant). Thus by using the

inequality

d�1/2(|s1

|+ |s2

|+ . . .+ |sd|) qs21

+ s22

+ . . .+ s2d (|s1

|+ |s2

|+ . . .+ |sd|).

we can show |c⌫(s)| �1+�(s) for any � > 0.

7

Remark 2.2 (Random vs Fixed design of locations) Formally, the methods in this paper are

developed under the assumption that the locations are random.

Of course, as a referee pointed out, it is unclear whether a design is random or not. However,

in the context of the current paper, ‘random design’ really refers to the locations being ‘irregular’ in

the sense of not being defined on or in the proximity of a lattice (such a design would be considered

‘near regular’). A simple way to check for this is to evaluate the Fourier transform of the locations

b�r =1

n

nX

j=1

exp(is0j!r).

In the case that the design is random b�r is an estimator of �r (the Fourier coe�cients of h(·)) andb�r ⇡ 0 for krk

1

large. On the other hand, if the design is ‘near regular’ {b�r} would be periodic

on the grid Zd. Therefore, by making a plot of {b�r; r 2 [�Kn1/d,Kn1/d]d} (for some K 2 Z+),

we can search for repetitions at shifts of n1/d, ie. b�r ⇡ b�r+n1/d where n1/d = (n1/d, . . . , n1/d). If

b�r ⇡ 0 for large krk1

, then the locations are su�ciently irregular such that we can treat them as

random and the results in this paper are valid. On the other hand repetitions suggest the locations

lie are ‘near regular’ and the results in this paper do not hold in this case.

In the case that spatial locations are defined on a regular lattice, then it is straightforward to

transfer the results for discrete time time series to spatial data on the grid (in this case a = � =

n1/d).

2.2 Assumptions for the analysis of Qa,�(·)

In order to asymptotically analyze the statistic, Qa,�(g✓; r), we will assume that the spatial domain

[��/2,�/2]d grows as the number of observations n ! 1 and we will mainly work under the mixed

domain framework described in the introduction, where �d/n ! 0 as � ! 1 and n ! 1.

Assumption 2.4 (Assumptions on g✓(·)) (i) If the frequency grid is bounded to [�C,C]d (thus

a = C�), then sup!2[�C,C]

d |g✓(!)| < 1 and for all 1 j d, sup!2[�C,C]

d |@g✓(!)

@!j| < 1.

(i) If the frequency grid is unbounded, then sup!2Rd |g✓(!)| < 1 and for all 1 j d,

sup!2Rd |@g✓(!)

@!j| < 1.

In the case that a = C� (where |C| < 1), the asymptotic properties of Qa,�(g✓; r) can be

mostly derived by using either Theorems 2.1 or 2.2. For example, in the case that the locations are

uniformly sampled, it follows from Theorem 2.1 that

E[Qa,�(g✓; r)] =

(I(g✓;C) +O(�

d

n + 1

�) r = 0

O( 1�) otherwise

where I(g✓; ·) is defined in (5), with �d/n ! 0 as � ! 1 and n ! 1 (the full details will be given

in Theorem 3.1). However, this analysis only holds in the case that a = O(�), in other words the

frequency grid is bounded. Situations where this statistic is of interest are rare, notable exceptions

include the Whittle likelihood and the spectral density estimator. But usually our aim is to estimate

8

the functional I(g✓;1), therefore the frequency grid {!k;k = (k1

, . . . , kd),�a ki a} should be

unbounded. In this case, using Theorems 2.1 or 2.2 to analyse E[Qa,�(g✓; r)] will lead to errors of

the order O( a�2 ), which cannot be controlled without placing severe restrictions on the expansion

of the frequency grid.

Below we state the assumptions required on the spatial process. Below each assumption we

explain where it is used. Note that Assumption 2.5(a) is used for the bounded frequency grid and

2.5(b) is (mainly) used for the nonbounded frequency grid (however the same assumption can also

be used for the bounded frequency, see the comments below 2.5(b)). It is unclear which Assumption

2.5(a) or (b) is stronger, since neither assumption implies the other.

Assumption 2.5 (Conditions on the spatial process) (a) |c(s)| �1+�(s)

Required for the bounded frequency grid, to obtain the DFT calculations.

(b) For some � > 0, f(!) ��(!)

Required for the unbounded frequency grid - using this assumption instead of (a) in the case

of an bounded frequency grid leads to slightly larger errors bounds in the derivation of the first

and second order moments. This assumption is also used to obtain the CLT result for both

the bounded and unbounded frequency grids.

(c) For all 1 j d and some � > 0, the partial derivatives satisfy |@f(!)

@!j| ��(!).

We use this condition to approximate sums with integral for both the bounded and unbounded

frequency grids. It is also used to make a series of approximations to derive the limiting

variance in the case that the frequency grid is unbounded.

(d) For some � > 0, | @df(!)

@!1,...,@!d| ��(!).

Required only in the proof of Theorem 3.1(ii)(b).

Remark 2.3 Assumption 2.5(b,c,d) appears quite technical, but it is satisfied by a wide range of

spatial covariance functions. For example, the spectral density of the Matern covariance defined in

Remark 2.1 is f⌫(!) = �(1 + k!k22

)�(⌫+d/2). It is straightforward to show that this spectral density

satisfies Assumption 2.5(b,c,d), noting that the � used to define �� will vary with ⌫, dimension d

and the order of the derivative f⌫(·).

We show in Appendix F, Subba Rao (2015b), that

E[Qa,�(g✓; r)] ! h�, �riI⇣g✓;

a

�

⌘+ E

⇥Z(sj)

2 exp(�is0!r)⇤

| {z }=c(0)�r

1

n

aX

k=�a

g(!k),

where I(g✓;a�) is defined in (5). We observe that in the case that the frequency grid is bounded then

the second term of the above is asymptotically negligible. However, in the case that the frequency

grid is unbounded then the second term above plays a role, indeed it can induce a bias unless the

9

rate of growth of the frequency grid is constrained such that ad/n ! 0. Therefore, the main focus

of this paper will be on a slight variant of the statistic Qa,�(g✓; r), defined as

eQa,�(g✓; r) = Qa,�(g✓; r)�1

n

aX

k=�a

g✓(!k)1

n

nX

j=1

Z(sj)2 exp(�is0j!r). (8)

eQa,�(g✓; r) avoids some of the bias issues when the frequency grid is unbounded. For the rest of this

paper our focus will be on eQa,�(g✓; r). Note that Matsuda and Yajima (2009) and Bandyopadhyay

et al. (2015) use a similar bias correction method.

The results for the non-bias corrected version (Qa,�(g✓; r)) can be found in Appendix F,

Subba Rao (2015b).

Below we give some examples of statistics of the form (4) and (8) which satisfy Assumption 2.4.

Example 2.1 (Fixed frequency grid) (i) The discretized Whittle likelihood given in equation

(3) can be written in terms of Qa,�(g✓; 0) where

1

�d

C�X

k1,...,kd=�C�

✓log f✓(!k) +

|Jn(!k)|2

f✓(!k)

◆= Qa,�(f

�1

✓ ; 0) +1

�d

C�X

k1,...,kd=�C�

log f✓(!k)

and {f✓(·); ✓ 2 ⇥} (where ⇥ has compact support) is a parametric family of spectral density

functions. The choice of a = C� depends on the rate of decay of f✓(!) to zero.

(ii) A nonparametric estimator of the spectral density function f(!) is

bf�,n(!) =X

k

Wb(! � !k)|Jn(!k)|2 �1

n

aX

k=�a

Wb(! � !k)1

n

nX

j=1

Z(sj)2 =

1

bdeQa,�(Wb, 0),

where Wb(!) = b�dQd

j=1

W (!j

b ) and W : [�1/2,�1/2] ! R is a spectral window. In this

case we set a = �/2.

Example 2.2 (Unbounded frequency grid) (i) In Bandyopadhyay and Subba Rao (2015) a

test statistic of the type

ean(g; r) = eQa,�(g; r) where r 2 Zd/{0},

is proposed. We show in Section 3.4 under the assumptions that the locations are uniformly

distributed and r 6= 0, that E[ eQa,�(g; r)] ⇡ 0. On the other hand, if the spatial process is

nonstationary, this property does not hold (see Bandyopadhyay and Subba Rao (2015) for the

details). Using this dichotomy, Bandyopadhyay and Subba Rao (2015) test for stationarity of

a spatial process.

(ii) A nonparametric estimator of the spatial (stationary) covariance function is

bcn(v) = T

✓2v

�

◆ 1

�d

aX

k=�a

|Jn(!k)|2 exp(iv0!k)

!= T

✓2v

�

◆Qa,�(e

iv0·;0) (9)

10

where T (u) =Qd

j=1

T (uj) and T : R :! R is the triangular kernel defined as T (u) = (1� |u|)for |u| 1 and T (u) = 0 for |u| > 1. It can be shown that the Fourier transform of cn(v) is

bf�(!) =

Z

Rdbcn(v) exp(�i!0v)dv =

aX

k=�a

|Jn(!k)|2Sinc2 [�(!k � !)]

where Sinc(�!/2) =Qd

j=1

sinc(�!j/2) and sinc(!) = sin(!)/!. Clearly, f�(!) is non-

negative, therefore, the sample covariance function {bcn(v)} is a non-negative definite function

and thus a valid covariance function.

Hall et al. (1994) also propose a nonparametric estimator of the spatial covariance function

which has a valid covariance function. Their scheme is a three-stage scheme; (i) a nonpara-

metric pre-estimator of the covariance function is evaluated (ii) the Fourier transform of the

pre-estimator covariance function is evaluated over Rd (iii) negative values of the Fourier

transform are placed to zero and the inverse Fourier transform of the result forms an estima-

tor of the spatial covariance function. The estimator (9) avoids evaluting Fourier transforms

over Rd and is simpler to compute.

(iii) We recall that Whittle likelihood can only be defined on a bounded frequency grid. This can

be an issue if the observed locations are dense on the spatial domain and thus contain a large

amount of high frequency information which would be missed by the Whittle likelihood. An

alternative method for parameter estimation of a spatial process is to use a di↵erent loss

function. Motivated by Rice (1979), consider the quadratic loss function

Ln(✓) =1

�d

aX

k1,...,kd=�a

�|Jn(!k)|2 � f✓(!k)

�2

, (10)

and let b✓n = argmin✓2⇥ Ln(✓) or equivalently solve r✓Ln(✓) = 0, where

r✓Ln(✓) = � 2

�d

aX

k1,...,kd=�a

<r✓f✓(!k)�|Jn(!k)|2 � f✓(!k)

+

= Qa,�(�2r✓f✓(·); 0) +2

�d

aX

k=�a

f✓(!k)r✓f✓(!k), (11)

and <X denotes the real part of the random variable X. It is well known that the distributional

properties of a quadratic loss function are determined by its first derivative. In particular, the

asymptotic sampling properties of b✓n are determined by Qa,�(�2<r✓f(·; ✓);0).

To minimize notation, from now onwards we will suppress the ✓ in eQa,�(g✓; r), using insteadeQa,�(g; r) (and only give the full form when necessary).

3 Sampling properties of

eQa,�(g; r)

In this section we show that under certain conditions eQa,�(g; r) is a consistent estimator of I(g; a�)

(or some multiple of it), where I(g; a�) is defined in (5). We note that I�g; a�

�! I(g;1) if a/� ! 1

as a ! 1 and � ! 1.

11

3.1 The mean of

eQa,�(g; r)

We start with the expectation of eQa,�(g; r). As we mentioned earlier, it is unclear how to choose

the number of frequencies a in the definition of eQa,�(g; r). We show if sup!2Rd |g(!)| < 1, the

choice of a does not play a significant role in the asymptotic properties of eQa,�(g; r). To show

this result, we note that by not constraining a, such that a = O(�), then the number of terms in

the sum Qa,�(g; r) grows at a faster rate than the standardisation �d. In this case, the analysis ofeQa,�(g; r) requires more delicate techniques than those used to prove Theorems 2.1 and 2.2. We

start by stating some pertinent features in the analysis of eQa,�(g; r), which gives a flavour of our

approach. By writing eQa,�(g; r) as a quadratic form it is straightforward to show that

E [Qa,�(g; r)] = c2

aX

k=�a

g(!k)1

�d

Z

[��/2,�/2]dc(s

1

�s2

) exp(i!0k(s1�s

2

)�is02

!r)h⇣s

1

�

⌘h⇣s

2

�

⌘ds

1

ds2

,

where c2

= n(n � 1)/n2. The proof of Theorems 2.1 and 2.2 are based on making a change of

variables v = s1

� s2

and then systematically changing the limits of the integral. This method

works if the frequency grid [�a/�, a/�]d is fixed for all �. However, if the frequency grid [�a/�, a/�]d

is allowed to grow with �, applying this brute force method to EheQa,�(g; r)

ihas the disadvantage

that it aggregrates the errors within the sum of EheQa,�(g; r)

i. Instead, to further the analysis,

we replace c(s1

� s2

) by its Fourier transform c(s1

� s2

) = 1

(2⇡)d

RRd f(!) exp(i!0(s

1

� s2

))d! and

focus on the case that the sampling design is uniform; h(s/�) = ��dI[��/2,�/2](s) (later we consider

general densities). This reduces the first term in EheQa,�(g; r)

ito the Fourier transforms of step

functions, which is the product of sinc functions, Sinc(�2

!) (noting that the Sinc function is defined

in Example 2.2). Specifically, we obtain

EheQa,�(g; r)

i=

c2

(2⇡)d

aX

k=�a

g(!k)

Z

Rdf(!)Sinc

✓�!

2+ k⇡

◆Sinc

✓�!

2+ (k + r)⇡

◆d!

=c2

⇡d

Z

RdSinc(y)Sinc(y + r⇡)

"1

�d

aX

k=�a

g(!k)f(2y

�� !k)

#dy,

where the last line above is due to a change of variables y = �!2

+ k⇡. Since the spectral density

function is absolutely integrable it is clear thath

1

�d

Pak=�a g(!k)f(

2y� � !k)

iis uniformly bounded

over y and that EheQa,�(g; r)

iis finite for all �. Furthermore, if f(2y� � !k) were replaced with

f(�!k), then what remains in the integral are two shifted sinc functions, which is zero if r 2 Zd/{0},i.e.

EheQa,�(g; r)

i=

c2

⇡d

Z


"1

�d

aX

k=�a

g(!k)f(2y

�� !k)

#dy +R,

where

R =c2

⇡d

Z


"1

�d

aX

k=�a

g(!k)

✓f(

2y

�� !k)� f(�!k)

◆#dy.

12

In the following theorem we show that under certain conditions on f , R is asymptotically negligible.

Let b = b(r) denote the number of zero elements in the vector r 2 Zd.

Theorem 3.1 Let I(g; ·) be defined as in (5). Suppose Assumptions 2.1(i) and 2.2 hold.

(i) If Assumptions 2.4(i) and 2.5(a,c) hold, then we have

EheQa,�(g; r)

i=

(O( 1

�d�b ) r 2 Zd/{0}I (g;C) +O( 1�) r = 0

(12)

(ii) Suppose Assumptions 2.4(ii) holds and

(a) Assumption2.5(b) holds, then supa

��EheQa,�(g; r)

i�� < 1.

(b) Assumption 2.5(b,c,d) holds and {m1

, . . . ,md�b} is the subset of non-zero values in

r = (r1

, . . . , rd), then we have

EheQa,�(g; r)

i=

(O⇣

1

�d�b

Qd�bj=1

(log �+ log |mj |)⌘

r 2 Zd/{0}I�g; a�

�+O

�log �� + 1

n

�r = 0

. (13)

(c) If only Assumption 2.5(b,c) holds, then the O⇣

1

�d�b

Qd�bj=1

(log �+ log |mj |)⌘term in (b)

is replaced with the slower rate O�1

� (log �+ log krk1

)�.

Note that the above bounds for (b) and (c) are uniform in a.

PROOF See Appendix A.1. ⇤

We observe that if r 6= 0, then eQa,�(g; r) is estimating zero. It would appear that these terms

don’t contain any useful information, however in Section 5 we show how these terms can be used

to estimate nuisance parameters.

In order to analyze EheQa,�(g; r)

iin the case that the locations are not from a uniform distri-

bution we return to (12) and replace c(s1

� s2

) by its Fourier representation and also replace the

sampling density h(s/�) by its Fourier representation h(s/�) =P

j2Zd �j exp(i2⇡j0s/�) to give

EheQa,�(g; r)

i

=c2

⇡d

X

j1,j22Z�j1�j2

1

�d

aX

k=�a

g(!k)

Z 1

�1f

✓2y

�� !k

◆Sinc(y)Sinc(y + (r � j

1

� j2

)⇡)dy.

This representation allows us to use similar techniques to those used in the uniform sampling case

to prove the following result.

Theorem 3.2 Let I(g; ·) be defined as in (5). Suppose Assumptions 2.1(i) and 2.3 hold.

(i) If in addition Assumptions 2.4(i) and 2.5(a,c) hold, then we have

E⇥ eQa,�(g; r)

⇤= h�, �riI

⇣g;

a

�

⌘+O(��1),

O(��1) is uniform over r 2 Zd.

13

(ii) If in addition Assumptions 2.4(ii) and 2.5(b,c) hold, then we have

E⇥ eQa,�(g; r)

⇤= h�, ��riI

⇣g;

a

�

⌘+O

✓log �+ I(r 6= 0) log krk

1

�

◆.

PROOF See Appendix B in (Subba Rao, 2015b). ⇤

We observe that by applying Theorem 3.2 to the case that h is uniform (using that �0

= 1 else

�j = 0) gives E⇥ eQa,�(g; r)

⇤= O(��1) for r 6= 0. Hence, in the case that the sampling is uniform,

Theorems 3.1 and 3.2 give similar results, though the bounds in Theorem 3.1 are sharper.

Remark 3.1 (Estimation ofP

j2Zd |�j |2) The above lemma implies that E⇥ eQa,�(g; 0)

⇤= h�, �0iI

�g; a�

�.

Therefore, to estimate I�g; a�

�we require an estimator of h�, �0i. To do this, we recall that

h�, �0i =X

j2Z|�j |2 =

1

�d

Z

[��/2,�/2]dh�(!)2d!.

Therefore one method for estimating the above integral is to define a grid on [��/2,�/2]d and

estimate h� at each point, then to take the average squared over the grid (see Remark 1, Matsuda

and Yajima (2009)). An alternative, computationally simpler method, is to use the method proposed

in Gine and Nickl (2008), that is

\h�, �0i =2

n(n� 1)b

X

1j1<j2n

K

✓sj1 � sj2

b

◆2

,

as an estimator of h�, �0i, where K : [�1/2, 1/2]d ! R is a kernel function. Note that multiplying

the above kernel with exp(�i!0rsj2) results in an estimator of h�, �ri. In the case d = 1 and

under certain regularity conditions, Gine and Nickl (2008) show if the bandwidth b is selected in an

appropriate way then \h�, �0i attains the classical O(n�1/2) rate under suitable regularlity conditions

(see, also, Bickel and Ritov (1988) and Laurent (1996)). It seems plausible a similar result holds

for d > 1 (though we do not prove it here). Therefore, an estimator of I�g; a�

�is eQa,�(g; r)/\h�, �0i.

3.2 The covariance and asymptotic normality

In the previous section we showed that the expectation of eQa,�(g; r) depends only the number of

frequencies a through the limit of the integral I(g; a�) (if sup!Rd |g(!)| < 1). In this section, we

show that a plays a mild role in the higher order properties of eQa,�(g; r). We focus on the case that

the random field is Gaussian and later describe how the results di↵er in the case that the random

field is non-Gaussian.

Theorem 3.3 Suppose Assumptions 2.1, 2.3 and

(i) Assumption 2.4(i) and 2.5(a,c) hold. Then uniformly for all 0 kr1

k1

, kr2

k1

C|a| (forsome finite constant C) we have

�dcovheQa,�(g; r1), eQa,�(g; r2)

i= U

1

(r1

, r2

;!r1 ,!r2) +O

✓1

�+

�d

n

◆

14

and


i= U

2

(r1

, r2

;!r1 ,!r2) +O

✓1

�+

�d

n

◆

(ii) Assumption 2.4(ii) and Assumption 2.5(b) hold. Then we have

�d supa,r1,r2

��covheQa,�(g; r1), eQa,�(g; r2)

i�� < 1 and �d supa,r1,r2


i�� < 1,

if �d/n ! c (where 0 c < 1) as � ! 1 and n ! 1.

(iii) Assumption 2.4(ii) and 2.5(b,c) hold. Then uniformly for all 0 kr1

k1

, kr2

k1

C|a| (forsome finite constant C) we have


i= U

1

(r1

, r2

;!r1 ,!r2) +O(`�,a,n)


i= U

2

(r1

, r2

;!r1 ,!r2) +O(`�,a,n),

where

`�,a,n = log2(a)

log a+ log �

�

�+

�d

n. (14)

Note, if we drop the restriction on r1

and r2

and simply let r1

, r2

2 Zd then the bound for

`�,a,n needs to include the additional term log2(a)(log kr1

k1

+ log kr2

k1

)/�.

U1

(·) and U2

(·) are defined as

U1

(r1

, r2

;!r1 ,!r2) = U1,1(r1, r2;!r1 ,!r2) + U

1,2(r1, r2;!r1 ,!r2)

U2

(r1

, r2

;!r1 ,!r2) = U2,1(r1, r2;!r1 ,!r2) + U

2,2(r1, r2;!r1 ,!r2) (15)

with

U1,1(r1, r2;!r1 ,!r2)

=1

2⇡

X

j1+...+j4=r1�r2

�j1�j2�j3�j4

Z

D(j1+j3)

g(!)g(! + !j1+j3)f(! + !j1)f(! + !r1�j2)d!

U1,2(r1, r2;!r1 ,!r2)

=1

2⇡

X

j1+j2+j3+j4=r1�r2

�j1�j2�j3�j4

Z

D(r1�j2�j3)

g(!)g(�! � !r2�j3�j2)f(! + !j1)f(! + !r1�j2)d!

U2,1(r1, r2;!r1 ,!r2)

=1

2⇡3

X

j1+j2+j3+j4=r1+r2

�j1�j2�j3�j4

Z

D(j1+j3)

g(!)g(!�j1�j3 � !)f(! + !j1)f(! + !r1�j2)d!

U2,2(r1, r2;!r1 ,!r2)

=1

2⇡

X

j1+j2+j3+j4=r1+r2

�j1�j2�j3�j4

Z

D(r1�j2�j3)

g(!)g(! � !j2+j3�r1)f(! + !j1)f(! + !r1�j2)d!,

where the integral is defined asRDr

=R2⇡min(a,a�r1)/�2⇡max(�a,�a�r1)/�

. . .R2⇡min(a,a�rd)/�2⇡max(�a,�a�rd)/�

.

15

PROOF See Appendix B in Subba Rao (2015b). ⇤

We now briefly discuss the above results. From Theorem 3.3(iii) we see that eQa,�(g; r) is a mean

squared consistent estimator of h�, �riI(g; a/�), i.e. E[ eQa,�(g; r)�h�, �riI(g; a�)]2 = O(��d+( log �� +

1

n)2) as a ! 1 and � ! 1. Indeed, if sup!2Rd |g(!)| < 1, then the rate that a/� ! 1 plays no

role.

However, in order to obtain an explicit expression for the variance additional conditions are

required. In the case that the frequency grid is bounded we obtain an expression for the variance

of eQa,�(g; r). On the other hand, we observe (see Theorem 3.3(iii)) that if the frequency grid is

unbounded we require some additional conditions on the spectral density function and some mild

constraints on the rate of grow of the frequency domain a. More precisely, a should be such that

a = O(�k) for some 1 k < 1. If these conditions are fulfilled, then the asymptotic variance ofeQa,�(g; r) (up to the limits of an integral) is the equivalent for both the bounded and unbounded

frequency grid.

By comparing Theorem 3.3(i) and (ii) we observe that in the case the frequency grid is bounded,

the same result can be proved Assumption 2.5(b,c) rather than 2.5(a,c), however the error bounds

would be slightly larger.

The expressions above are rather unwieldy, however, some simplifications can be made if

kr1

k1

<< � and kr2

k1

<< �.

Corollary 3.1 Suppose Assumptions 2.3, 2.4 and 2.5(a,c) or 2.5(b,c) hold. Then we have

U1

(r1

, r2

;!r1 ,!r2) = �r1�r2C1

+O

✓kr

1

k1

+ kr2

k1

+ 1

�

◆

U2

(r1

, r2

;!r1 ,!r2) = �r1+r2C2

+O

✓kr

1

k1

+ kr2

k1

+ 1

�

◆

where �r =P

j1+j2+j3+j4=r �j1�j2�j3�j4 and

C1

=1

(2⇡)d

Z

2⇡[�a/�,a/�]df(!)2

h|g(!)|2 + g(!)g(�!)

id!

C2

=1

(2⇡)d

Z

2⇡[�a/�,a/�]df(!)2 [g(!)g(�!) + g(!)g(!)] d!.

Using the above expressions for �dcovheQa,�(g; r1), eQa,�(g; r2)

iand �dcov

heQa,�(g; r1), eQa,�(g; r2)

i

the variance of the real and imaginary parts of eQa,�(g; r) can easily be deduced.

In the following theorem we derive bounds for the cumulants of eQa,�(g; r), which are subse-

quently used to show asymptotical normality of eQa,�(g; r).

Theorem 3.4 Suppose Assumptions 2.1, 2.3, 2.4 and 2.5(b) hold. Then for all q � 3 and uniform

in r1

, . . . , rq 2 Zd we have

cumq

heQa,�(g, r1), . . . , eQa,�(g, rq)

i= O

✓log2d(q�2)(a)

�d(q�1)

◆(16)

16

if �d

n log

2d(a)

! 0 as n ! 1, a ! 1 and � ! 1.

PROOF See Appendix B in Subba Rao (2015b). ⇤

From the above theorem we see that if �d

n log

2d(a)

! 0 and log2(a)/�1/2 ! 0 as � ! 1, n ! 1

and a ! 1, then we have �dq/2cumq( eQa,�(g, r1), . . . , eQa,�(g, rq)) ! 0 for all q � 3. Using this

result we show asymptotic Gaussianity of eQa,�(g, r).

Theorem 3.5 Suppose Assumptions 2.1, 2.3, 2.4 and 2.5(b,c) hold. Let C1

and C2

, be defined as

in Corollary 3.1. Under these conditions we have

�d/2��1/2

0

@<⇣eQa,�(g, r1)� h�, ��riI(g; a�)

⌘

=⇣eQa,�(g, r1)� h�, ��riI(g; a�)

⌘

1

A D! N (0, I2

),

where <X and =X denote the real and imaginary parts of the random variable X, and

� =1

2

<(�

0

C1

+ �2rC2

) �=(�0

C1

+ �2rC2

)

�=(�0

C1

+ �2rC2

) <(�0

C1

� �2rC2

)

!

with log

2(a)

�1/2 ! 0 as � ! 1, n ! 1 and a ! 1.

PROOF See Appendix D, Subba Rao (2015b). ⇤

It is likely that the above result also holds when the assumption of Gaussianity of the spa-

tial random field is relaxed and replaced with the conditions stated in Theorem 3.6 (below) to-

gether with some mixing-type assumptions. We leave this for future work. However, in the fol-

lowing theorem, we obtain an expression for the variance of eQa,�(g; r) for non-Gaussian random

fields. We make the assumption that {Z(s); s 2 Rd} is fourth order stationary, in the sense that

E[Z(s)] = µ, cov[Z(s1

), Z(s2

)] = c(s1

� s2

), cum[Z(s1

), Z(s2

), Z(s3

)] = 2

(s1

� s2

, s1

� s3

) and

cum[Z(s1

), Z(s2

), Z(s3

), Z(s4

)] = 4

(s1

� s2

, s1

� s3

, s1

� s4

), for some functions 3

(·) and 4

(·)and all s

1

, . . . , s4

2 Rd.

Theorem 3.6 Let us suppose that {Z(s); s 2 Rd} is a fourth order stationary spatial random field

that satisfies Assumption 2.1(i). Furthermore, we suppose that Assumptions 2.3, 2.4 and 2.5(a,c) or

2.5(b,c) are satisfied. Let f4

(!1

,!2

,!3

) =RR3d 4(s1, s2, s3) exp(�i

P3

j=1

s0j!j)d!1

d!2

d!3

. We

assume that for some � > 0 the spatial tri-spectral density function is such that |f4

(!1

,!2

,!3

)| ��(!1

)��(!2

)��(!3

) and |@f4(!1,...,!3d)

@!j| ��(!1

)��(!2

)��(!3

) (where ��(·) is defined in (6)).

If krk1

, krk2

<< �, then we have


i= �r1�r2(C1

+D1

) +O

✓`�,a,n +

(a�)d

n2

+kr

1

k1

+ kr2

k1

�

◆(17)


i= �r1+r2(C2

+D2

) +O

✓`�,a,n +

(a�)d

n2

+kr

1

k1

+ kr1

k1

�

◆, (18)

17

where C1

and C2

are defined as in Corollary 3.1 and

D1

=1

(2⇡)2d

Z

2⇡[�a/�,a/�]2dg(!

1

)g(!2

)f4

(�!1

,�!2

,!2

)d!1

d!2

D2

=1

(2⇡)2d

Z

2⇡[�a/�,a/�]2dg(!

1

)g(!2

)f4

(�!1

,�!2

,!2

)d!1

d!2

.

Note that we can drop the restriction that krk1

, krk2

<< � and let r1

, r2

2 Zd, under this more

general assumption we obtain expressions that are similar to those for U1

and U2

(given in Theorem

3.3) and we need to include log2(a)[krk1

+ kr2

k1

]/� in the bounds.

PROOF See Appendix B in Subba Rao (2015b).

We observe that to ensure the term (a�)d

n2 ! 0 we need to choose a such that ad = o(n2/�d). We

recall that for Gaussian random fields a = O(�k) for some 1 k < 1 was su�cient for obtaining

an expression for the variance and asymptotic normality. However, in contrast, in the case that the

spatial random field is non-Gaussian stronger assumptions are required on a (indeed if (a�)d

n2 ! 1the variance �dvar[ eQa,�(g; r)] is not bounded).

3.3 Alternative frequency grids and asymptotics

In this subsection we discuss two issues related to the estimator eQa,�,↵(g; r).

3.3.1 Frequency grids

In a recent paper, Bandyopadhyay et al. (2015) define the spatial empirical likelihood within the

frequency domain. They use the spatial ‘periodogram’ {|Jn(!↵,k)|2;k 2 Zd and k 2 C[��⌘,�⌘]d}with !↵,k = 2⇡�↵k

� as the building blocks of the empirical likelihood. The way in which they

define their frequency grid is di↵erent to the frequency grid defined in this paper. More precisely,

whereas {!k = 2⇡k� ;k = (k

1

, . . . , kd),�a kj a} is used to construct eQa,�(g; r), Bandyopadhyay

et al. (2015) use the frequency grid {!↵,k = 2⇡�↵k� ;k = (k

1

, . . . , kd),��⌘ kj �⌘} where

0 < (1�⌘) < ↵ < 1. In other words, by using !↵,k to define their frequency grid they ensure that the

frequencies are su�ciently far apart to induce the near uncorrelatedness (indeed near independence)

of the {|Jn(!↵,k)|2;k 2 Zd and k 2 C[��⌘,�⌘]d} even in the case that the distribution of the

locations in not uniform (see the discussion below Theorem 2.2). This construction is crucial in

ensuring that the statistics used to define the empirical likelihood are asymptotically pivotal.

In this section we consider the sampling properties of eQa,�(g; r) under the {!↵,k = 2⇡�↵k� ;k =

(k1

, . . . , kd),�a kj a} frequency grid. To do this, we note that under this frequency grid eQ is

defined as

eQa,�,↵(g; r) =1

�d(1�↵)

aX

k1,...,kd=�a

g(!↵,k)Jn(!↵,k)Jn(!↵,k+r)��d↵

n

aX

k=�a

g(!↵,k)1

n

nX

j=1

Z(sj)2 exp(�is0j!↵,r),(19)

where we assume �↵ 2 Z+. Comparing eQa,�,↵(g; r) with eQa,�(g; r) we see that they are the same

when we set ↵ = 0. The methods developed in this paper can easily be extended to the analysis of

18

eQa,�,↵(g; r). More precisely, using the same method discussed in Section 3.1 it can be shown that

EheQa,�,↵(g; r)

i

=c2

⇡d

X

j1,j22Z�j1�j2

1

�d(1�↵)

aX

k=�a

g(!↵,k)

Z 1

�1f

✓2y

�� !↵,k

◆Sinc(y)Sinc(y + �↵(r � j

1

� j2

)⇡)dy.

And by using the same method used to prove Lemma 3.2 we have

E⇥ eQ�,a,↵(g; r)

⇤= h�, ��↵riI

⇣g;

a

�1�↵

⌘+O

✓log �+ log krk

1

�+

1

�1�↵

◆. (20)

Similarly it can be shown that

�d(1�↵)covheQa,�,↵(g; r1), eQa,�,↵(g; r2)

i

= ��↵(r1�r2)

eC1

+O

✓`�,a,n +

(a�)d

n2

+1

�↵

�IS=NG +

kr1

k1

+ kr2

k1

+ 1

�1�↵

◆(21)

where IS=NG denotes the indicator variable, such that IS=NG = 1 if the spatial process {Z(s); s 2Rd} is non-Gaussian else it is zero, and eC

1

is defined in exactly the same way as C1

, but the integralR2⇡[�a/�1�↵,a/�1�↵

]

d replacesR2⇡[�a/�,a/�]d . Note, when using this grid, for non-Gaussian random

fields the the term involving the fourth order cumulant is asymptotically negligible (however, this

is not true for the case ↵ = 0). Bandyopadhyay et al. (2015), Lemma 7.4, prove the same result

for �d(1�↵)varheQa,�,↵(g; r)

i.

In many respects, the results above are analogous to those in the case that the locations come

from the uniform distribution (see Section 3.4, below). More precisely, by using a frequency grid

where the frequencies are defined su�ciently far apart eQa,�,↵(g; 0) is a consistent estimator of

h�, ��0

iI⇣g;

a

�1�↵

⌘

whereas for r 6= 0, eQa,�,↵(g; r) is a consistent estimator of zero. Furthermore, { eQa,�,↵(g; r); r 2Zd} is a near uncorrelated sequence. However, unlike the case that the locations are uniformly

distributed it is unclear how { eQa,�,↵(g; r); r 6= 0} can be exploited to estimate the variance ofeQa,�,↵(g; 0).

3.3.2 Mixed Domain verses Pure Increasing Domain asymptotics

The asymptotics in this paper are done using mixed domain asymptotics, that is, as the domain

� ! 1, the number of locations observed grows at a faster rate than �, in other words �d/n ! 0

as n ! 1. However, as rightly pointed out by a referee, for a given application it may be di�cult

to disambiguate Mixed Domain (MD) from the Pure Increasing Domain (PID), where �d/n ! c

(0 < c < 1) set-up. We briefly discuss how the results change under PID asymptotics and the

implications of this. We find that the results point to a rather intriguing di↵erence for spatial

processes that are Gaussian and non-Gaussian.

19

As mentioned above, in the case of PID asymptotic �d/n is not asymptotically negligible. This

means that the covariance �dcovheQa,�(g; r1), eQa,�(g; r2)

iwill involve extra terms. These additional

terms can be deduced from the proof, but are long so we do not state them here. Interestingly,

in the case that the locations are uniformly distributed the procedure described in Section 5 can

still be used to estimate the variance without any adjustments to the method (in other words, this

method is robust to the asymptotics used).

This brings us to the most curious di↵erence between the MID and PD case. Comparing

Theorems 3.3 and 3.6 we observe the following:

• In the case that spatial process is Gaussian, using both MD and PID asymptotics we have

�dvar[ eQa,�(g; r)] = O(1) (see Theorem 3.3(i)). Furthermore, an asymptotic expression for

the variance is

�dvar[ eQa,�(g; r)] = �0

[C1

+ E1

] +O

✓log2(a)

log a+ log �

�

�+

krk1

+ 1

�

◆

where E1

is the additional term that is asymptotically not negligible under PID asymptotics;

E = O(�d/n). From the above we see that if we choose a such that a = O(�k) for some

1 < k < 1 then the frequency grid is unbounded and similar results as those stated in

Sections 3.1 and 3.2 hold under PID asymptotics.

• In the case that the process is non-Gaussian, using Theorem 3.6 we have

�dvar[ eQa,�(g; r)] = �0

[C1

+D1

+ E1

+ F1

] +O

✓log2(a)

log a+ log �

�

�+

(a�)d

n2

+krk

1

+ 1

�

◆,

where E1

+ F1

are asymptotically negligible under MD asymptotics but not negligible under

PID asymptotics; E1

+F1

= O(�d/n). Now a plays a vital role in the properties of the variance.

Indeed from the proof of Theorem 3.6, we can show that if (a�)d

n2 ! 1 as a,�, n ! 1, then

�dvar[ eQa,�(g; r)] is not bounded.

In the case of MD asymptotics we choose a such that (a�)d/n2 ! 0 and log3(a)/� ! 1.

Under these two conditions the frequency grid can be unbounded and grow at the rate a/�

as � ! 1. However, under PID asymptotics (where � = O(n1/d)) in order to ensure that

(a�)d/n2 = O(1) we require a = O(n1/d) = O(�). This constrains the frequency grid to be

bounded. In other words, in the case that the spatial process is non-Gaussian, in order that

var[ eQa,�(g; r)] = O(��d), the frequency grid must be bounded.

However, it is interesting to note that it is still possible to have an unbounded frequency grid

for non-Gaussian random processes under PID asymptotics if we use the estimator eQa,�,↵(g; r)

(defined in (19) with 0 < ↵ < 1). In this case let a = O(�) (thus the condition (a�)d/n2 =

O(1) is satisfied), which gives rise to the frequency grid {!↵,k = 2⇡�↵k� ;k = (k

1

, . . . , kd),�� kj �}, thus the frequency grid grows at the rate a/�1�↵ = �↵. Using this estimator we

have �dvar[ eQa,�(g; r)] = O(1). We note that these conditions are very similar to those given

in Bandyopadhyay et al. (2015) (see condition C.4(i)).

20

3.4 Uniformly sampled locations

In many situations it is reasonable to suppose that the locations are uniformly distributed over a

region. In this case, many of the results stated above can be simplified. These simplifications allow

us to develop a simple method for estimating nuisance parameters in Section 5.

We first recall that if the locations are uniformly sampled then h�(s) = ��dI[��/2,�/2]d(s),

therefore the Fourier coe�cients of h�(·) are simply �0 = 1 and �j = 0 if j 6= 0. This implies that

�r (defined in Corollary 3.1) is such that �r = 0 if r 6= 0 and �0 = 1. Therefore if {Z(s); s 2 Rd}is a stationary spatial random process that satisfies the assumptions in Theorem 3.6 (with uniform

sampling of the locations) and krk1

<< �

�dcovh< eQa,�(g; r1),< eQa,�(g; r2)

i=

8<

:

1

2

[C1

+D1

] +O⇣`�,a,n + (a�)2IS=NG

n2 + krk1�

⌘r1

= r2

(= r)

O(`�,a,n + (a�)2IS=NG

n2 ) r1

6= r2

or � r2

,

�dcovh= eQa,�(g; r1),= eQa,�(g; r2)

i=

8<

:

1

2

[C1

+D1

] +O⇣`�,a,n + (a�)2IS=NG

n2 + krk1�

⌘r1

= r2

(= r)

O(`�,a,n + (a�)2IS=NG

n2 ) r1

6= r2

or � r2

(22)

and �dcovh< eQa,�(g; r1),= eQa,�(g; r2)

i= O(`�,a,n+

(a�)2IS=NG

n2 ), where IS=NG

denotes the indicator

variable and is one if the random field is Gaussian, else it is zero (see Appendix A.2). From the

above we observe that {< eQa,�(g; r),= eQa,�(g; r); r 2 Zd} is a ‘near uncorrelated’ sequence where in

the case r 6= 0, < eQa,�(g; r) and = eQa,�(g; r) are consistent estimators of zero. It is this property we

exploit in Section 5 for estimating the variance of eQa,�(g; 0). Below we show asymptotic normality

in the case of uniformly distributed locations.

Theorem 3.7 [CLT on real and imaginary parts] Suppose Assumptions 2.1, 2.2, 2.5(b,c) and

2.4(i) or 2.4(ii) hold. Let C1

and C2

, be defined as in Corollary 3.1. We define the m-dimension

complex random vectors eQm = ( eQa,�(g, r1), . . . , eQa,�(g, rm)), where r1

, . . . , rm are such that ri 6=�rj and ri 6= 0. Under these conditions we have

2�d/2

C1

✓C1

C1

+ <C2

< eQa,�(g, 0),<eQm,=eQm

◆P! N

�0, I

2m+1

�(23)

with log

2(a)

�1/2 ! 0 as � ! 1, n ! 1 and a ! 1.

PROOF See Appendix D, Subba Rao (2015b). ⇤

4 Statistical inference

In this section we apply the results derived in the previous section to some of the examples discussed

in Section 2.2.

We will assume the locations are uniformly distributed and the random field is Gaussian and

Assumption 2.5(b,c) is satisfied. All the statistics considered below are bias corrected and based

on eQa,�(g; 0). Since we make a bias correction, let b�2

n = 1

n

Pnj=1

Z(sj)2.

In Appendix F, Subba Rao (2015b), we state the results for the non-biased corrected statistics.

21

4.1 The Whittle likelihood

We first consider the Whittle likelihood described in Example 2.1(i), but with the ‘bias’ term

removed. Let e✓n = argmin✓2⇥ eLn(✓), where

eLn(✓) =1

�d

C�X

k=�C�

✓log f✓(!k) +

|Jn(!k)|2

f✓(!k)

◆� b�2

n

n

C�X

k=�C�

1

f✓(!k)

= eQa,�(f✓(!k)�1; 0) +

C�X

k=�C�

log f✓(!k)

and ⇥ is a compact parameter space. Let ✓0

= argmin✓2⇥ L(✓), where

L(✓) = 1

(2⇡)d

Z

2⇡[�C,C]

d

log f✓(!) +

f(!)

f✓(!)

�d!.

We will assume that ✓0

is such that for all ! 2 Rd, f(!) = f✓0(!), where f(!) denotes the true

spectral density of the stationary spatial random process and there does not exist another ✓ 2 ⇥ such

that for some ! 2 2⇡[�C,C]d f✓0(!) = f✓(!). Under the assumption that sup✓2⇥ sup!22⇡[�C,C]

d |r✓f✓(!)�1| <1 and sup✓2⇥ sup!22⇡[�C,C]

d |r3

✓f✓(!)�1| < 1 and by using Theorem 3.1(i) and (22) we can show

equicontinuity in probability of eLn(✓) and r3

✓eLn(✓). These two conditions imply consistency i.e.

e✓nP! ✓

0

and r2

✓eL(✓n)

P! V for any ✓n that lies between e✓n and ✓0

and

V =2

(2⇡)d

Z

2⇡[�C,C]

d[r✓ log f✓(!)] [r✓ log f✓(!)]

0c✓=✓0d!.

Using these preliminary results we now derive the limiting distribution of e✓n.By making a Taylor expansion of eLn about ✓

0

is clear that the asymptotic sampling results are

determined by the the first derivative of eLn(✓) at the true parameter

r✓eLn(✓0) = eQa,�(r✓f✓0(·)�1; 0) +

1

�d

aX

k=�a

1

f✓0(!k)r✓f✓0(!k),

where the second term on the right hand side of the above is deterministic and

eQa,�(r✓f✓0(·)�1; 0) =1

�d

C�X

k=�C�

r✓f✓(!k)�1|Jn(!k)|2 �

b�2

n

n

aX

k=�a

r✓f✓(!k)�1.

By applying Theorem 3.7 to eQa,�(r✓f✓0(·)�1; 0), we have �d/2r✓eLn(✓)

D! N (0, V ). Altogether this

gives

�d/2(e✓n � ✓0

)P! N (0, V �1),

with �d/n ! 0 and log

2(a)

�1/2 ! 0 as a ! 1 and � ! 1. In the non-Gaussian case the limiting

variance will change to V �1WV �1, where an expression for W can be deduced from Theorem A.2.

Note that the asymptotic sampling properties of the discretized Whittle likelihood estimator e✓nare identical to the asymptotic sampling properties of the integrated Whittle likelihood estimator

derived in Matsuda and Yajima (2009).

22

4.2 The nonparametric covariance estimator

We now consider the bias corrected version of nonparametric estimator defined in Example 2.2(ii).

Let

bcn(v) = T

✓2v

�

◆ecn(v)

where ecn(v) =

1

�d

aX

k=�a

|Jn(!k)|2 exp(iv0!k)�b�2

n

n

aX

k=�a

exp(iv0!k)

!,

and T is the d-dimensional triangle kernel. It is clear the asymptotic sampling properties of bcn(v)are determined by ecn(v). Therefore, we first consider ecn(v). We observe that ecn(v) = eQa,�(eiv

0·; 0),

thus we use the results in Section 3 to derive the asymptotic sampling properties of ecn(v). By using

Theorem 3.1(ii) and Assumption 2.5 we have

E[ecn(v)] = I⇣eiv·;

a

�

⌘+O

✓log �

�

◆

=1

(2⇡)d

Z

Rdf(!) exp(iv0!)d! +O

✓�

a

◆�

+log �

�

!= c(v) +O

✓�

a

◆�

+log �

�

!, (24)

where we use the condition f(!) ��(!) to replace I�eiv·; a�

�with I

�eiv·;1

�. Therefore, if

��+d/2/a� ! 0, �d/n ! 0 as a ! 1 and � ! 1, then by using Theorem 3.7 we have

�d/2 [ecn(v)� c(v)]D! N (0,�(v)2) (25)

where

�(v)2 =1

(2⇡)d

Z

2⇡[�a/�,a/�]df(!)2

⇥1 + exp(i2v0!)

⇤d!.

We now derive the sampling properties of bcn(v). By using properties of the triangle kernel we have

E[bcn(v)] = c(v) +O

✓��

a�+

log �

�

�✓1� min

1id |vi|�

◆+

max1id |vi|�

◆

and

�d/2

⇥bcn(v)� T

�2v�

�c(v)

⇤

T�2v�

��(v)

D! N (0, 1),

with ��+d/2/a� ! 0, �d

n ! 0 and log

2(a)

�1/2 ! 0 as a ! 1 and � ! 1.

We mention that the Fourier transform of the bias corrected nonparametric covariance estimator

isRRd ecn(v)e�iv0!dv =

Pak=�a

�|Jn(!k)|2 � b�2

n

Sinc2[�(!k � !)]. Therefore, when a bias correc-

tion is made the corresponding sample covariance may not be a non-negative definite function.

The sampling properties of the non-negative definite nonparametric covariance function estimator

defined in Example 2.2(ii), can be found in Appendix F, Subba Rao (2015b).

23

4.3 Parameter estimation using an L2

criterion

In this section we consider the asymptotic sampling properties of the parameter estimator using the

L2

criterion. We will assume that there exists a ✓0

2 ⇥, such that for all ! 2 Rd, f✓0(!) = f(!) and

there does not exist another ✓ 2 ⇥ such that for all ! 2 Rd f✓0(!) = f✓(!). In addition, we assume

sup✓2⇥RRd kr✓f(!; ✓)k2

1

d! < 1, sup✓2⇥ sup!2Rd |r✓f✓(!)| < 1 and sup✓2⇥ sup!2Rd |r3

✓f✓(!)| <1.

We consider the bias corrected version of the L2

criterion given in Example 2.2(iii). That is

Ln(✓) =1

�d

aX

k1,...,kd=�a

�|Jn(!k)|2 � b�2

n � f✓(!k)�2

,

and let b✓n = argmin✓2⇥ Ln(✓) (⇥ is a compact set).

By adapting similar methods to those developed in this paper to quartic forms we can show that

pointwise Ln(✓)P! L(✓), where L(✓) =

RRd [f(!)�f✓(!)]dd!+

RRd f(!)2d!, which is minimised at

✓0

. Under the stated conditions we can show uniform convergence of Ln(✓), which implies b✓nP! ✓

0

.

Using this result the sampling properties of b✓n are determined by r✓Ln(✓) (given in equation (11))

and in particular its random component eQa,�(�2r✓f✓(·); 0), where

r✓Ln(✓) = � 2

�d

aX

k1,...,kd=�a

<r✓f✓(!k)�|Jn(!k)|2 � f✓(!k)

+

2b�2

n

�d

aX

k1,...,kd=�a

<r✓f✓(!k)

= eQa,�(�2r✓f✓(·); 0) +2

�d

aX

k=�a

f✓(!k)r✓f✓(!k).

Using the assumptions stated above and Theorem 3.3 we can show that for any ✓n between ✓0

and b✓n we have r2

✓Ln(✓n)P! A where

A =2

(2⇡)d

Z

Rd[r✓f(!; ✓

0

)][r✓f(!; ✓0

)]0d!, (26)

as �d/n ! 0 with � ! 1 and n ! 1. Thus �d/2(b✓n � ✓0

) = A�1�d/2rLn(✓0) + op(1), and it is

clear the asymptotic sampling properties of b✓n are determined r✓Ln(✓0) = eQa,�(�2r✓f✓0(·); 0) +1

�d

Pak=�ar✓f✓0(!k)2.

Thus by using Theorem 3.7 we have �d/2r✓Ln(✓0)D! N (0, B), where

B =4

(2⇡)d

Z

Rdf(!)2 [r✓f✓(!)] [r✓f✓(!)]0c✓=✓0d!.

Therefore, by using the above we have

�d/2(b✓n � ✓0

)P! N (0, A�1BA�1)

with �d

n ! 0 and log

2(a)

�1/2 ! 0 as a ! 1 and � ! 1.

24

5 An estimator of the variance of

eQa,�(g; 0)

The expression for the variance eQa,�(g; 0) given in the examples above, is rather unwieldy and

di�cult to estimate directly. For example, in the case that the random field is Gaussian, one

can estimate C1

by replacing the integral with the sumPa

k=�a and the spectral density function

with the squared bias corrected periodogram (|Jn(!k)|2 � b�2

n)2 (see Bandyopadhyay et al. (2015),

Lemma 7.5, though their estimator also holds for non-Gaussian spatial processes when eQa,�,↵ is

defined using the frequency grid described in Section 3.3.1).

In this section we describe an alternative, very simple method for estimating the variance ofeQa,�(g; 0) under the assumption the locations are uniformly distributed. For ease of presentation

we will assume the spatial random field is Gaussian. However, exactly the same methodology holds

when the spatial random fields is non-Gaussian. The methodology is motivated by the method

of orthogonal samples for time series proposed in Subba Rao (2015a), where the idea is to define

a sample which by construction shares some of the properties of the statistic of interest. In this

section we show that { eQa,�(g; r); r 6= 0} is an orthogonal sample associated with eQa,�(g; r).

We first focus on estimating the variance of eQa,�(g; 0). By using Theorem 3.1 we have E[ eQa,�(g; 0)] =

I�g; a�

�+ o(1) and

�dvar[ eQa,�(g; 0)] = C1

+O(`�,a,n).

In contrast, we observe that if no elements of the vector r are zero, then by Theorem 3.1 E[ eQa,�(g; r)] =

O(Qd

i=1

[log �+log |ri|]/�d) (slightly slower rates are obtained when r contains some zero). In other

words, eQa,�(g; r) is estimating zero. On the other hand, by (22) we observe that


i=

(1

2

C1

+O⇣`�,a,n + krk1

�

⌘r1

= r2

(= r)

O (`�,a,n) r1

6= r2

, r1

6= �r2

.

A similar result holds for {= eQa,�(g; r)}, furthermore we have �dcov[< eQa,�(g; r1),= eQa,�(g; r2)] =

O(`�,a,n).

In summary, if krk1

is not too large, then {< eQa,�(g; r),= eQa,�(g; r)} are ‘near uncorrelated’

random variables whose variance is approximately the same as eQa,�(g; 0)/p2. This suggests that

we can use {< eQa,�(g; r),= eQa,�(g; r); r 2 S} to estimate var[ eQa,�(g; 0)], where the set S is defined

as

S = {r; krk1

M, r1

6= r2

and all elements of r are non-zero} . (27)

This leads to the following estimator

eV =�d

2|S|X

r2S

⇣2|< eQa,�(g; r)|2 + 2|= eQa,�(g; r)|2

⌘=

�d

|S|X

r2S| eQa,�(g; r)|2, (28)

where |S| denotes the cardinality of the set S. Note that we specifically select the set S such that no

element r contains zero, this is to ensure that E[ eQa,�(g; r)] is small and does not bias the variance

estimator.

In the following theorem we obtain a mean squared bound for eV .

25

Theorem 5.1 Let eV be defined as in (28), where S is defined in (27). Suppose Assumptions 2.1,

2.2 and 2.5(a,b,c) hold and either Assumption 2.4(i) or (ii) holds. Then we have

E⇣eV � �dvar[ eQa,�(g; 0)]

⌘2

= O(|S|�1 + |M |��1 + `�,a,n + ��d log4d(a))

as � ! 1, a ! 1 and n ! 1 (where `a,�,n is defined in (35)).

PROOF. See Subba Rao (2015b), Section E. ⇤

Thus it follows from the above result that if the set S grows at a rate such that |M |��1 ! 0 as

� ! 1, then eV is a mean square consistent estimators of �dvar[ eQa,�(g; 0)].

We note that the same estimator can be used in the case that the random field is non-Gaussian,

in this case to show consistency we need to place some conditions on the higher order cumulants

of the spatial random process. However, it is a trickier to relax the assumption that the locations

are uniformly distributed. This is because in the case of a non-uniform design E[ eQa,�(g; r)] (r 6= 0)

will not, necessarily, be estimating zero. We note that the same method can be used to estimate

the variance of the non-bias corrected statistic Qa,�(g; 0).

Acknowledgments

This work has been partially supported by the National Science Foundation, DMS-1106518 and

DMS-1513647. The author gratefully acknowledges Gregory Berkolaiko for many very useful dis-

cussions. Furthermore, the author gratefully acknowledges the editor, associate editor and three

anonymous referees who made substantial improvements to every part of the manuscript.

A Appendix

A.1 Proof of Theorem 3.1

It is clear from the motivation at the start of Section 3.1 that the sinc function plays an important

role in the analysis of Qa,�(g; r) and eQa,�(g; r). Therefore we now summarise some of its properties.

It is well known that 1

�

R �/2��/2 exp(ix!)dx = sinc(�!

2

) and

Z 1

�1sinc(u)du = ⇡ and

Z 1

�1sinc2(u)du = ⇡. (29)

We next state a well know result that is an important component of the proofs in this paper.

Lemma A.1 [Orthogonality of the sinc function]Z 1

�1sinc(u)sinc(u+ x)du = ⇡sinc(x) (30)

and if s 2 Z/{0} thenZ 1

�1sinc(u)sinc(u+ s⇡)du = 0. (31)

26

PROOF In Appendix C, Subba Rao (2015b). ⇤

We use the above results in the proof below.

PROOF of Theorem 3.1. We first prove (i). By using the same proof used to prove Theorem

2.1 (see Bandyopadhyay and Subba Rao (2015), Theorem 1) we can show that

E

2

4Jn(!k1)Jn(!k2)��d

n

nX

j=1

Z(sj)2eis

0j(!k1

�!k2)

3

5 =

(f(!k) +O( 1�) k

1

= k2

(= k),

O( 1

�d�b ) k1

� k2

6= 0.,

where b = b(k1

�k2

) denotes the number of zero elements in the vector r. Therefore, since a = O(�),

by taking expectations of eQa,�(g; r) we use the above to give

EheQa,�(g; r)

i=

(1

�d

Pak=�a g(!k)f(!k) +O( 1� ) r = 0

O( 1

�d�b ) r 6= 0.

Therefore, by replacing the summand with the integral we obtain (12).

The above method cannot be used to prove (ii) since a/� ! 1, this leads to bounds which

may not converge. Therefore, as discussed in Section 3.1 we consider an alternative approach. To

do this we expand eQa,�(g; r) as a quadratic form to give

eQa,�(g; r) =1

n2

nX

j1,j2=1

j1 6=j2

aX

k=�a

g(!k)Z(sj1)Z(sj2) exp(i!0k(sj1 � sj2)) exp(�i!0

rsj2).

Taking expectations gives

EheQa,�(g; r)

i= c

2

aX

k=�a

g(!k)E⇥c(s

1

� s2

) exp(i!0k(s1 � s

2

)� is02

!r)⇤

where c2

= n(n� 1)/2. In the case that d = 1 the above reduces to

EheQa,�(g; r)

i= c

2

aX

k=�a

g(!k)E [c(s1

� s2

) exp(i!k(s1 � s2

)� is2

!r)]

=c2

�2

aX

k=�a

g(!k)

Z �/2

��/2

Z �/2

��/2c(s

1

� s2

) exp(i!k(s1 � s2

)� is2

!r)ds1ds2. (32)

Replacing c(s1

� s2

) with the Fourier representation of the covariance function gives

EheQa,�(g; r)

i=

c2

2⇡

aX

k=�a

g(!k)

Z 1

�1f(!)sinc

✓�!

2+ k⇡

◆sinc

✓�!

2+ (k + r)⇡

◆d!.

By a change of variables y = �!/2 + k⇡ and replacing the sum with an integral we have

EheQa,�(g; r)

i=

c2

⇡�

aX

k=�a

g(!k)

Z 1

�1f(

2y

�� !k)sinc(y)sinc(y + r⇡)dy

=c2

⇡

Z 1

�1sinc(y)sinc(y + r⇡)

✓1

�

aX

k=�a

g(!k)f(2y

�� !k)

◆dy

=c2

2⇡2

Z 1


Z2⇡a/�

�2⇡a/�g(u)f(

2y

�� u)dudy +O

✓1

�

◆,

27

where the O(��1) comes from Lemma E.1(ii) in Subba Rao (2015b). Replacing f(2y� � u) with

f(�u) gives

EheQa,�(g; r)

i=

c2

2⇡2

Z 1


Z2⇡a/�

�2⇡a/�g(u)f(�u)dudy +Rn +O(��1),

where

Rn =c2

2⇡2

Z 1


✓Z2⇡a/�

�2⇡a/�g(u)

✓f(

2y

�� u)� f(�u)

◆dudy.

By using Lemma C.2 in Subba Rao (2015b), we have |Rn| = O( log �+I(r 6=0) log |r|� ). Therefore, by

using Lemma A.1, replacing c2

with one (which leads to the error O(n�1)) and (29) we have

EheQa,�(g; r)

i

=c2

2⇡2

Z 1


Z2⇡a/�

�2⇡a/�g(u)f(�u)dudy +O

✓log �+ I(r 6= 0) log |r|

�

◆

=I(r = 0)

2⇡

Z2⇡a/�

�2⇡a/�g(u)f(u)dudy +O

✓1

�+

log �+ I(r 6= 0) log |r|�

◆,

which gives (13) in the case d = 1.

To prove the result for d > 1, we will only consider the case d = 2, as it highlights the di↵erence

from the d = 1 case. By substituting the spectral representation into (12) we have

EheQa,�(g; (r1, r2))

i

=c2

⇡2�2

aX

k1,k2=�a

g(!k1 ,!k2)

Z

R2f

✓2u

1

�� !k1 ,

2u2

�� !k2

◆sinc(u

1

)sinc(u1

+ r1

⇡)sinc(u2

)sinc(u2

+ r2

⇡)du1

du2

.

In the case that either r1

= 0 or r2

= 0 we can use the same proof given in the case d = 1 to give

E[ eQa,�(g; r)] =1

(2⇡)2⇡2�2

Z

[�a/�,a/�]2g(!

1

,!2

)f (!1

,!2

)

Z

Rdsinc(u

1

)sinc(u1

+ r1

⇡)sinc(u2

)sinc(u2

+ r2

⇡)du1

du2

+Rn,

where |Rn| = O( log �+I(krk1 6=0) log krk1� + n�1), which gives the desired result. However, in the case

that both r1

6= 0 and r2

6= 0, we can use Lemma C.2, equation (49), Subba Rao (2015b) to obtain

EheQa,�(g; (r1, r2))

i=

c2

⇡2�2

aX

k1,k2=�a

g(!k1 ,!k2)

Z

Rdf

✓2u

1

�� !k1 ,

2u2

�� !k2

◆⇥

sinc(u1

)sinc(u1

+ r1

⇡)sinc(u2

)sinc(u2

+ r2

⇡)du1

du2

= O

Q2

i=1

[log �+ log |ri|]�2

!

thus we obtain the faster rate of convergence.

It is straightforward to generalize these arguments to d > 2. ⇤

28

A.2 Variance calculations in the case of uniformly sampled locations

We start by proving the results in Section 3 for the case that the locations are uniformly distributed.

The proof here forms the building blocks for the case of non-uniform sampling of the locations,

which can found in Section B, Subba Rao (2015b).

In Lemma E.2, Appendix E, Subba Rao (2015b) we show that if the spatial process is Gaussian

and the locations are uniformly sampled then for a bounded frequency grid we have


i=

(C1

(!r) +O( 1� + �d

n ) r1

= r2

(= r)

O( 1� + �d

n ) r1

6= r2

and


i=

(C2

(!r) +O( 1� + �d

n ) r1

= �r2

(= r)

O( 1� + �d

n ) r1

6= �r2

where

C1

(!r) = C1,1(!r) + C

1,2(!r) and C2

(!r) = C2,1(!r) + C

2,2(!r), (33)

with

C1,1(!r) =

1

(2⇡)d

Z

2⇡[�a/�,a/�]df(!)f(! + !r)|g(!)|2d!,

C1,2(!r) =

1

(2⇡)d

Z

Dr

f(!)f(! + !r)g(!)g(�! � !r)d!,

C2,1(!r) =

1

(2⇡)d

Z

2⇡[�a/�,a/�]df(!)f(! + !r)g(!)g(�!)d!,

C2,2(!r) =

1

(2⇡)d

Z

Dr

f(!)f(! + !r)g(!)g(! + !r)d!. (34)

On the other other hand, if the frequency grid is unbounded, then under Assumptions 2.4(ii) and

2.5(b) we have


i= A

1

(r1

, r2

) +A2

(r1

, r2

) +O(�d

n),


i= A

3

(r1

, r2

) +A4

(r1

, r2

) +O(�d

n),

where

A1

(r1

, r2

) =1

⇡2d�d

2aX

m=�2a

min(a,a+m)X

k=max(�a,�a+m)

Z

R2df(

2u

�� !k)f(

2v

�+ !k + !r1)

g(!k)g(!k � !m)Sinc(u�m⇡)Sinc(v + (m+ r1

� r2

)⇡)Sinc(u)Sinc(v)dudv

29

and expressions for A2

(r1

, r2

), . . . , A4

(r1

, r4

) can be found in Lemma E.2, Subba Rao (2015b). In-

terestingly supa |A1

(r1

, r2

)| < 1, supa |A2

(r1

, r2

)| < 1, supa |A3

(r1

, r2

)| < 1 and supa |A4

(r1

, r2

)| <1. Hence if �d/n ! c (c < 1) we have

�d supa


i�� < 1 and �d supa


i�� < 1,

We now obtain simplified expressions for the terms A1

(r1

, r2

), . . . , A4

(r1

, r2

) under the slightly

stronger condition that Assumption 2.5(c) also holds.

Lemma A.2 Suppose Assumptions 2.4(ii) and 2.5(b,c) hold. Then for 0 |r1

|, |r2

| C|a| (whereC is some finite constant) we have

A1

(r1

, r2

) =

(C1,1(!r) +O(`�,a,n) r

1

= r2

(= r)

O(`�,a,n) r1

6= r2

A2

(r1

, r2

) =

(C1,2(!r) +O(`�,a,n) r

1

= r2

(= r)

O(`�,a,n) r1

6= r2

A3

(r1

, r2

) =

(C2,1(!r) +O(`�,a,n) r

1

= �r2

(= r)

O(`�,a,n) r1

6= �r2

and

A4

(r1

, r2

) =

(C2,2(!r) +O(`�,a,n) r

1

= �r2

(= r)

O(`�,a,n) r1

6= �r2

,

where C1,1(!r), . . . , C2,2(!r) (using d = 1) and `�,a,n are defined in Lemma E.2.

PROOF. We first consider A1

(r1

, r2

) and write it as

A1

(r1

, r2

) =

1

⇡2

Z 1

�1

Z 1

�1

2aX

m=�2a

sinc(u)sinc(v)sinc(u�m⇡)sinc(v + (m+ r1

� r2

)⇡)Hm,�

✓2u

�,2v

�; r

1

◆dudv,

where

Hm,�

✓2u

�,2v

�; r

1

◆=

1

�

min(a,a+m)X

k=max(�a,�a+m)

f

✓�2u

�+ !k

◆f

✓2v

�+ !k + !r

◆g(!k)g(!k � !m)

noting that f(2u� � !) = f(! � 2u� ).

If f(�2u� + !k) and f(2v� + !k + !r) are replaced with f(!k) and f(!k + !r) respectively, then

we can exploit the orthogonality property of the sinc functions. This requires the following series

of approximations.

30

(i) We start by defining a similar version of A1

(r1

, r2

) but with the sum replaced with an integral.

Let

B1

(r1

� r2

; r1

)

=1

⇡2

Z 1

�1

Z 1

�1

2aX

m=�2a

sinc(u)sinc(u�m⇡)sinc(v)sinc(v + (m+ r1

� r2

)⇡)Hm

✓2u

�,2v

�; r

1

◆dudv,

where

Hm

✓2u

�,2v

�; r

1

◆=

1

2⇡

Z2⇡min(a,a+m)/�

2⇡max(�a,�a+m)/�f(! � 2u

�)f(

2v

�+ ! + !r1)g(!)g(! � !m)d!.

By using Lemma C.4, Subba Rao (2015b) we have

|A1

(r1

, r2

)�B1

(r1

� r2

; r1

)| = O

✓log2 a

�

◆.

(ii) Define the quantity

C1

(r1

� r2

; r1

)

=1

⇡2

Z 1

�1

Z 1

�1

2aX

m=�2a

sinc(u)sinc(u�m⇡)sinc(v)sinc(v + (m+ r1

� r2

)⇡)Hm(0, 0; r1

)dudv.

By using Lemma C.5, Subba Rao (2015b), we can replace B1

(r1

� r2

; r1

) with C1

(r1

� r2

; r1

)

to give the replacement error

|B1

(r1

� r2

; r1

)� C1

(r1

� r2

; r1

)| = O

✓log2(a)

(log a+ log �)

�

�◆.

(iii) Finally, we analyze C1

(r1

� r2

; r1

). Since Hm(0, 0; r1

) does not depend on u or v we take it

out of the integral to give

C1

(r1

� r2

; r1

)

=1

⇡2

2aX

m=�2a

Hm(0, 0; r1

)

✓Z

Rsinc(u)sinc(u�m⇡)du

◆✓Z

Rsinc(v)sinc(v + (m+ r

1

� r2

)⇡)dv

◆

=1

⇡2

H0

(0, 0; r1

)

✓Z

Rsinc2(u)du

◆✓Z

Rsinc(v)sinc(v + (r

1

� r2

)⇡)dv

◆,

where the last line of the above is due to orthogonality of the sinc function (see Lemma A.1).

If r1

6= r2

, then by orthogonality of the sinc function we have C1

(r1

� r2

; r1

) = 0. On the

other hand if r1

= r2

we have

C1

(0; r) =1

2⇡

Z2⇡a/�

�2⇡a/�f(!)f(! + !r)|g(!)|2d! = C

1,1(!r).

The proof for the remaining terms A2

(r1

, r2

), A3

(r1

, r2

) and A4

(r1

, r2

) is identical, thus we omit

the details. ⇤

31

Theorem A.1 Suppose Assumptions 2.1, 2.2, 2.4(ii) and 2.5(b,c) hold. Then for 0 |r1

|, |r2

| C|a| (where C is some finite constant) we have


i=

(C1

(!r) +O(`�,a,n) r1

= r2

(= r)

O(`�,a,n) r1

6= r2


i=

(C2

(!r) +O(`�,a,n) r1

= �r2

(= r)

O(`�,a,n) r1

6= �r2

,

where C1

(!r) and C2

(!r) are defined in (79) and

`�,a,n = log2(a)

log a+ log �

�

�+

�d

n. (35)

PROOF. By using Lemmas E.2 and A.2 we immediately obtain (in the case d = 1)

covheQa,�(g; r1), eQa,�(g; r2)

i=

(C1

(!r) +O(`�,a,n) r1

= r2

(= r)

O(`�,a,n) r1

6= r2

and

covheQa,�(g; r1), eQ�,a,n(g; r2)

i=

(C2

(!r) +O(`�,a,n) r1

= �r2

(= r)

O(`�,a,n) r1

6= �r2

.

This gives the result for d = 1. To prove the result for d > 1 we use the same procedure outlined

in the proof of Lemma A.2 and the above. ⇤

The above theorem means that the variance for both the bounded and unbounded frequency

grid are equivalent (up to the limits of an integral).

Using the above results and the Lipschitz continuity of g(·) and f(·) we can show that

Cj(!r) = Cj +O

✓krk

1

�

◆,

where C1

and C2

are defined in Corollary 3.1.

We now derive an expression for the variance for the non-Gaussian case.

Theorem A.2 Let us suppose that {Z(s); s 2 Rd} is a fourth order stationary spatial random

field that satisfies Assumption 2.1(i). Suppose all the assumptions in Theorem 3.6 hold with the

exception of Assumption 2.3 which is replaced with Assumption 2.2 (i.e. we assume the locations

are uniformly sampled). Then for krk1

, kr2

k1

<< � we have


i=

(C1

+D1

+O(`�,a,n + (a�)d

n2 + krk1� ) r

1

= r2

(= r)

O(`�,a,n + (a�)d

n2 ) r1

6= r2

(36)


i=

(C2

+D2

+O(`�,a,n + (a�)d

n2 + krk1� ) r

1

= �r2

(= r)

O(`�,a,n + (a�)d

n2 ) r1

6= �r2

, (37)

32

where C1

and C2

are defined in Corollary 3.1 and

D1

=1

(2⇡)2d

Z

2⇡[�a/�,a/�]2dg(!

1

)g(!2

)f4

(�!1

,�!2

,!2

)d!1

d!2

D2

=1

(2⇡)2d

Z

2⇡[�a/�,a/�]2dg(!

1

)g(!2

)f4

(�!1

,�!2

,!2

)d!1

d!2

.

PROOF We prove the result for the notationally simple case d = 1. By using indecomposable

partitions, conditional cumulants and (81) in Subba Rao (2015b) we have

�covheQa,�(g; r1), eQa,�(g; r2)

i= A

1

(r1

, r2

) +A2

(r1

, r2

) +B1

(r1

, r2

) +B2

(r1

, r2

) +O

✓�

n

◆, (38)

where A1

(r1

, r2

) and A2

(r1

, r2

) are defined below equation (81) and

B1

(r1

, r2

) = �c4

aX

k1,k2=�a

g(!k1)g(!k2)E

4

(s2

� s1

, s3

� s1

, s4

� s1

)eis1!k1e�is2!k1+r1e�is3!k2eis4!k2+r2

�

B2

(r1

, r2

) =�

n4

X

j1,...,j42D3

aX

k1,k2=�a

g(!k1)g(!k2)⇥

E

4

(sj2 � sj1 , sj3 � sj1 , sj4 � sj1)eisj1!k1e�isj2!k1+r1e�isj3!k2eisj4!k2+r2

�

with c4

= n(n� 1)(n� 2)(n� 3)/n2

D3

= {j1

, . . . , j4

; j1

6= j2

and j3

6= j4

but some j’s are in common}. The limits of A1

(r1

, r2

) and

A2

(r1

, r2

) are given in Lemma A.2, therefore, all that remains is to derive bounds for B1

(r1

, r2

)

and B2

(r1

, r2

). We will show that B1

(r1

, r2

) is the dominating term, whereas by placing su�cient

conditions on the rate of growth of a, we will show that B2

(r1

, r2

) ! 0.

In order to analyze B1

(r1

, r2

) we will use the resultZ

R3sinc(u

1

+ u2

+ u3

+ (r2

� r1

)⇡)sinc(u2

)sinc(u3

)du1

du2

du3

=

(⇡3 r

1

= r2

0 r1

6= r2

., (39)

which follows from Lemma A.1. In the following steps we will make a series of approximations

which will allow us to apply (39).

We start by substituting the Fourier representation of the cumulant function

4

(s1

� s2

, s1

� s3

, s1

� s4

) =1

(2⇡)3

Z

R3f4

(!1

,!2

,!3

)ei(s1�s2)!1ei(s1�s3)!2ei(s1�s4)!4d!1

d!2

d!3

,

into B1

(r1

, r2

) to give

B1

(r1

, r2

)

=c4

(2⇡)3�3

aX

k1,k2=�a

g(!k1)g(!k2)

Z

R3f4

(!1

,!2

,!3

)

Z

[��/2,�/2]4eis1(!1+!2+!3+!k1

)

e�is2(!1+!k1+r1)e�is3(!2+!k2

)eis4(�!3+!k2+r2)ds

1

ds2

ds3

ds4

d!1

d!2

d!3

=c4

�

(2⇡)3

aX

k1,k2=�a

g(!k1)g(!k2)

Z

R3f4

(!1

,!2

,!3

)sinc

✓�(!

1

+ !2

+ !3

)

2+ k

1

⇡

◆sinc

✓�!

1

2+ (k

1

+ r1

)⇡

◆

⇥sinc

✓�!

2

2+ k

2

⇡

◆sinc

✓�!

3

2� (k

2

+ r2

)⇡

◆d!

1

d!2

d!3

.

33

Now we make a change of variables and let u1

= �!12

+ (k1

+ r1

), u2

= �!22

+ k2

⇡ and u3

=�!32

� (k2

+ r2

)⇡, this gives

B1

(r1

, r2

) =c4

⇡3�2

aX

k1,k2=�a

Z

R3g(!k1)g(!k2)f4

✓2u

1

�� !k1+r1 ,

2u2

�� !k2 ,

2u3

�+ !k2+r2

◆⇥

⇥sinc (u1

+ u2

+ u3

+ (r2

� r1

)⇡) sinc(u1

)sinc(u2

)sinc(u3

)du1

du2

du3

.

Next we exchange the summand with a double integral and use Lemma E.1(iii) together with

Lemma C.1, equation (44) (in Subba Rao (2015a)) to obtain

B1

(r1

, r2

) =c4

(2⇡)2⇡3

Z

2⇡[�a/�,a/�]2

Z

R3g(!

1

)g(!2

)f4

✓2u

1

�� !

1

� !r1 ,2u

2

�� !

2

,2u

3

�+ !

2

+ !r2

◆⇥

sinc (u1

+ u2

+ u3

+ (r2

� r1

)⇡) sinc(u1

)sinc(u2

)sinc(u3

)du1

du2

du3

d!1

d!2

+O

✓1

�

◆.

By using Lemma C.3, we replace f4

(2u1� � !

1

� !r1 ,2u2� � !

2

, 2u3� + !

2

+ !r2) in the integral with

f4

(�!1

� !r1 ,�!2

,!2

+ !r2), this gives

B1

(r1

, r2

)

=c4

(2⇡)2⇡3

Z

2⇡[�a/�,a/�]2g(!

1

)g(!2

)f4

(�!1

� !r1 ,�!2

,!2

+ !r2)⇥Z

R3sinc(u

1

+ u2

+ u3

+ (r2

� r1

)⇡)sinc(u1

)sinc(u2

)sinc(u3

)du1

du2

du3

d!1

d!2

+O

✓log3 �

�

◆

=c4

Ir1=r2

(2⇡)2

Z

2⇡[�a/�,a/�]2g(!

1

)g(!2

)f4

(�!1

� !r1 ,�!2

,!2

+ !r2)d!1

d!2

+O

✓log3 �

�

◆,

where the last line follows from the orthogonality relation of the sinc function in equation (39).

Finally we make one further approximation. In the case r1

= r2

we replace f4

(�!1

�!r1 ,�!2

,!2

+

!r2) with f4

(�!1

,�!2

,!2

) which by using the Lipschitz continuity of f4

and Lemma C.1, equation

(44) gives

B1

(r1

, r1

) =c4

(2⇡)2

Z

2⇡[�a/�,a/�]2g(!

1

)g(!2

)f4

(�!1

,�!2

,!2

)d!1

d!2

+O

✓log3 �

�+

|r1

|�

◆.

Altogether this gives

B1

(r1

, r2

)

=

(1

(2⇡)2

R2⇡[�a/�,a/�]2 g(!1

)g(!2

)f4

(�!1

,�!2

,!2

)d!1

d!2

+O( log3 �� + |r1|+|r2|

� ) r1

= r2

O( log3 �� ) r

1

6= r2

..

Next we show that B2

(r1

, r2

) is asymptotically negligible. To bound B2

(r1

, r2

) we decompose D3

into six sets, the set D3,1 = {j

1

, . . . , j4

; j1

= j3

, j2

and j4

are di↵erent)}, D3,2 = {j

1

, . . . , j4

; j1

= j4

,

j2

and j3

are di↵erent), D3,3 = {j

1

, . . . , j4

; j2

= j3

, j1

and j4


1

, . . . , j4

; j2

=

j4

, j1

and j3


1

, . . . , j4

; j1

= j3

and j2

= j4

}, D2,2 = {j

1

, . . . , j4

; j1

= j4

and

34

j2

= j3

}. Using this decomposition we have B2

(r1

, r2

) =P

4

j=1

B2,(3,j)(r1, r2)+

P2

j=1

B2,(2,j)(r1, r2),

where

B2,(3,1)(r1, r2) =

|D3,1|�n4

aX

k1,k2=�a

g(!k1)g(!k2)⇥

E

4

(sj2 � sj1 , 0, sj4 � sj1)eisj1!k1e�isj2!k1+r1e�isj1!k2eisj4!k2+r2

�

for j = 2, 3, 4, B2,(3,j)(r1, r2) are defined similarly,

B2,(2,1)(r1, r2) =

|D2,1|�n4

aX

k1,k2=�a

g(!k1)g(!k2)⇥

E

4

(sj2 � sj1 , 0, sj2 � sj1)eisj1!k1e�isj2!k1+r1e�isj1!k2eisj2!k2+r2

�,

B2,(2,2)(r1, r2) is defined similarly and | · | denotes the cardinality of a set. By using identical

methods to those used to bound B1

(r1

, r2

) we have

|B2,(3,1)(r1, r2)|

C�

n(2⇡)3

aX

k1,k2=�a

|g(!k1)g(!k2)|Z

R3|f

4

(!1

,!2

,!3

)|��sinc

✓�(!

1

+ !3

)

2+ (k

2

� k1

)⇡

◆��⇥

��sinc✓�!

1

2� (k

1

+ r1

)⇡

◆��⇥��sinc

✓�!

3

2+ (k

2

+ r2

)⇡

◆�� d!1

d!2

d!3

= O

✓�

n

◆.

Similarly we can show |B2,(3,j)(r1, r2)| = O(�n) (for 2 j 4) and

|B2,(2,1)(r1, r2)|

�

(2⇡)3n2

aX

k1,k2=�a

|g(!k1)g(!k2)|Z

R3|f

4

(!1

,!2

,!3

)|��sinc

✓�(!

1

+ !2

)

2+ (k

2

� k1

)⇡

◆⇥

sinc

✓�(!

1

+ !2

)

2+ (k

2

� k1

+ r2

� r1

)⇡

◆ ��d!1

d!2

d!3

= O

✓a�

n2

◆.

This immediately gives us (36). To prove (37) we use identical methods. Thus we obtain the result.

⇤

Note that the term B2,(2,1)(r1, r2) in the proof above is important as it does not seem possible

to improve on the bound O(a�/n2).

References

Bandyopadhyay, B., & Subba Rao, S. (2015). A test for stationarity for irregularly spaced spatial

data. To appear in the Journal of the Royal Statistical Society (B).

35

Bandyopadhyay, S., & Lahiri, S. (2009). Asymptotic properties of discrete fourier transforms for

spatial data. Sankhya, Series A, 71, 221-259.

Bandyopadhyay, S., Lahiri, S., & Norman, D. (2015). A frequency domain empirical likelihood

method for irregular spaced spatial data. Annals of Statistics, 43, 519-545.

Beutler, F. J. (1970). Alias-free randomly timed sampling of stochastic processes. IEEE Transac-

tions on Information Theory, 16, 147-152.

Bickel, J. P., & Ritov, Y. (1988). Estimating integrated squared density derivatives: Sharp best

order of convergence. Sankhya Ser A, 50, 381-393.

Brillinger, D. (1969). The calculation of cumulants via conditioning. Annals of the Institute of

Statistical Mathematics, 21, 215-218.

Brillinger, D. (1981). Time series, data analysis and theory. San Francisco: SIAM.

Can, S., Mikosch, T., & Samorodnitsky, G. (2010). Weak convergence of the function-indexed

integrated periodogram for infinite variance processes. Bernoulli, 16, 995-1015.

Chen, W. W., Hurvich, C., & Lu, Y. (2006). On the correlation matrix of the discrete Fourier

Transform and the fast solution of large toeplitz systems for long memory time series. Journal

of the American Statistical Association, 101, 812-821.

Cressie, N., & Huang, H.-C. (1999). Classes of nonseperable spatiotemporal stationary covariance

functions. Journal of the American Statistical Association, 94, 1330-1340.

Dahlhaus, R. (1989). E�cient parameter estimation for self-similar processes. Annals of Statistics,

17, 749-1766.

Dahlhaus, R., & Janas, D. (1996). Frequency domain bootstrap for ratio statistics in time series

analysis. Annals of Statistics, 24, 1934-1963.

Deo, R. S., & Chen, W. W. (2000). On the integral of the squared periodogram. Stochastic

processes and their applications, 85, 159-176.

Dunsmuir, W. (1979). A central limit theorem for parameter estimation in stationary vector time

series and its application to models for a signal observed with noise. Annals of Statistics, 7,

490-506.

Fox, R., & Taqqu, M. (1987). Central limit theorems for quadratic forms in random variables

having long-range dependence. Probability Theory and Related Fields, 74, 213-240.

Fuentes, M. (2007). Approximate likelihood for large irregular spaced spatial data. Journal of the

American Statistical Association, 136, 447-466.

36

Gine, G., & Nickl, R. (2008). A simple adaptive estimator of the integrated square of a density.

Bernoulli, 14, 47-61.

Giraitis, L., & Surgailis, D. (1990). A central limit theorem for quadratic forms in strongly

dependent linear variables and its application to asymptotical normality of whittle’s estimate.

Probability Theory and Related Fields, 86, 87-104.

Hall, P., Fisher, N. I., & Ho↵man, B. (1994). On the nonparametric estimation of covariance

functions. Annals of Statistics, 22, 2115-2134.

Hall, P., & Patil, P. (1994). Properties of nonparametric estimators of autocovariance for stationary

random fields. Probability Theory and Related Fields, 99, 399-424.

Hannan, E. J. (1971). Non-linear time series regression. Advances in Applied Probability, 8, 767-780.

Kawata, T. (1959). Some convergence theorems for stationary stochastic processes. The Annals of

Mathematical Statistics, 30, 1192-1214.

Lahiri, S. (2003). Central limit theorems for weighted sums of a spatial process under a class of

stochastic and fixed designs. Sankhya, 65, 356-388.

Laurent, B. (1996). E�cient estimation of integral functions of a density. Annals of Statistics, 24,

659-681.

Masry, E. (1978). Poisson sampling and spectral estimation of continuous time processes. IEEE

Transactions on Information Theory, 24, 173-183.

Matsuda, Y., & Yajima, Y. (2009). Fourier analysis of irregularly spaced data on Rd. Journal of

the Royal Statistical Society (B), 71, 191-217.

Niebuhr, T., & Kreiss, J.-P. (2014). Asymptotics for autocovariances and integrated periodograms

for linear processes observed at lower frequencies. International Statistical Review, 82, 123-

140.

Rice, J. (1979). On the estimation of the parameters of a power spectum. Journal of Multivariate

Analysis, 9, 378-392.

Shapiro, H., & Silverman, R. A. (1960). Alias free sampling of random noise. J SIAM, 8, 225-248.

Stein, M., Chi, Z., & Welty, L. J. (2004). Approximating likelihoods for large spatial data sets.

Journal of the Royal Statistical Society (B), 66, 1-11.

Stein, M. L. (1994). Fixed domain asymptotics for spatial periodograms. Journal of the American

Statistical Association, 90, 1277-1288.

Stein, M. L. (1999). Interpolation of spatial data: Some theory for kriging. New York: Springer.

37

Subba Rao, S. (2015a). Proxy samples: A simple method for estimating nuisance parameters in

time series. Preprint.

Subba Rao, S. (2015b). Supplementary material: Fourier based statistics for irregular spaced

spatial data.

Taqqu, M., & Peccati, G. (2011). Wiener chaos: moments, cumulants and diagrams. a survey with

computer implementation. Milan: Springer.

Vecchia, A. V. (1988). Estimation and model identification for continuous spatial processes. Journal

of the Royal Statistical Society (B), 50, 297-312.

Walker, A. M. (1964). Asymptotic properties of least suqares estimates of paramters of the spectrum

of stationary non-deterministic time series. Journal of the Australian Mathematical Society,

4, 363-384.

38

Supplementary material

B Proofs in the case of non-uniform sampling

In this section we prove the results in Section 3. Most of the results derived here are based on the

methodology developed in the uniform sampling case.

PROOF of Theorem 2.2 We prove the result for d = 1. It is straightforward to show

cov [Jn(!k1), Jn(!k2)] =c2

�

1X

j1,j2=�1�j1�j2

Z �/2

��/2

Z �/2

��/2c(s

1

� s2

)eis1!j1eis2!j2eis1!k1�is2!k2ds

1

ds2

+c(0)�k1�k2�

n.

Thus, by using the same arguments as those used in the proof of Theorem 2.1, Bandyopadhyay

and Subba Rao (2015), we can show that

cov [Jn(!k1), Jn(!k2)] =c2

�

1X

j1,j2=�1�j1�j2

Z �/2

��/2e�is(!k2

�!j2�!j1�!k1)ds

! Z �/2

��/2c(t)eit(!j1+!k1

)dt

!

+c(0)�k1�k2�

n+O

✓1

�

◆,

where to obtain the remainder O( 1�) we use thatP1

j=�1 |�j | < 1. Next, by using the identity

Z �/2

��/2e�is(!k2

�!j2�!j1�!k1)ds =

(0 k

2

� j2

6= k1

+ j1

� k2

� j2

= k1

+ j1

we have

cov [Jn(!k1), Jn(!k2)] = c2

1X

j=�1�j�k2�k1�j

Z �/2

��/2c(t)eit(!j1+!k1

)dtds+c(0)�k1�k2�

n+O(��1)

=1X

j=�1�j�k2�k1�jf(!k1+j) +

c(0)�k1�k2�

n+O

✓1

�+

1

n

◆.

Finally, we replace f(!k1+j) with f(!k1) and use the Lipschitz continiuity of f(·) to give

��

1X

j=�1�j�k2�k1�j [f(!k1)� f(!k1+j)]

�� C

�

1X

j=�1|j| · |�j�k2�k1�j | = O

✓1

�

◆,

where the last line follows from |�j | C|j|�(1+�)I(j 6= 0). Altogether, this gives

cov [Jn(!k1), Jn(!k2)] = f(!k1)1X

j=�1�j�k2�k1�j +

c(0)�k1�k2�

n+O

✓1

�

◆.

This completes the proof for d = 1, the proof for d > 1 is the same. ⇤

39

PROOF of Theorem 3.2 We prove the result for the case d = 1. Using the same method

used to prove Theorem 3.1 (see the arguments in Section 3.1) we obtain

EheQa,�(g; r)

i

=c2

2⇡

1X

j1,j2=�1�j1�j2

aX

k=�a

g(!k)

Z 1

�1f(!)sinc

✓�!

2+ (k + j

1

)⇡

◆sinc

✓�!

2+ (k + r � j

2

)⇡

◆d!.

By the change of variables y = �!2

+ (k + j1

)⇡ we obtain

EheQa,�(g; r)

i

=c2

�⇡

1X

j1,j2=�1�j1�j2

aX

k=�a

g(!k)

Z 1

�1f

✓2y

�� !k

◆sinc(y)sinc(y + (r � j

1

� j2

)⇡)dy.

Replacing sum with an integral and using Lemma E.1(ii) gives

EheQa,�(g; r)

i

=c2

2⇡2

1X

j1,j2=�1�j1�j2

Z2⇡a/�

�2⇡a/�g(!)f

✓2y

�� !

◆d!

Z 1

�1sinc(y)sinc(y + (r � j

1

� j2

)⇡)dy +O

✓1

�

◆.

Next, replacing f(2y� � !) with f(�!) and we have

EheQa,�(g; r)

i

=c2

2⇡2

1X

j1,j2=�1�j1�j2

Z2⇡a/�

�2⇡a/�g(!)f

✓2y

�� !

◆d!

Z 1


1

� j2

)⇡)dy +Rn +O

✓1

�

◆.

where

Rn =c2

2⇡2

1X

j1,j2=�1�j1�j2

Z2⇡a/�

�2⇡a/�g(!)

f

✓2y

�� !

◆� f(�!)

� Z 1


1

� j2

)⇡)dyd!.

By using Lemma C.2, Subba Rao (2015b) we have

|Rn| C1X

j1,j2=�1|�j1 | · |�j2 |

log �+ log |r � j1

� j2

|�

C1X

j1,j2=�1|�j1 | · |�j2 |

log �+ log |r|+ log |j1

|+ log |j2

|�

,

noting that C is a generic constant that changes between inequalities. This gives

EheQa,�(g; r)

i

=c2

2⇡2

1X

j1,j2=�1�j1�j2

Z2⇡a/�

�2⇡a/�g(!)f(�!)d!

Z 1


1

� j2

)⇡)dy +O

✓log �+ log |r|

�

◆.

Finally, by the orthogonality of the sinc function at integer shifts (and f(�!) = f(!)) we have

EheQa,�(g; r)

i=

1

2⇡

1X

j=�1�j�r�j

Z2⇡a/�

�2⇡a/�g(!)f(!)d! +O

✓log �+ log |r|

�+

1

n

◆,

40

thus we obtain the desired result. ⇤

PROOF of Theorem 3.3(i) To prove (i) we use Theorem 2.2 and Lemma D.1 which imme-

diately gives the result.

PROOF of Theorem 3.3(ii) We first note that by using Lemma D.1 (generalized to non-

uniform sampling), we can show that


i= A

1

(r1

, r2

) +A2

(r1

, r2

) +O

✓�

n

◆(40)

where

A1

(r1

, r2

) = �aX

k1,k2=�a

g(!k1)g(!k2)cov⇥Z(s

1

) exp(is1

!k1), Z(s3

) exp(is3

!k2)⇤⇥

cov⇥Z(s

2

) exp(�is2

!k1+r1), Z(s4

) exp(�is4

!k2+r2)⇤

A2

(r1

, r2

) = �aX

k1,k2=�a


1

) exp(is1

!k1), Z(s4

) exp(�is4

!k2+r2)⇤⇥

cov⇥Z(s

2

) exp(�is2

!k1+r1), Z(s3

) exp(is3

!k2)⇤.

We first analyze A1

(r1

, r2

). Conditioning on the locations s1

, . . . , s4

gives

A1

(r1

, r2

)

=1X

j1,...,j4=�1�j1�j2�j3�j4

aX

k1,k2=�a

g(!k1)g(!k2)1

�3

Z

[��/2,�/2]4c(s

1

� s3

)c(s2

� s4

)

eis1!k1�is3!k2e�is2!k1+r1

+is4!k2+r2ei(s1!j1+s2!j2+s3!j3+s4!j4 )ds1

ds2

ds3

ds4

.

By using the spectral representation theorem and integrating out s1

, . . . , s4

we can write the above

as

A1

(r1

, r2

)

=�

(2⇡)2

1X

j1,...,j4=�1�j1�j2�j3�j4

aX

k1,k2=�a

g(!k1)g(!k2)

Z 1

�1

Z 1

�1f(x)f(y)sinc

✓�x

2+ (k

1

+ j1

)⇡

◆

sinc

✓�y

2� (r

1

+ k1

� j2

)⇡

◆sinc

✓�x

2+ (k

2

� j3

)⇡

◆sinc

✓�y

2� (r

2

+ k2

+ j4

)⇡

◆dxdy

=1

⇡2�

1X

j1,...,j4=�1�j1�j2�j3�j4

aX

k1,k2=�a

g(!k1)g(!k2)

Z 1

�1

Z 1

�1f(

2u

�� !k1+j1)f(

2v

�+ !k1+r1�j2)

⇥sinc(u)sinc(u+ (k2

� k1

� j1

� j3

)⇡)sinc(v)sinc(v � (r2

+ k2

+ j4

� r1

� k1

+ j2

)⇡)dudv. (41)

By making a change of variables m = k1

� k2

we have

|A1

(r1

, r2

)|

1

⇡2�

1X

j1,...,j4=�1�j1�j2�j3�j4

1X

m=�1

aX

k1=�a

g(!k1)g(!m�k1)

Z 1

�1

Z 1

�1|f(2u

�� !k1+j1)f(

2v

�+ !k1+r1�j2)|

⇥sinc(u)sinc(u+ (m+ j1

+ j3

)⇡)sinc(v)sinc(v + (m� r2

� j4

+ r1

� j2

)⇡)dudv.

41

Now by following the same series of bounds used to prove Lemma E.2(iii) we have

|A1

(r1

, r2

)| 1

⇡2�

1X

j1,...,j4=�1|�j1�j2�j3�j4 | sup

!|g(!)|2kfk2

2

⇥1X

m=�1

Z

R2d|sinc(u� (m+ j

1

+ j3

)⇡sinc(v + (m� r2

+ r1

� j4

� j2

)⇡)sinc(u)sinc(v)|dudv

< 1.

Similarly we can bound A2

(r1

, r2

) and �covheQa,�(g; r1), eQa,�(g; r2)

i, thus giving the required result.

⇤

PROOF of Theorem 3.3(iii) The proof uses the expansion (40). Using this as a basis, we

will show that

A1

(r1

, r1

) = U1

(r1

, r2

;!r1 ,!r2) +O(`�,a,n)

A2

(r1

, r1

) = U2

(r1

, r2

;!r1 ,!r2) +O(`�,a,n).

We find an approximation for A1

(r1

, r2

) starting with the expansion given in (41). We use the same

proof as that used to prove Lemma A.2 to approximate the terms inside the sumP

j1,...,j4. More

precisely we let m = k1

� k2

, replace !k1 with ! and by using the same methodology given in the

proof of Lemma A.2 (and thatP

j |�j | < 1), we have

A1

(r1

, r2

)

=1

2⇡3

1X

j1,...,j4=�1�j1�j2�j3�j4

2aX

m=�2a

Z2⇡min(a,a+m)/�

2⇡max(�a,�a+m)/�f(�! � !j1)f(! + !r1�j2)g(!)g(! � !m)d!

⇥Z

R2sinc(u)sinc(u+ (m+ j

1

+ j3

)⇡)sinc(v)sinc(v � (m� r2

� j4

+ r1

� j2

)⇡)dudv +O(`�,a,n).

By orthogonality of the sinc function we see that the above is zero unless m = �j1

� j3

and

m = r2

� r1

+ j2

+ j4

(and using that f(�! � !j1) = f(! + !j1)), therefore

A1

(r1

, r2

)

=1

2⇡

X

j1+...+j4=r1�r2

�j1�j2�j3�j4

Z2⇡min(a,a�j1�j3)/�

2⇡max(�a,�a�j1�j3)/�g(!)g(! + !j1+j3)f(! + !j1)f(! + !r1�j2)d! +O(`�,a,n).

This gives us U1,1(r1, r2;!r1 ,!r2). Next we consider A

2

(r1

, r2

)

A2

(r1

, r2

) =1

�3

X

j1,j2,j3,j42Z�j1�j2�j3�j4

aX

k1,k2=�a

g(!k1)g(!k2)

Z

[��/2,�/2]4c(s

1

� s4

)c(s2

� s3

) exp(is1

!k1+j1)

exp(is4

!k2+r2+j4) exp(�is2

!k1+r1�j2) exp(�is3

!k2�j3)ds1ds2ds3ds4

= �X

j1,j2,j3,j42Z

aX

k1,k2=�a

g(!k1)g(!k2)

Z

R2f(x)f(y)sinc

✓�x

2+ (k

1

+ j1

)⇡

◆⇥

sinc

✓�x

2� (k

2

+ r2

+ j4

)⇡

◆sinc

✓�y

2� (k

1

+ r1

� j2

)⇡

◆sinc

✓�y

2+ (k

2

� j3

)⇡

◆dxdy.

42

Making a change of variables u = �x2

+ (k1

+ j1

)⇡ and v = �y2

� (k1

+ r1

� j2

)⇡ we have

A2

(r1

.r2

) =1

⇡2�

X

j1,j2,j3,j42Z�j1�j2�j3�j4

aX

k1,k2=�a

g(!k1)g(!k2)

Z

R2f

✓2u

�� !k1+j1

◆f

✓2v

�+ !k1+r1�j2

◆⇥

sinc(u)sinc (u� (k2

+ r2

+ j4

+ k1

+ j1

)⇡) sinc(v)sinc (v + (k2

� j3

+ k1

+ r1

� j2

)⇡) dudv.

Again by using the same proof as that given in Lemma A.2 to approximate the terms inside the

sumP

j1,...,j4(setting m = k

1

+ k2

and replacing !k1 with !), we can approximate A2

(r1

, r2

) with

A2

(r1

, r2

)

=1

2⇡3

X

j1,j2,j3,j42Z�j1�j2�j3�j4

2aX

m=�2a

Z2⇡min(a,a+m)/�

2⇡(max(�a,�a+m)/�g(!)g(�! + !m)

Z

R2f(�! � !j1)f(! + !r1�j2)⇥

sinc(u)sinc (u� (m+ r2

+ j4

+ j1

)⇡) sinc(v)sinc (v + (m� j3

+ r1

� j2

)⇡) dudvd! +O(`�,a,n).

Using the orthogonality of the sinc function, the inner integral is non-zero when m�j3

+r1

�j2

= 0

and m+ r2

+ j4

+ j1

= 0. Setting m = �r1

+ j2

+ j3

, this implies

A2

(r1

, r2

)

=1

2⇡

X

j1+j2+j3+j4=r1�r2

�j1�j2�j3�j4

Z2⇡min(a,a�r1+j2+j3)/�

2⇡max(�a,�a�r1+j2+j3)/�g(!)g(�! � !r1�j3�j2)

⇥f(! + !j1)f(! + !r1�j2)d! +O(`�,a,n),

thus giving us U1,2(r1, r2;!r1 ,!r2).

By using Lemma D.1, we can show that


i= A

3

(r1

, r2

) +A4

(r1

, r2

) +O

✓�

n

◆

where

A3

(r1

, r2

) = �aX

k1,k2=�a


1

) exp(is1

!k1), Z(s3

) exp(�is3

!k2)⇤⇥

cov⇥Z(s

2

) exp(�is2

!k1+r1), Z(s4

) exp(is4

!k2+r2)⇤

A4

(r1

, r2

) = �aX

k1,k2=�a


1

) exp(is1

!k1), Z(s4

) exp(is4

!k2+r2)⇤⇥

cov⇥Z(s

2

) exp(�is2

!k1+r1), Z(s3

) exp(�is3

!k2)⇤.

43

By following the same proof as that used to prove A1

(r1

, r2

) we have

A3

(r1

, r2

)

=�

(2⇡)2

1X

j1,...,j4=�1�j1�j2�j3�j4

aX

k1,k2=�a

g(!k1)g(!k2)

Z 1

�1

Z 1

�1f(x)f(y)sinc

✓�x

2+ (k

1

+ j1

)⇡

◆

sinc

✓�x

2� (k

2

+ j3

)⇡

◆sinc

✓�y

2� (r

1

+ k1

� j2

)⇡

◆sinc

✓�y

2+ (r

2

+ k2

� j4

)⇡

◆dxdy

=1

⇡2�

1X

j1,...,j4=�1�j1�j2�j3�j4

aX

k1,k2=�a

g(!k1)g(!k2)

Z 1

�1

Z 1

�1f(

2u

�� !k1+j1)f(

2v

�+ !k1+r1�j2)

⇥sinc(u)sinc(u� (k2

+ k1

+ j1

+ j3

)⇡)sinc(v)sinc(v + (r2

+ k2

� j4

+ r1

+ k1

� j2

)⇡)dudv

=1

⇡2�

1X

j1,...,j4=�1�j1�j2�j3�j4

2aX

m=�2a

min(a,a+m)X

k1=max(�a,�a+m)

g(!k1)g(!m�k1)

Z 1

�1

Z 1

�1f(

2u

�� !k1+j1)f(

2v

�+ !k1+r1�j2)

⇥sinc(u)sinc(u� (m+ j1

+ j3

)⇡)sinc(v)sinc(v + (r2

+m� j4

+ r1

� j2

)⇡)dudv.

Again using the method used to bound A1

(r1

, r2

) gives

A3

(r1

, r2

) =1

2⇡3

X

j1+j2+j3+j4=r1+r2

�j1�j2�j3�j4X

m

Z2⇡min(a,a�j1�j3)/�

2⇡max(�a,�a�j1�j3)/�g(!)g(!�j1�j3 � !)⇥

f(! + !j1)f(! + !r1�j2)d! +O(`�,a,n) = U1,2(r1, r2;!r1 ,!r2) +O(`�,a,n).

Finally we consider A4

(r1

, r2

). Using the same expansion as the above we have

A4

(r1

, r2

)

=1

�3

X

j1,j2,j3,j42Z�j1�j2�j3�j4

aX

k1,k2=�a

g(!k1)g(!k2)

Z

[��/2,�/2]4c(s

1

� s4

)c(s2

� s3

) exp(is1

!k1+j1)

exp(�is4

!k2+r2�j4) exp(�is2

!k1+r1�j2) exp(is3!k2+j3)ds1ds2ds3ds4

= �X

j1,j2,j3,j42Z

aX

k1,k2=�a

g(!k1)g(!k2)

Z

R2f(x)f(y)sinc

✓�x

2+ (k

1

+ j1

)⇡

◆⇥

sinc

✓�x

2+ (k

2

+ r2

� j4

)⇡

◆sinc

✓�y

2� (k

1

+ r1

� j2

)⇡

◆sinc

✓�y

2� (k

2

+ j3

)⇡

◆dxdy

=1

⇡2�

X

j1,j2,j3,j42Z�j1�j2�j3�j4

aX

k1,k2=�a

g(!k1)g(!k2)

Z

R2f(

2u

�� !k1+j1)f(

2v

�+ !k1+r1�j2)⇥

sinc(u)sinc (u+ (k2

+ r2

� j4

� k1

� j1

)⇡) sinc(v)sinc (v � (k2

+ j3

� k1

� r1

+ j2

)⇡) dudv

=1

2⇡3

X

j1,j2,j3,j42Z�j1�j2�j3�j4

2aX

m=�2a

Z2⇡min(a,a+m)

2⇡max(�a,�a+m)/�g(!)g(! � !m)

Z

R2f(�! � !j1)f(! + !r1�j2)⇥

sinc(u)sinc (u+ (r2

� j4

�m� j1

)⇡) sinc(v)sinc (v � (j3

�m� r1

+ j2

)⇡) dudvd! +O(`�,a,n)

=1

2⇡

X

j1+j2+j3+j4=r1+r2

�j1�j2�j3�j4

Z2⇡min(a,a�r1+j2+j3)/�

max(�a,a�r1+j2+j3)/�g(!)g(! � !j2+j3�r2)

⇥f(! + !j1)f(! + !r1�j2)d! +O(`�,a,n).

44

This gives the desired result. ⇤

We now obtain and approximation of U1

and U2

.

PROOF of Corollary 3.1 By using Lipschitz continuity of g(·) and f(·) and |�j | CQd

i=1

|ji|�(1+�)I(ji 6=0) we obtain the result. ⇤

PROOF of Theorem 3.6We prove the result for the case d = 1 and usingA1

(r1

, r2

), . . . , A4

(r1

, r2

)

defined in proof of Theorem 3.3. The proof is identical to the proof of Theorem A.2. Following the

same notation in proof of Theorem A.2 we have


i= A

1

(r1

, r2

) +A2

(r1

, r2

) +B1

(r1

, r2

) +B2

(r1

, r2

) +O

✓�

n

◆,

with |B2

(r1

, r2

)| = O((a�)2/n2) and the main term involving the trispectral density is

B1

(r1

, r2

)

= �c4

aX

k1,k2=�a

g(!k1)g(!k2)E

4

(s1

� s2

, s1

� s3

, s1

� s4

)eis1!k1e�is2!k1+r1e�is3!k2eis4!k2+r2

�

=c4

(2⇡)3�3

1X

j1,...,j4=�1�j1�j2�j3�j4

aX

k1,k2=�a

g(!k1)g(!k2)

Z

R3f4

(!1

,!2

,!3

)

Z

[��/2,�/2]4eis1(!1+!2+!3+!k1+j1

)

e�is2(!1+!k1+r1�j2)e�is3(!2+!k2�j3

)eis4(�!3+!k2+r2+j4)ds

1

ds2

ds3

ds4

d!1

d!2

d!3

=c4

�

(2⇡)3

1X

j1,...,j4=�1�j1�j2�j3�j4

aX

k1,k2=�a

g(!k1)g(!k2)

Z

R3f4

(!1

,!2

,!3

)sinc

✓�(!

1

+ !2

+ !3

)

2+ (k

1

+ j1

)⇡

◆

⇥sinc

✓�!

1

2+ (k

1

+ r1

� j2

)⇡

◆sinc

✓�!

2

2+ (k

2

� j3

)⇡

◆sinc

✓�!

3

2� (k

2

+ r2

+ j4

)⇡

◆d!

1

d!2

d!3

.

Now we make a change of variables and let u1

= �!12

+ (k1

+ r1

� j2

), u2

= �!22

+ (k2

� j3

)⇡ and

u3

= �!32

� (k2

+ r2

+ j4

)⇡, this gives

B1

(r1

, r2

)

=c4

⇡3�2

1X

j1,...,j4=�1�j1�j2�j3�j4

aX

k1,k2=�a

Z

R3g(!k1)g(!k2)f4

✓2u

1

�� !k1+r1�j2 ,

2u2

�� !k2�j3 ,

2u3

�+ !k2+r2+j4

◆

⇥sinc (u1

+ u2

+ u3

+ (r2

� r1

+ j1

+ j2

+ j3

+ j4

)⇡) sinc(u1

)sinc(u2

)sinc(u3

)du1

du2

du3

.

Next we exchange the summand with a double integral and use Lemma E.1(iii) together with

Lemma C.1, equation (44) to obtain

B1

(r1

, r2

)

=c4

⇡3(2⇡)2

1X

j1,...,j4=�1�j1�j2�j3�j4

Z

2⇡[�a/�,a/�]2g(!

1

)g(!2

)⇥

Z

R3f4

✓2u

1

�� !

1

� !r1�j2 ,2u

2

�� !

2

� !j3 ,2u

3

�+ !

2

+ !r2+j4

◆⇥

sinc (u1

+ u2

+ u3

+ (r2

� r1

+ j1

+ j2

+ j3

+ j4

)⇡) sinc(u1

)sinc(u2

)sinc(u3

)du1

du2

du3

d!1

d!2

+O

✓1

�

◆.

45

By using Lemma C.3, we replace f4

�2u1� � !

1

� !r1�j2 ,2u2� � !

2

� !j3 ,2u3� + !

2

+ !r2+j4

�with

f4

(�!1

� !r1�j2 ,�!2

� !j3 ,!2

+ !r2+j4), to give

B1

(r1

, r2

)

=c4

⇡3(2⇡)2

1X

j1,...,j4=�1�j1�j2�j3�j4

Z

2⇡[�a/�,a/�]2g(!

1

)g(!2

)⇥

Z

R3f4

(�!1

� !r1�j2 ,�!2

� !j3 ,!2

+ !r2+j4) sinc (u1 + u2

+ u3

+ (r2

� r1

+ j1

+ j2

+ j3

+ j4

)⇡)

⇥sinc(u1

)sinc(u2

)sinc(u3

)du1

du2

du3

d!1

d!2

+O

✓log3(�)

�

◆

=c4

(2⇡)2

X

j1+j2+j3+j4=r2�r1

�j1�j2�j3�j4

Z

2⇡[�a/�,a/�]2g(!

1

)g(!2

)⇥

f4

(�!1

� !r1�j2 ,�!2

� !j3 ,!2

+ !r2+j4) d!1

d!2

+O

✓log3(�)

�

◆,

where the last line follows from (39).

To obtain an expression for �covheQa,�(g; r1), eQa,�(g; r2)

iwe note that


i= A

3

(r1

, r2

) +A4

(r1

, r2

) +B3

(r1

, r2

) +B4

(r1

, r2

) +O

✓�

n

◆,

just as in the proof of Theorem A.2 we can show that |B4

(r1

, r2

)| = O((�a)d/n2) and the leading

term involving the trispectral density is

B3

(r1

, r2

)

= �c4

aX

k1,k2=�a

g(!k1)g(!k2)E

4

(s1

� s2

, s1

� s3

, s1

� s4

)eis1!k1e�is2!k1+r1eis3!k2e�is4!k2+r2

�

=c4

(2⇡)3�3

1X

j1,...,j4=�1�j1�j2�j3�j4

aX

k1,k2=�a

g(!k1)g(!k2)

Z

R3f4

(!1

,!2

,!3

)

Z

[��/2,�/2]4eis1(!1+!2+!3+!k1+j1

)

e�is2(!1+!k1+r1�j2)e�is3(!2�!k2+j3

)eis4(�!3�!k2+r2�j4)ds

1

ds2

ds3

ds4

d!1

d!2

d!3

=c4

�

(2⇡)3

1X

j1,...,j4=�1�j1�j2�j3�j4

aX

k1,k2=�a

g(!k1)g(!k2)

Z

R3f4

(!1

,!2

,!3

)sinc

✓�(!

1

+ !2

+ !3

)

2+ (k

1

+ j1

)⇡

◆

⇥sinc

✓�!

1

2+ (k

1

+ r1

� j2

)⇡

◆sinc

✓�!

2

2� (k

2

+ j3

)⇡

◆sinc

✓�!

3

2+ (k

2

+ r2

� j4

)⇡

◆d!

1

d!2

d!3

.

We make a change of variables u1

= �!12

+ (k1

+ r1

� j2

)⇡ u2

= �!22

� (k2

+ j3

)⇡ and u3

=�!32

+ (k2

+ r2

� j4

)⇡. This gives

B3

(r1

, r2

)

=c4

�2(2⇡)3

1X

j1,...,j4=�1�j1�j2�j3�j4

aX

k1,k2=�a

g(!k1)g(!k2)

Z

R3f4

(2u

1

�� !k1+r1�j2 ,

2u2

�+ !k2+j3 ,

2u3

�� !k2�r2+j4)

⇥sinc (u1

+ u2

+ u3

+ (j1

+ j2

+ j3

+ j4

� r1

� r2

)⇡) sinc(u1

)sinc(u2

)sinc(u3

)du1

du2

du3

=1

(2⇡)2

X

j1+j2+j3+j4=r1+r2

�j1�j2�j3�j4

Z

2⇡[�a/�,a/�]2g(!

1

)g(!2

)f4

(�!1

� !r1�j2 ,!2

+ !j3 ,�!2

� !�r2+j4)d!1

d!2

.

46

Finally, by replacing f4

(�!1

� !r1�j2 ,�!2

� !j3 ,!2

+ !r2+j4) with f4

(�!1

,�!2

,!2

) inB1

(r1

, r2

)

and f4

(�!1

�!r1�j2 ,!2

+!j3 ,�!2

�!�r2+j4) with f4

(�!1

,!2

,�!2

) and using the pointwise Lips-

chitz continuity of f4

and that |�j | CI(j 6= 0)|j|�(1+�) we obtain B1

(r1

, r2

) = D1

+O⇣log

3(�)

� + 1

�

⌘

and B3

(r1

, r2

) = D3

+O⇣log

3(�)

� + 1

�

⌘. Thus giving the required result. ⇤

C Technical Lemmas

We first prove Lemma A.1, then state four lemmas, which form an important component in the

proofs of this paper. Through out this section we use C to denote a finite generic constant. It is

worth mentioning that many of these results build on the work of T. Kawata (see Kawata (1959)).

PROOF of Lemma A.1 We first prove (30). By using partial fractions and the definition of

the sinc function we haveZ 1

�1sinc(u)sinc(u+ x)du =

1

x

Z 1

�1sin(u) sin(u+ x)

✓1

u� 1

u+ x

◆du

=1

x

Z 1

�1

sin(u) sin(u+ x)

udu�

Z 1

�1

sin(u) sin(u+ x)

u+ xdu.

For the second integral we make a change of variables u0 = u+ x, this gives

Z 1

�1sinc(u)sinc(u+ x)du =

1

x

Z 1

�1

sin(u) sin(u+ x)

udu� 1

x

Z 1

�1

sin(u0) sin(u0 � x)

u0du0

=1

x

Z 1

�1

sin(u)

u

✓sin(u+ x)� sin(u� x)

◆du

=2 sin(x)

x

Z 1

�1

cos(u) sin(u)

udu =

⇡ sin(x)

x.

To prove (31), it is clear that for x = s⇡ (with s 2 Z{0}) ⇡ sin(s⇡)s⇡ = 0, which gives the result. ⇤

The following result is used to obtain bounds for the variance and higher order cumulants.

Lemma C.1 Define the function `p(x) = C/e for |x| e and `p(x) = C logp |x|/|x| for |x| � e.

(i) We have

Z 1

�1

| sin(x) sin(x+ y)||x(x+ y)| dx

(C log |y|

|y| |y| � e

C |y| < e= `

1

(y), (42)

Z 1

�1|sinc(x)|`p(x+ y)dx `p+1

(y) (43)

and

Z

Rp

��sinc

0

@pX

j=1

xj

1

ApY

j=1

sinc(xj)

��dx

1

. . . dxp C, (44)

47

(ii)

aX

m=�a

Z 1

�1

��sin2(x)

x(x+m⇡)

�� dx C log2 a

(iii)

Z 1

�1

Z 1

�1

aX

m=�a

sin2(x)

|x(x+m⇡)|sin2(y)

|y(y +m⇡)|dxdy C, (45)

(iv)

aX

m1,...,mq�1=�a

Z

Rq

��q�1Y

j=1

sinc(xj)sinc(xj +mj⇡)⇥

sinc(xq)sinc(xq + ⇡q�1X

j=1

mj)

��qY

j=1

dxj C log2(q�2)(a),

where C is a finite generic constant which is independent of a.

PROOF. We first prove (i), equation (42). It is clear that for |y| e thatR1�1

| sin(x) sin(x+y)||x(x+y)| dx C.

Therefore we now consider the case |y| > e, without loss of generality we prove the result for y > e.

Partitioning the integral we have

Z 1

�1

| sin(x) sin(x+ y)||x(x+ y)| dx = I

1

+ I2

+ I3

+ I4

+ I5

,

where

I1

=

Z y

0

| sin(x) sin(x+ y)||x(x+ y)| dx I

2

=

Z0

�y


I3

=

Z �y

�2y

| sin(x) sin(x+ y)||x(x+ y)| dx I

4

=

Z �2y

�1


I5

=

Z 1

y

| sin(x) sin(x+ y)||x(x+ y)| dx.

To bound I1

we note that for y > 1 and x > 0 that | sin(x+ y)/(x+ y)| 1/y, thus

I1

=1

y

Z y

0

| sin(x)||x| dx C

log y

y.

To bound I2

, we further partition the integral

I2

=

Z �y/2

�y

| sin(x) sin(x+ y)||x(x+ y)| dx+

Z0

�y/2


2

y

Z �y/2

�y

| sin(x+ y)||(x+ y)| dx+

2

y

Z0

�y/2

| sin(x)||x| dx C

log y

y.

48

To bound I3

, we use the bound

I3

1

y

Z �y

�2y

| sin(x+ y)||(x+ y)| dx C

log y

y.

To bound I4

we use that for y > 0,R1y x�2dx C|y|�1, thus

I4

Z �y

�1

1

x2dx C|y|�1

and using a similar argument we have I5

C|y|�1. Altogether, this gives (42).

We now prove (43). It is clear that for |y| e thatR1�1 |sinc(x)|`p(x + y)|dx C. Therefore

we now consider the case |y| > e, without loss of generality we prove the result for y > e. As in

(42) we partition the integralZ 1

�1|sinc(x)`p(x+ y)|dx = II

1

+ . . .+ II5

,

where II1

, . . . , II5

are defined in the same way as I1

, . . . , I5

just with |sinc(x)`p(x + y)| replacing| sin(x) sin(x+y)|

|x(x+y)| . To bound II1

we note that

II1

=

Z y

0

|sinc(x)`p(x+ y)|dx logp(y)

y

Z y

0

|sinc(x)|dx logp+1(y)

y,

we use similar method to show II2

C log

p+1(y)

y and II3

C log

p+1(y)

y . Finally to bound II4

and

II5

we note that by using a change of variables x = yz, we have

II5

=

Z 1

y

| sin(x)| logp(x+ y)

x(x+ y)dx

Z 1

y

logp(x+ y)

x(x+ y)dx =

1

y

Z 1

1

[log(y) + log(z + 1)]p

z(z + 1)dz C

logp(y)

y.

Similarly we can show that II4

C log

p(y)

y . Altogether, this gives the result.

To prove (44) we recursively apply (43) to give

Z

Rp|sinc(x

1

+ . . .+ xp)|pY

j=1

|sinc(xj)|dx1 . . . dxp Z

Rp�1|`1

(x1

+ . . .+ xp�1

)p�1Y

j=1

sinc(xj)|dx1 . . . dxp�1

Z

R|`p�1

(x1

)sinc(x1

)|dx1

= O(1),

thus we have the required the result.

To bound (ii), without loss of generality we derive a bound overPa

m=1

, the bounds forPm

�a is

identical. Using (42) we have

aX

m=1

Z 1

�1

sin2(x)

|x(x+m⇡)|dx =aX

m=1

Z 1

�1

| sin(x) sin(x+m⇡)||x(x+m⇡)| dx

aX

m=1

`1

(m⇡) = CaX

m=1

log(m⇡)

m⇡

C log(a⇡)aX

m=1

1

m⇡= C log(a⇡) log(a) C log2 a.

49

Thus we have shown (ii).

To prove (iii) we use (42) to give

aX

m=�a

✓Z 1

�1

sin2(x)

|x(x+m⇡)|dx◆✓Z 1

�1

sin2(y)

|y(y +m⇡)|dy◆

CaX

m=�a

� logmm

�2 C

1X

m=�1

� logmm

�2 C.

To prove (iv) we apply (42) to each of the integrals this gives

aX

m1,...,mq�1=�a

Z

Rq

��q�1Y

j=1

sinc(xj)sinc(xj +mj⇡)sinc(xq)sinc(xq + ⇡q�1X

j=1

mj)

��qY

j=1

dxj

aX

m1,...,mq�1=�a

`1

(m1

⇡) . . . `1

(mq�1

⇡)`1

(mq�1

⇡)`1

(⇡q�1X

j=1

mj) C log2(q�2) a,

thus we obtain the desired result. ⇤.

The proofs of Theorem 3.1, Theorems A.1 (ii), Theorem 2.2 involve integrals of sinc(u)sinc(u+

m⇡), where |m| a + |r1

| + |r2

| and that a ! 1 as � ! 1. In the following lemma we obtain

bounds for these integrals. We note that all these results involve an inner integral di↵erence of the

formZ b

�bg(!)

h

✓! +

2u

�

◆� h(!)

�d!du

��.

By using the mean value theorem heuristically it is clear that O(��1) should come out of the

integral however the 2u in the numerator of h�! + 2u

�

�makes the analysis quite involved.

Lemma C.2 Suppose h is a function which is absolutely integrable and |h0(!)| �(!) (where � is

a montonically decreasing function that is absolutely integrable) and m 2 Z and g(!) is a bounded

function. b can take any value, and the bounds given below are independent of b. Then we have��Z 1

�1sinc(u)sinc(u+m⇡)

Z b

�bg(!)

h

✓! +

2u

�

◆� h(!)

�d!du

��

Clog(�) + I(m 6= 0) log(|m|)

�, (46)

where C is a finite constant independent of m and b. If g(!) is a bounded function with a bounded

first derivative, then we have��Z 1


Z b

�bh(!)

✓g(! +

2u

�)� g(!)

◆d!du

��

Clog(�) + I(m 6= 0) log(|m|)

�. (47)

In the case of double integrals, we assume h(·, ·) is such thatRR2 |h(!1

,!2

)|d!1

d!2

< 1, and

|@h(!1,!2)

@!1| �(!

1

)�(!2

), |@h(!1,!2)

@!2| �(!

1

)�(!2

) and |@2h(!1,!2)

@!1@!2| �(!

1

)�(!2

) (where � is a

50

montonically decreasing function that is absolutely integrable) and g(·, ·) is a bounded function.

Then if m1

6= 0 and m2

6= 0 we have��Z 1

�1

Z 1

�1sinc(u

1

)sinc(u1

+m1

⇡)sinc(u2

)sinc(u2

+m2

⇡)

Z b

�b

Z b

�bg(!

1

,!2

)⇥h

✓!1

+2u

1

�,!

2

+2u

2

�

◆� h(!

1

,!2

)

�d!

1

d!2

du1

du2

�� C

Q2

i=1

[log(�) + log(|mi|)]�2

. (48)

and��Z 1

�1

Z 1

�1sinc(u

1

)sinc(u1

+m1

⇡)sinc(u2

)sinc(u2

+m2

⇡)1

�2

b�X

�b�

b�X

�b�

g(!k1 ,!k2)⇥

h

✓!k1 +

2u1

�,!k2 +

2u2

�

◆� h(!k1 ,!k2)

�du

1

du2

�� C

Q2

i=1

[log(�) + log(|mi|)]�2

. (49)

where a ! 1 as � ! 1.

PROOF. To simplify the notation in the proof, we’ll prove (46) for m > 0 (the proof for m 0 is

identical).

The proof is based on considering the cases that |u| � and |u| > � separately. For |u| �

we apply the mean value theorem to the di↵erence h(! + 2u� ) � h(!) and for |u| > � we exploit

that the integralRu>|�| |sinc(u)sinc(u +m⇡)|du decays as � ! 1. We now make these argument

precise. We start by partitioning the integralZ 1


Z b

�bg(!)

✓h(! +

2u

�)� h(!)

◆d!du = I

1

+ I2

, (50)

where

I1

=

Z

|u|>�sinc(u)sinc(u+m⇡)

Z b

�bg(!)

✓h(! +

2u

�)� h(!)

◆d!du

I2

=

Z �

��sinc(u)sinc(u+m⇡)

Z b

�bg(!)

✓h(! +

2u

�)� h(!)

◆d!du.

We further partition the integral I1

= I11

+ I12

+ I13

, where

I11

=

Z 1

�sinc(u)sinc(u+m⇡)

Z b

�bg(!)

✓h(! +

2u

�)� h(!)

◆d!du

I12

=

Z ��

��m⇡sinc(u)sinc(u+m⇡)

Z b

�bg(!)

✓h(! +

2u

�)� h(!)

◆d!du

I13

=

Z ��m⇡


Z b

�bg(!)

✓h(! +

2u

�)� h(!)

◆d!du

and partition I2

= I21

+ I22

+ I23

, where

I21

=

Z �


Zmin(4|u|/�,b)

�min(4|u|/�,b)g(!)

✓h(! +

2u

�)� h(!)

◆d!du

I22

=

Z �


Z �min(4|u|/�,b)

�bg(!)

✓h(! +

2u

�)� h(!)

◆d!du

I23

=

Z �


Z b

min(4|u|/�,b)g(!)

✓h(! +

2u

�)� h(!)

◆d!du.

51

We start by bounding I1

. Taking absolutes of I11

, and using that h(!) is absolutely integrable we

have

|I11

| 2�

Z 1

�

sin2(u)

u(u+m⇡)du,

where � = supu |g(u)|R10

|h(u)|du. Sincem > 0, it is straightforward to show thatR1�

sin

2(u)

u(u+m⇡)du C��1, where C is some finite constant. This implies |I

11

| 2C��1. Similarly it can be shown

that |I13

| 2C��1. To bound I12

we note that

|I12

| 2�

�

Z ��

��m⇡

sin2(u)

|u+m⇡|du 2�

�⇥(

log⇣

��m⇡

⌘m⇡ < �

log �+ log(m⇡ � �) m⇡ > �.

Thus, we have |I12

| 2��1

⇥log �+ logm

⇤. Altogether, the bounds for I

11

, I12

, I13

give

|I1

| C(log �+ logm)

�.

To bound I2

we apply the mean value theorem to h(! + 2u� )� h(!) = 2u

� h0(! + ⇣(!, u)2u� ), where

0 ⇣(!, u)| 1. Substituting this into I23

gives

|I23

| 2

�

Z �

��

sin2(u)

|u+m⇡|

Z b

min(4|u|/�,b)

��h0(! + ⇣(!, u)

2u

�)

�� d!du.

Since the limits of the inner integral are greater than 4u/�, and the derivative is bounded by �(!),

this means |h0(! + ⇣(!, u)2u� )| max[�(!),�(! + 2u� )] = �(!). Altogether, this gives

|I23

| 2

�

Z b

min(4|u|/�,b)�(!)d!

!Z �

��

sin2(u)

|u+m⇡|du 2� log(�+m⇡)

�.

Using the same method we obtain I22

2� log(�+m⇡)� . Finally, to bound I

21

, we cannot bound

h0(! + ⇣(!, u)2u� ) by a monotonic function since ! and ! + 2u� can have di↵erent signs. Therefore

we simply bound h0(! + ⇣(!, u)2u� ) with a constant, this gives

|I21

| 8C

�2

Z �

��

|u| sin2(u)|u+m⇡| du 16C

�.

Altogether, the bounds for I21

, I22

, I23

give

|I2

| C�log �+ log(m) + log(�+m⇡)

�.

Finally, we recall that if � > 2 and m⇡ > 2, then (� + m⇡) < �m⇡, thus log(� + m⇡) log � +

log(m⇡), Therefore, we obtain (46). The proof of (47) is similar, but avoids some of the awkward

details that are required to prove (46).

To prove (48) we note that both m1

6= 0 and m2

6= 0 (if one of these values were zero the slower

bound given in (46) only holds). Without loss of generality we will prove the result for m1

> 0 and

52

m2

> 0. We first note since m1

6= 0 and m2

6= 0, by orthogonality of the sinc function at integer

shifts (see Lemma A.1) we have

Z 1

�1

Z 1

�1sinc(u

1

)sinc(u1

+m1

⇡)sinc(u2

)sinc(u2

+m2

⇡)

Z b

�b

Z b

�bg(!

1

,!2

)⇥h

✓!1

+2u

1

�,!

2

+2u

2

�

◆� h(!

1

,!2

)

�d!

1

d!2

du1

du2

=

Z 1

�1sinc(u

2

)sinc(u2

+m2

⇡)

Z 1

�1sinc(u

1

)sinc(u1

+m1

⇡)

Z b

�b

Z b

�bg(!

1

,!2

)

⇥h

✓!1

+2u

1

�,!

2

+2u

2

�

◆d!

1

du1

d!2

du2

, (51)

since in the last line h(!1

,!2

) comes outside the integral over u1

and u2

. To further the proof, we

note that for m2

6= 0 we have

Z 1

�1sinc(u

2

)sinc(u2

+m2

⇡)

Z

|u1|>�sinc(u

1

)sinc(u1

+m1

⇡)

Z b

�b

Z b

�bg(!

1

,!2

)

⇥h

✓!1

+2u

1

�,!

2

◆d!

1

du1

d!2

du2

= 0. (52)

We use this zero-equality at relevant parts in the proof.

We subtract (52) from (51), and use the same decomposition and notation used in (50) we

decompose the integral in (51) over u1

intoR|u1|>�+

R|u1|�, to give

=

Z 1

�1sinc(u

2

)sinc(u2

+m2

⇡)

✓Z 1

�1sinc(u

1

)sinc(u1

+m1

⇡)

Z b

�b

Z b

�bg(!

1

,!2

)⇥h

✓!1

+2u

1

�,!

2

+2u

2

�

◆� h

✓!1

+2u

1

�,!

2

◆�d!

1

du1

◆d!

2

du2

=

Z 1

�1sinc(u

2

)sinc(u2

+m2

⇡)

Z b

�b

✓Z 1

�1sinc(u

1

)sinc(u1

+m1

⇡)

Z b

�bg(!

1

,!2

)⇥h

✓!1

+2u

1

�,!

2

+2u

2

�

◆� h

✓!1

+2u

1

�,!

2

◆�d!

1

du1

◆d!

2

du2

=

Z 1

�1sinc(u

2

)sinc(u2

+m2

⇡)

Z b

�b[I1

(u2

,!2

) + I2

(u2

,!2

)] du2

d!2

= J1

+ J2

,

where

I1

(u2

,!2

) =

Z

|u1|>�sinc(u

1

)sinc(u1

+m1

⇡)⇥Z b

�bg(!

1

,!2

)

h

✓!1

+2u

1

�,!

2

+2u

2

�

◆� h

✓!1

+2u

1

�,!

2

◆�d!

1

du1

I2

(u2

,!2

) =

Z

|u1|�sinc(u

1

)sinc(u1

+m1

⇡)

Z b

�bg(!

1

,!2

)@h

@!1

✓!1

+ ⇣1

(!1

, u1

)2u

1

�,!

2

+2u

2

�

◆d!

1

du1

and |⇣1

(!1

, u1

)| 1. Note that the expression for I2

(u2

,!2

) applies the mean value theorem to

the di↵erence h�!1

+ 2u1� ,!

2

+ 2u2�

�� h

�!1

+ 2u1� ,!

2

�. Next to bound J

1

we decompose the outer

53

integral over u2

intoR|u2|>� and

R|u2|� to give J

1

= J11

+ J12

, where

J11

=

Z

|u1|>�

Z

|u2|>�sinc(u

1

)sinc(u1

+m1

⇡)sinc(u2

)sinc(u2

+m2

⇡)⇥Z b

�b

Z b

�bg(!

1

,!2

)

h

✓!1

+2u

1

�,!

2

+2u

2

�

◆� h

✓!1

+2u

1

�,!

2

◆�d!

1

du1

du2

d!2

J12

=

Z

|u1|>�

Z b

�bsinc(u

1

)sinc(u1

+m1

⇡)⇥Z

|u2|�sinc(u

2

)sinc(u2

+m2

⇡)

Z b

�bg(!

1

,!2

)@h

@!2

✓!1

+2u

1

�,!

2

+ ⇣2

(!2

, u2

)2u

2

�

◆d!

1

du1

du2

d!2

,

and |⇣2

(!2

, u2

)| < 1. Note that the expression for J12

is obtained by applying the mean value

theorem to h�!1

+ 2u1� ,!

2

+ 2u2�

�� h

�!1

+ 2u1� ,!

2

�. Now by using the same methods used to

bound I1

and I2

in (50) we can show that |J11

| = O(��2), |J12

| = O([log(�) + log |m2

|]/�2) and

|J21

| = O(��2). To bound J2

we again use that m2

6= 0 and orthogonality of the sinc function to

add the additional term @h@!1

�!1

+ ⇣1

(!1

, u1

)2u1� ,!

2

�(whose total contribution is zero) to J

2

to give

J2

=

Z 1

�1

Z b

�bsinc(u

2

)sinc(u2

+m2

⇡)

Z

|u1|�sinc(u

1

)sinc(u1

+m1

⇡)⇥Z b

�bg(!

1

,!2

)

@h

@!1

✓!1

+ ⇣1

(!1

, u1

)2u

1

�,!

2

+2u

2

�

◆� @h

@!1

✓!1

+ ⇣1

(!1

, u1

)2u

1

�,!

2

◆�d!

1

d!2

du1

du2

.

By decomposing the outer integral of J2

over u2

intoR|u2|>� and

R|u2|� we have J

2

= J21

+ J22

,

where

J21

=

Z

|u2|>�

Z b

�bsinc(u

2

)sinc(u2

+m2

⇡)

Z

|u1|�sinc(u

1

)sinc(u1

+m1

⇡)⇥Z b

�bg(!

1

,!2

)

@h

@!1

✓!1

+ ⇣1

(!1

, u1

)2u

1

�,!

2

+2u

2

�

◆� @h

@!1

✓!1

+ ⇣1

(!1

, u1

)2u

1

�,!

2

◆�d!

1

d!2

du1

du2

and

J22

=

Z

|u2|�

Z b

�bsinc(u

2

)sinc(u2

+m2

⇡)

Z

|u1|�sinc(u

1

)sinc(u1

+m1

⇡)⇥Z b

�bg(!

1

,!2

)@2h

@!1

@!2

✓!1

+ ⇣1

(!1

, u1

)2u

1

�,!

2

+ ⇣3

(!2

, u2

)2u

2

�

◆d!

1

d!2

du1

du2

with |⇣3

(!2

, u2

)| < 1. Note that the expression for J22

is obtained by applying the mean value

theorem to @h@!1

�!1

+ ⇣1

(!1

, u1

)2u1� ,!

2

+ 2u2�

�� @h

@!1

�!1

+ ⇣1

(!1

, u1

)2u1� ,!

2

�.

Again by using the methods used to bound I1

and I2

in (50) we can show that |J21

| = O([log(�)+

log |m1

|]/�2) and |J22

| = O(Q

2

i=1

[log(�) + log |mi|]/�2). Altogether this proves (48).

The proof of (49) follows exactly the same method used to prove (48) the only di↵erence is that

the summand rather than the integral makes the notation more cumbersome. For this reason we

omit the details. ⇤

54

The following result is used in to obtain expression for the fourth order cumulant term (in the

case that the spatial random field is not Gaussian). It is used in the proofs of Theorems 3.6 and

A.2.

Lemma C.3 Suppose h is a function which is absolutely integrable and |h0(!)| �(!) (where �

is a monotonically decreasing function that is absolutely integrable), m 2 Z and g(!) is a bounded

function. Then we have

Z

R3sinc(u

1

+ u2

+ u3

+m⇡)sinc(u1

)sinc(u2

)sinc(u3

)

Z a/�

�a/�g(!)

h

✓2u

1

�� !

◆� h(�!)

�d!du

1

du2

du3

= O

✓[log(�)|+ log |m|]3

�

◆. (53)

Z

R2sinc(u

1

+ u2

+m⇡)sinc(u1

)sinc(u2

)

Z a/�

�a/�g(!)

h

✓2u

1

�� !

◆� h(�!)

�d!du

1

du2

= O

✓[log(�) + log |m|]3

�

◆. (54)

PROOF. The proof of (53) is very similar to the proof of Lemma C.2. We start by partitioning the

integral over u1

intoR|u1|� and

R|u1|>�

Z

R3sinc(u

1

+ u2

+ u3

+m⇡)sinc(u1

)sinc(u2

)sinc(u3

)⇥Z a/�

�a/�g(!)

✓h(

2u1

�� !)� h(�!)

◆d!du

1

du2

du3

= I1

+ I2

,

where

I1

=Z

|u1|>�

Z

R3sinc(u

1

+ u2

+ u3

+m⇡)sinc(u1

)sinc(u2

)sinc(u3

)

⇥Z a/�

�a/�g(!)

✓h(

2u1

�� !)� h(�!)

◆d!du

1

du2

du3

I2

=Z

|u1|�

Z

R3sinc(u

1

+ u2

+ u3

+m⇡)sinc(u1

)sinc(u2

)sinc(u3

)

⇥Z a/�

�a/�g(!)

✓h(

2u1

�� !)� h(�!)

◆d!du

1

du2

du3

.

Taking absolutes of I1

and using Lemma C.1, equations (42) and (43) we have

|I1

| 4�

Z

|u1|>�|sinc(u

1

+m⇡)|`2

(u1

)du1

C

Z

|u|>�

log2(u)

|u| ⇥ |sinc(u+m⇡)||u+m⇡| du,

55

where � = sup! |g(!)|R10

|h(!)|d! and C is a finite constant (which has absorbed �). Decomposing

the above integral we have

|I1

| C

Z

|u|>�

log2(u)

|u| ⇥ |sinc(u+m⇡)||u+m⇡| du = I

11

+ I12

,

where

I11

=

Z

�<|u|�(1+|m|)

log2(u)

|u| ⇥ |sinc(u1

+m⇡)||u

1

+m⇡| du

I12

=

Z

|u|>�(1+|m|)

log2(u)

|u| ⇥ |sinc(u1

+m⇡)||u

1

+m⇡| du.

We first bound I11

I11

log2[�(1 + |m|⇡)]�

Z

�<|u|�(1+|m|)

|sinc(u1

+m⇡)||u

1

+m⇡| du

2C log2[�(1 + |m|⇡)]�

⇥ log(�+m⇡) = C(log |m|+ log �)3

�

To bound I12

we make a change of variables u = �z, the above becomes

I12

C

�

Z

|z|>1+|m|�

[log �+ log z]2

z(z + m� )

dz = O

✓log2(�)

�

◆.

Altogether the bounds for I11

and I12

give |I1

| C(log |m|+ log �)3/�.

To bound I2

, just as in Lemma C.2, equation (46), we decompose it into three parts I2

=

I21

+ I22

+ I23

, where using Lemma C.1, equations (42) and (43) we have the bounds

|I21

| Z

|u|�|sinc(u +m⇡)|`

2

(u)

Zmin(a,4u)/�

�min(a,4|u|)/�|g(!)|

��h✓2u

�� !

◆� h(�!)

�� d!du

|I22

| Z

|u|�|sinc(u +m⇡)|`

2

(u)

Z �min(a,4|u|)/�

�a/�|g(!)|

��h✓2u

�� !

◆� h(�!)

�� d!du

|I23

| Z

|u|�|sinc(u +m⇡)|`

2

(u)

Z a/�

min(a,4|u|)/�|g(!)|

��h✓2u

�� !

◆� h(�!)

�� d!du.

Using the same method used to bound |I21

|, |I22

|, |I23

| in Lemma C.2, we have |I21

|, |I22

|, |I23

| C[log(�) + log(|m|)]3/�. Having bounded all partitions of the integral, we have the result.

The proof of (54) is identical and we omit the details. ⇤

C.1 Lemmas required to prove Lemma A.2 and Theorem A.1

In this section we give the proofs of the three results used in Lemma A.2 (which in turn proves

Theorem A.1).

Lemma C.4 Suppose Assumptions 2.4(ii) and 2.5(b,c) holds. Then for r1

, r2

2 Z we have

|A1

(r1

, r2

)�B1

(r1

� r2

; r1

)| = O

✓log2(a)

�

◆

56

PROOF. To obtain a bound for the di↵erence we use Lemma E.1(ii) to give

|A1

(r1

, r2

)�B1

(r1

� r2

; r1

)|

=

Z 1

�1

Z 1

�1

2aX

m=�2a

|sinc(u)sinc(u�m⇡)sinc(v)sinc(v + (m+ r1

� r2

)⇡)|

��Hm,�(2u

�,2v

�; r

1

)�Hm(2u

�,2v

�; r

1

)

��| {z }

C/�

dudv

C

�

2aX

m=�2a

Z 1

�1|sinc(u)sinc(u�m⇡)|du

Z 1

�1|sinc(v)sinc(v + (m+ r

1

� r2

)⇡)|dv| {z }

<1

C

�

2aX

m=�2a

Z 1

�1|sinc(u)sinc(u�m⇡)|du (from Lemma C.1(ii))

= O(log2 a

�),

thus giving the desired result. ⇤

Lemma C.5 Suppose Assumptions 2.4(ii) and 2.5(b,c) holds (with 0 |r1

|, |r2

| < C|a|). Then we

have

|B1

(s; r)� C1

(s; r)| = O

✓log2(a)

log a+ log �

�

�◆.

Note, if we relax the assumption on r1

, r2

to r1

, r2

2 Z then the above bound requires the additional

term log2(a)[I(r1

6= 0) log |r1

|+ I(r2

0) log |r2

|]/�.

PROOF. Taking di↵erences, it is easily seen that

B1

(s, r)� C1

(s, r)

=

Z

R2

2aX

m=�2a

sinc(u)sinc(u�m⇡)sinc(v)sinc(v + (m+ s)⇡)

⇥ 1

(2⇡)

Z2⇡min(a,a+m)/�

2⇡max(�a,�a+m)/�g(!)g(! + !m)

f(! � 2u

�)f(! +

2v

�+ !r)� f(!)f(! + !r)

�d!dvdu

= I1

+ I2

where

I1

=

Z

R2

2aX

m=�2a


⇥ 1

(2⇡)

Z2⇡min(a,a+m)/�

2⇡max(�a,�a+m)/�g(!)g(! + !m)f(! +

2v

�+ !r)

f(! � 2u

�)� f(!)

�d!dvdu

=

Z

R

2aX

m=�2a

sinc(v)sinc(v + (m+ s)⇡)Dm(v)dv

57

and

I2

=

Z

R2

2aX

m=�2a


⇥ 1

(2⇡)

Z2⇡min(a,a+m)/�

2⇡max(�a,�a+m)/�g(!)g(! + !m)f(!)

f(! +

2v

�+ !r)� f(! + !r)

�d!dvdu

=2aX

m=�2a

dm

Z

Rsinc(u)sinc(u�m⇡)du

with

Dm(v) =Z

Rsinc(u)sinc(u�m⇡)

1

(2⇡)

Z2⇡min(a,a+m)/�

2⇡max(�a,�a+m)/�g(!)g(! + !m)f(! +

2v

�+ !r)

⇥f(! � 2u

�)� f(!)

�d!du

and

dm =Z

Rsinc(v)sinc(v + (m+ s)⇡)

1

(2⇡)

Z2⇡min(a,a+m)/�

2⇡max(�a,�a+m)/�g(!)g(! + !m)f(!)

⇥f(! +

2v

�+ !r)� f(! + !r)

�d!dv.

Since the functions f(·) and g(·) satisfy the conditions stated in Lemma C.2, the lemma can be

used to show that

max|m|a

supv

|Dm(v)| C

✓log �+ log a

�

◆

and

max|m|a

|dm| C

✓log �+ log a

�

◆.

Substituting these bounds into I1

and I2

give

|I1

| C

✓log �+ log a

�

◆2aX

m=�2a

Z 1

�1|sinc(v)sinc(v + (m+ s)⇡)|dv

|I2

| C

✓log �+ log a

�

◆2aX

m=�2a

Z 1

�1|sinc(u)sinc(u�m⇡)|du.

Therefore, by using Lemma C.1(ii) we have

|I1

| and |I2

| = O

✓log2(a)

log a+ log �

�

◆.

Since |B1

(s; r)� C1

(s; r)| |I1

|+ |I2

| this gives the desired result. ⇤

58

D Approximations to the covariance and cumulants of

eQa,�(g; r)

In this section, our objective is to obtain bounds for cumq

� eQa,�(g; r1), . . . , eQa,�(g; rq)�, these results

will be used to prove the asymptotic expression for the variance of eQa,�(g; r) (given in Section A.2)

and asymptotic normality of eQa,�(g; r). Fox and Taqqu (1987), Dahlhaus (1989), Giraitis and

Surgailis (1990) (see also Taqqu and Peccati (2011)) have developed techniques for dealing with

the cumulants of sums of periodograms of Gaussian (discrete time) time series, and one would have

expected that these results could be used here. However, in our setting there are a few di↵erences

that we now describe (i) despite the spatial random being Gaussian the locations are randomly

sampled, thus the composite process Z(s) is not Gaussian (we can only exploit the Gaussianity

when we condition on the location) (ii) the random field is defined over Rd (not Zd) (iii) the number

of terms in the sums eQa,�(·) is not necessarily the sample size. Unfortunately, these di↵erences

make it di�cult to apply the above mentioned results to our setting. Therefore, in this section we

consider cumulant based results for spatial data observed at irregular locations. In order to reduce

cumbersome notation we focus on the case that the locations are from a uniform distribution.

As a simple motivation we first consider var[ eQa,�(1, 0)]. By using indecomposable partitions we

have

var[ eQa,�(1, 0)]

=1

n4

nX

j1,j2,j3,j4=1

j1 6=j2,j3 6=j4

aX

k1,k2=�a

cov [Z(sj1)Z(sj2) exp(i!k1(sj1 � sj2)), Z(sj3)Z(sj4) exp(i!k2(sj3 � sj4))]

=1

n4

nX

j1,j2,j3,j4=1

r1 6=r2,r3 6=r4

aX

k1,k2=�a

✓cum

⇥Z(sj1)e

isj1!k1 , Z(sj3)e�isj3!k2

⇤cum

⇥Z(sj2)e

�isj2!k1 , Z(sj4)eisj4!k2

⇤

+cum⇥Z(sj1)e

isj1!k1 , Z(sj4)eisj4!k2

⇤cum

⇥Z(sj2)e

�isj2!k1 , Z(sj3)e�isj3!k2

⇤

+cum⇥Z(sj1)e

isj1!k1 , Z(sj2)e�isj2!k1 , Z(sj3)e


⇤◆. (55)

In order to evaluate the covariances in the above we condition on the locations {sj}. To evaluate

the fourth order cumulant of the above we appeal to a generalisation of the conditional variance

method. This expansion was first derived in Brillinger (1969), and in the general setting it is stated

as

cum(Y1

, Y2

, . . . , Yq) =X

⇡

cum [cum(Y⇡1 |s1, . . . , sq), . . . , cum(Y⇡b |s1, . . . , sq)] , (56)

where the sum is over all partitions ⇡ of {1, . . . , q} and {⇡1

, . . . ,⇡b} are all the blocks in the par-

tition ⇡. We use (56) to evaluate cum[Z(sj1)eisj1!k1 , . . . , Z(sjq)e

isjq!kq ], where Yi = Z(sji)eisji!ki

and we condition on the locations {sj}. Using this decomposition we observe that because the

spatial process is Gaussian, cum[Z(sj1)eisj1!k1 , . . . , Z(sjq)e

isjq!kq ] can only be composed of cumu-

lants of covariances conditioned on the locations. Moreover, if s1

, . . . , sq are independent then

by using the same reasoning we see that cum[Z(s1

)eis1!k1 , . . . , Z(sq)eisq!kq ] = 0. Therefore,

59

cum⇥Z(sj1)e


isj3!k2 , Z(sj4)e�isj4 (!k2

⇤will only be non-zero if some el-

ements of sj1 , sj2 , sj3 , sj4 are dependent. Using these rules we have

var[ eQa,�(1, 0)]

=1

n4

X

j1,j2,j3,j42D4

aX

k1,k2=�a

✓cum

⇥Z(sj1)e


⇤cum

⇥Z(sj2)e


⇤

+cum⇥Z(sj1)e


⇤cum

⇥Z(sj2)e


⇤◆

+1

n4

X

j1,j2,j3,j42D3

aX

k1,k2=�a

✓cum

⇥Z(sj1)e


⇤cum

⇥Z(sj2)e


⇤

+cum⇥Z(sj1)e


⇤cum

⇥Z(sj2)e


⇤◆

+1

n4

X

j1,j2,j3,j42D3

aX

k1,k2=�a

cum⇥Z(sj1)e



⇤, (57)

where D4

= {j1

, . . . , j4

= all js are di↵erent}, D3

= {j1

, . . . , j4

; two js are the same but j1

6=j2

and j3

6= j4

} (noting that by definition of eQa,�(1, 0) more than two elements in {j1

, . . . , j4

}cannot be the same). We observe that |D

4

| = O(n4) and |D3

| = O(n3), where | · | denotes the

cardinality of a set. We will show that the second and third terms are asymptotically negligible

with respect to the first term. To show this we require the following lemma.

Lemma D.1 Suppose Assumptions 2.1, 2.2, 2.4 and 2.5(b,c) hold (note we only use Assumption

2.5(c) to get ‘neater expressions in the proofs’ it is not needed to obtain the same order). Then we

have

supa

aX

k1,k2=�a

cum[Z(s1

)eis1!k1 , Z(s2

)e�is2!k1 , Z(s3

)e�is3!k2 , Z(s1

)eis1!k2 ] = O(1) (58)

supa

aX

k1,k2=�a

cum⇥Z(s

1

)eis1!k1 , Z(s1

)eis1!k2⇤cum

⇥Z(s

2

)e�is2!k1 , Z(s3

)e�is3!k2⇤= O(1). (59)

PROOF. To show (58) we use conditional cumulants (see (56)). By using the conditional cumulant

60

expansion and Gaussianity of Z(s) conditioned on the location we have

aX

k1,k2=�a

cum[Z(s1

)eis1!k1 , Z(s2

)e�is2!k1 , Z(s3

)e�is3!k2 , Z(s1

)eis1!k2 ]

=aX

k1,k2=�a

cum[c(s1

� s2

)eis1!k1�is2!k1 , c(s

1

� s3

)e�is3!k2+is1!k2 ] +

aX

k1,k2=�a

cum[c(s1

� s3

)eis1!k1�is3!k2 , c(s

1

� s2

)e�is2!k1+is1!k2 ] +

aX

k1,k2=�a

cum[c(0)eis1(!k1+!k2

), c(s2

� s3

)e�is2!k1�is3!k2 ]

| {z }=0 (since s1 is independent of s2 and s3)

= I1

+ I2

.

Writing I1

in terms of expectations and using the spectral representation of the covariance we have

I1

=aX

k1,k2=�a

✓E⇥c(s

1

� s2

)c(s1

� s3

)eis1(!k1+!k2

)e�is3!k2e�is2!k1⇤� E

⇥c(s

1

� s2

)eis1!k1�is2!k1

⇤

⇥E⇥c(s

1

� s3

)eis1!k2�is3!k2

⇤◆

=1

(2⇡)2

aX

k1,k2=�a

Z Zf(x)f(y)sinc

✓�

2(x+ y) + (k

1

+ k2

)⇡

◆sinc

✓�

2x+ k

1

⇡

◆sinc

✓�

2y + k

2

⇡

◆dxdy �

1

(2⇡)2

aX

k1,k2=�a

Z Zf(x)f(y)sinc

✓�

2x+ k

2

⇡

◆sinc

✓�

2x+ k

2

⇡

◆sinc

✓�

2y + k

1

⇡

◆sinc

✓�

2y + k

1

⇡

◆dxdy

= E1

� E2

.

To bound E1

we make a change of variables u = �x2

+ k1

⇡, v = �y2

+ k2

⇡, and replace sum with

integral (and use Lemma E.1) to give

E1

=1

⇡2�2

Z Z aX

k1,k2=�a

f(2u

�� !k1)f(

2v

�� !k2)sinc(u+ v)sinc(u)sinc(v)dudv

=1

(2⇡)2⇡2

Z Zsinc(u+ v)sinc(u)sinc(v)

✓Z2⇡a/�

�2⇡a/�

Z2⇡a/�

�2⇡a/�f(

2u

�� !

1

)f(2v

�� !

2

)d!1

d!2

◆dudv +O(

1

�).

Let G(2u� ) = 1

2⇡

R2⇡a/��2⇡a/� f(

2u� �!)d!, then substituting this into the above and using equation (44)

in Lemma C.1 we have

E1

=

Z Zsinc(u+ v)sinc(u)sinc(v)G

✓2u

�

◆G

✓2v

�

◆dudv +O(

1

�) = O(1).

To bound E2

we use a similar technique and Lemma C.1(iii) to give E2

= O( 1�). Altogether, this

gives I1

= O(1). The same proof can be used to show that I2

= O(1). Altogether this gives (58).

61

To bound (59), we observe that if k1

6= �k2

, then cum⇥Z(s

1

)eis1!k1 , Z(s1

)eis1!k2⇤= E[c(0)eis(!k1

+!k2)] =

0 otherwise cum⇥Z(s

1

)eis1!k , Z(s1

)e�is1!k⇤= c(0). Using this, (59) can be reduced to

aX

k1,k2=�a

cum⇥Z(s

1

)eis1!k1 , Z(s1

)eis1!k2⇤cum

⇥Z(s

2

)e�is2!k1 , Z(s3

)e�is3!k2⇤

= c(0)aX

k=�a

cum⇥Z(s

2

)eis2!k , Z(s3

)e�is3!k⇤

=c(0)

2⇡

Z 1

�1

aX

k=�a

f(x)sinc

✓�x

2+ k⇡

◆sinc

✓�x

2+ k⇡

◆dx

✓change variables using ! =

�x

2+ k⇡

◆

= c(0)

Z 1

�1

1

⇡�

aX

k=�a

f

✓2!

�� !k

◆sinc2(!)d!

=c(0)

2⇡2

Z 1

�1sinc2(!)

✓Z2⇡a/�

�2⇡a/�f

✓2!

�� x

◆dx

◆d! +O

✓1

�

◆= O(1),

thus proving (59). ⇤

We now derive an expression for var[ eQa,�(1, 0)], by using Lemma D.1 we have

var[ eQa,�(1, 0)]

=1

n4

X

j1,j2,j3,j42D4

aX

k1,k2=�a

✓cum

⇥Z(sj1)e


⇤cum

⇥Z(sj2)e


⇤

+cum⇥Z(sj1)e


⇤cum

⇥Z(sj2)e


⇤◆+O

✓1

n

◆. (60)

In Lemma E.2 we have shown that the covariance terms above are of order O(��1), thus dominating

the fourth order cumulant terms which is of order O(n�1) (so long as � << n).

Lemma D.2 Suppose that {Z(s); s 2 Rd} is a Gaussian random process and {sj} are iid random

variables. Then the following hold:

(i) As we mentioned above, by using that the spatial process is Gaussian and (56),

cum[Z(sj1)eisj1!k1 , . . . , Z(sjq)e

isjq!kq ] can be written as the sum of products of cumulants of

the spatial covariance conditioned on location. Therefore, it is easily seen that the odd order

cumulant

cum2q+1

[Z(sj1) exp(isj1!k1), . . . , Z(sj2q+1 exp(isj2q+1!k2q+1))] = 0 for all q and regardless of

{sj} being dependent or not.

(ii) By the same reasoning given above, we observe that if more than (q + 1) locations {sj ; j =

1, . . . , 2q} are independent, then

cum2q[Z(sj1) exp(isj1!k1), . . . , Z(sj2q) exp(isj2q!k2q))] = 0.

62

Lemma D.3 Suppose Assumptions 2.1, 2.2 2.4 and 2.5(b) are satisfied, and d = 1. Then we have

cum3

[ eQa,�(g, r)] = O

✓log2(a)

�2

◆(61)

with �d/(log2(a)n) ! 0 as � ! 1, n ! 1 and a ! 1.

PROOF. We prove the result for cum3

[ eQa,�(1, 0)], noting that the proof is identical for general g

and r. We first expand cum3

[Qa,�(1, 0)] using indecomposable partitions. Using Lemma D.2(i) we

note that the third order cumulant is zero, therefore

cum3

[Qa,�(1, 0)]

=1

n6

X

j2D

aX

k1,k2,k3=�a

cum⇥Z(sj1)Z(sj2) exp(i!k1(sj1 � sj2)),

Z(sj3)Z(sj4) exp(i!k2(sj3 � sj4)), Z(sj5)Z(sj6) exp(i!k3(sj5 � sj6))⇤

=1

n6

X

j2D

X

⇡(2,2,2),2P2,2,2

Aj

2,2,2(⇡(2,2,2)) +1

n6

X

j2D

X

⇡4,22P4,2

Aj

4,2(⇡(4,2)) +1

n6

X

j2DA

j

6

= B2,2,2 +B

4,2 +B6

,

where D = {j1

, . . . , j6

2 {1, . . . , n} but j1

6= j2

, j3

6= j4

, j5

6= j6

}, Aj

2,2,2 consists of only the

product of cumulants of order two and P2,2,2 is the set of all cumulants of order two from the set

of indecomposable partitions of {(1, 2), (3, 4), (5, 6)}, Aj

4,2 consists of only the product of 4th and

2nd order cumulants and P4,2 is the set of all 4th order and 2nd order cumulant indecomposable

partitions of {(1, 2), (3, 4), (5, 6)}, finally Aj

6

is the 6th order cumulant. Examples of A’s are given

below

Aj

2,2,2(⇡(2,2,2),1) =aX

k1,k2,k3=�a

cum⇥Z(sj1)e


⇤cum

⇥Z(sj2)e


⇤

⇥cum⇥Z(sj4)e


⇤

=aX

k1,k2,k3=�a

E[c(sj1 � sj3)ei(sj1!k1

+sj3!k2)]E[c(sj2 � sj5)e

i(�sj2!k1+sj5!k3

)]⇥

E[c(sj4 � sj6)e�i(sj4!k2

+sj6!k3)], (62)

Aj

4,2(⇡(4,2),1) =aX

k1,k2,k3=�a

cum[Z(sj4) exp(�isj4!k2), Z(sj6) exp(�isj6!k3)]

cum[Z(sj1) exp(isj1!k1), Z(sj2) exp(�isj2!k1), Z(sj3) exp(isj3!k2), Z(sj5) exp(isj5!k3)],

(63)

Aj

6

=aX

k1,k2,k3=�a

cum[Z(sj1) exp(isj1!k1), Z(sj2) exp(�isj2!k1), Z(sj3) exp(isj3!k2),

Z(sj4) exp(�isj4!k2), Z(sj5) exp(isj5!k3), Z(sj6) exp(�isj6!k3)], (64)

where j = (j1

, . . . , j6

).

63

Bound for B222

We will show that B222

is the leading term in cum3

( eQa,�(g; 0)). The set D is split into four sets, D6

where all the elements of j are di↵erent, and for 3 i 5, Di where i elements in j are di↵erent,

such that

B2,2,2 =

1

n6

3X

i=0

X

j2D6�i

X

⇡(2,2,2)2P(2,2,2)

Aj

2,2,2(⇡(2,2,2)).

We start by bounding the partition given in (62), we later explain how the same bounds can be

obtained for other indecomposable partitions in P2,2,2. By using the spectral representation of the

covariance and that |D6

| = O(n6), it is straightforward to show that

1

n6

X

j2D6

Aj

2,2,2(⇡(2,2,2),1)

=X

k1,k2,k3

c6

(2⇡)3�6

Z

R3

Z

[��/2,�/2]6f(x)f(y)f(z) exp(i!k1(s1 � s

2

))⇥

exp(i!k2(s3 � s4

)) exp(i!k3(s5 � s6

))

exp(ix(s1

� s3

)) exp(iy(s4

� s6

)) exp(iz(s2

� s5

))3Y

j=1

ds2j�1

ds2jdxdydz

=c6

(2⇡)3

X

k1,k2,k3

Z

R3f(x)f(y)f(z)sinc

✓�x

2+ k

1

⇡

◆sinc

✓�z

2� k

1

⇡

◆⇥

sinc

✓�x

2� k

2

⇡

◆sinc

✓�y

2� k

2

⇡

◆sinc

✓�z

2� k

3

⇡

◆sinc

✓�y

2+ k

3

⇡

◆dxdydz, (65)

where c6

= n(n � 1) . . . (n � 5)/n6. By changing variables x = �x2

+ k1

⇡, y = �y2

� k2

⇡ and

z = �z2

� k3

⇡ we have

1

n6

X

j2D6

Aj

2,2,2(⇡(2,2,2),1)

=c6

(2⇡)3

X

k1,k2,k3

1

�3

Z

R3f(

2x

�� !k1)f(

2y

�+ !k2)f(

2z

�+ !k3)sinc(x)sinc(z)sinc(y)

sinc(x� (k2

+ k1

)⇡)sinc(y + (k3

+ k2

)⇡)sinc(z � (k1

� k3

)⇡)dxdydz. (66)

In order to understand how this case can generalise to other partitions in P1

, we represent the ks

inside the sinc function using the the linear equations

0

B@�1 �1 0

�1 0 1

0 1 1

1

CA

0

B@k1

k2

k3

1

CA , (67)

64

where we observe that the above is a rank two matrix. Based on this we make the following change

of variables k1

= k1

, m1

= k2

+ k1

and m2

= k1

� k3

, and rewrite the sum as

1

n6

X

j2D6

Aj

2,2,2(⇡(2,2,2),1)

=c6

(2⇡)3

X

k1,m1,m2

1

�3

Z

R3f

✓2x

�� !k1

◆f

✓2y

�+ !m1�k1

◆f

✓2z

�+ !k1�m2

◆sinc(x)sinc(z)sinc(y)

sinc(x�m1

⇡)sinc(y � (m2

�m1

)⇡)sinc(z �m2

⇡)dxdydz

=c6

�2⇡2

X

m1,m2

Z

R3sinc(x)sinc(x�m

1

⇡)sinc(y)sinc(y + (m1

�m2

)⇡)sinc(z)sinc(z �m2

⇡)| {z }contains two independent m terms

⇥ 1

�

X

k1

f

✓2x

�� !k1

◆f

✓2y

�+ !k1�m1

◆f

✓2z

�� !k1�m1

◆dxdydz. (68)

Finally, we apply Lemma C.1(iv) to give

1

n6

X

j2D6

Aj

2,2,2(⇡(2,2,2),1) = O

✓log2 a

�2

◆. (69)

The above only gives the bound for one partition of P1

, but we now show that the same bound

applies to all the other partitions. Looking back at (66) and comparing with (68), the reason that

only one of the three �s in the denominator of (66) gets ‘swallowed’ is because the matrix in (67) has

rank two. Therefore, there are two independent ms in the sinc function of (66), thus by applying

Lemma C.1(iv) the sumP

m1,m2only grows with rate O(log2 a). Moreover, it can be shown that

all indecomposable partitions of P2,2,2 correspond to rank two matrices (for a proof see equation

(A.13) in Deo and Chen (2000)). Thus all indecomposable partitions in P2,2,2 will have the same

order, which altogether gives

1

n6

X

j2D6

X

⇡(2,2,2)2P1

Aj

2,2,2(⇡(2,2,2)) = O

✓log2 a

�2

◆.

Now we consider the case that j 2 D5

. In this case, there are two ‘typical’ cases j = (j1

, j2

, j3

, j4

, j1

, j6

),

which gives

Aj

2,2,2(⇡(2,2,2),1) =

cum⇥Z(sj1)e


⇤cum

⇥Z(sj2)e


⇤cum

⇥Z(sj4)e


⇤

and j = (j1

, j2

, j1

, j4

, j5

, j6

), which gives

Aj

2,2,2(⇡(2,2,2),1) =

cum⇥Z(sj1)e


⇤cum

⇥Z(sj2)e


⇤cum

⇥Z(sj4)e


⇤.

Using the same method used to bound (69), when j = (j1

, j2

, j3

, j4

, j1

, j6

) Aj

2,2,2(⇡(2,2,2),1) =

O( log2 a

�2 ). However, when j = (j1

, j2

, j1

, j4

, j5

, j6

) we use the same proof used to prove (59) to

65

give Aj

2,2,2(⇡(2,2,2),1) = O( 1�). As we get similar expansions for all j 2 D5

and |D5

| = O(n5) we have

1

n6

X

j2D5

X

⇡(2,2,2)2P2,2,2

Aj

2,2,2(⇡(2,2,2)) = O

✓1

�n+

log2(a)

n�2

◆.

Similarly we can show that

1

n6

X

j2D4

X

⇡(2,2,2)2P2,2,2

Aj

2,2,2(⇡(2,2,2)) = O

✓1

�n2

+log2(a)

n2�2

◆.

and

1

n6

X

j2D3

X

⇡(2,2,2),2P2,2,2

Aj

2,2,2(⇡(2,2,2)) = O

✓1

n3

+log2(a)

n3�2

◆.

Therefore, if n >> �/ log2(a) we have B2,2,2 = O( log

2(a)

�2 ).

Bound for B4,2

To bound B4,2 we consider the ‘typical’ partition given in (63). Since A

j

4,2(⇡(4,2),1) involves fourth

order cumulants by Lemma D.2(ii) it will be zero in the case that the j are all di↵erent. Therefore,

only a maximum of five terms in j can be di↵erent, which gives

1

n6

X

j2DA

j

4,2(⇡(4,2),1) =1

n6

3X

i=1

X

j2D6�i

Aj

4,2(⇡(4,2),1).

We will show that for j 2 D5

, Aj

4,2(⇡(4,2),1) will not be as small as O(log2(a)/�2), however, this will

be compensated by |D5

| = O(n5) (noting that |D6

| = O(n6)). Let j = (j1

, j2

, j3

, j4

, j1

, j6

), then

expanding the fourth order cumulant in Aj

4,2(⇡(4,2),1) and using conditional cumulants (see (56))

we have

Aj

4,2(⇡(4,2),1)

=aX

k1,k2,k3=�a

cum[Z(sj4) exp(�isj4!k2), Z(sj6) exp(�isj6!k3)]

cum[Z(sj1) exp(isj1!k1), Z(sj2) exp(�isj2!k1), Z(sj3) exp(isj3!k2), Z(sj5) exp(isj5!k3)]

=aX

k1,k2,k3=�a

⇢cum

⇥c(s

1

� s2

)ei!k1(s1�s2), c(s

3

� s1

)eis3!k2+is1!k3

⇤

+cum⇥c(s

1

� s3

)ei(s1!k1+s3!k2

), c(s1

� s2

)eis1!k3�is2!k1

⇤

+cum⇥c(0)eis1!k1

+is1!k3 , c(s2

� s3

)e�is2!k1+is3!k2

⇤�E⇥c(s

4

� s6

)e�is4!k2�is6!k3

⇤

= Aj

4,2(⇡(4,2),1,⌦1

) +Aj

4,2(⇡(4,2),1,⌦2

) +Aj

4,2(⇡(4,2),1,⌦3

), (70)

66

where we use the notation ⌦ to denote the partition of the fourth order cumulant into it’s conditional

cumulants. To bound each term we expand the covariances as expectations, this gives

Aj

4,2(⇡(4,2),1,⌦1

)

=aX

k1,k2,k3=�a

⇢E⇥c(s

1

� s2

)c(s3

� s1

)ei!k1(s1�s2)eis3!k2

+is1!k3⇤E⇥c(s

4

� s6


⇤�

E⇥c(s

1

� s2

)ei!k1(s1�s2)

⇤E⇥c(s

3

� s1

)eis3!k2+is1!k3

⇤E⇥c(s

4

� s6


⇤�

= Aj

4,2(⇡(4,2),1,⌦1

,⇧1

) +Aj

4,2(⇡(4,2),1,⌦1

,⇧2

),

where we use the notation ⇧ to denote the expansion of the cumulants of the spatial covariances

expanded into expectations. To bound Aj

4,2(⇡(4,2),1,⌦1

,⇧1

) we use the spectral representation

theorem to give

Aj

4,2(⇡(4,2),1,⌦1

,⇧1

) =

1

(2⇡)3

Z

R3

aX

k1,k2,k3=�a

f(x)f(y)f(z)sinc

✓�(x� y)

2+ (k

1

+ k3

)⇡

◆sinc

✓�x

2+ k

1

⇡

◆⇥

sinc

✓�y

2+ k

2

⇡

◆sinc

✓�z

2� k

2

⇡

◆sinc

✓�z

2+ k

3

⇡

◆dxdydz.

By changing variables

Aj

4,2(⇡(4,2),1,⌦1

,⇧1

) =1

⇡3�3

aX

k1,k2,k3=�a

Z

R3f

✓2u

�� !k1

◆f

✓2v

�� !k2

◆f

✓2w

�� !k3

◆⇥

sinc(u� v + (k2

+ k3

)⇡)sinc(u)sinc(v)sinc(w)sinc(w � (k2

+ k3

)⇡)dudvdw.

Just as in the bound for B2,2,2, we represent the ks inside the sinc function as a set of linear

equations

0

B@0 1 1

0 0 0

0 1 1

1

CA

0

B@k1

k2

k3

1

CA , (71)

observing the matrix has rank one. We make a change of variables m = k2

+ k3

, k1

= k1

and

k2

= k2

to give

Aj

4,2(⇡(4,2),1,⌦1

,⇧1

)

=1

⇡3�

Z

R3

1

�2

aX

k1,k2=�a

f

✓2u

�� !k1

◆f

✓2v

�� !k2

◆f

✓2w

�� !m1�k2

◆⇥

X

m

sinc(u� v +m⇡)sinc(u)sinc(v)sinc(w)sinc(w �m⇡)dudvdw

=23

�

Z

R3

X

m

G�,m

✓2u

�,2v

�,2w

�

◆sinc(u� v +m⇡)sinc(u)sinc(v)sinc(w)sinc(w �m⇡)dudvdw,

67

where G�,m(2u� , 2v� , 2w� ) = 1

�2

Pak1,k2=�a f(

2u� � !k1)f(

2v� � !k2)f(

2w� � !m�k2). Taking absolutes

gives

|Aj

4,2(⇡(4,2),1,⌦1

,⇧1

)| C

�

Z

R3

X

m

|sinc(u� v +m⇡)sinc(w �m⇡)sinc(u)sinc(v)sinc(w)|dudvdw

Since the above contains m in the sinc function we use Lemma C.1(i), equations (42) and (43), to

show

|Aj

4,2(⇡(4,2),1,⌦1

,⇧1

)| C

�

Z

R3

X

m

|sinc(u� v +m⇡)sinc(w �m⇡)sinc(u)sinc(v)sinc(w)|dudvdw

C

�

X

m

`1

(m⇡)`2

(m⇡) = O(��1),

where the functions `1

(·) and `2

(·) are defined in Lemma C.1(i). Thus |Aj

4,2(⇡(4,2),1,⌦1

,⇧1

)| = O( 1�).

We use the same method used to bound (69) to show that Aj

4,2(⇡(4,2),1,⌦1

,⇧2

) = O( log2(a)

�2 ) and

|Aj

4,2(⇡(4,2),1,⌦2

)| = O( 1�). Furthermore, it is straightforward to see that by the independence of s1

and s2

and s3

that Aj

4,2(⇡(4,2),1,⌦3

) = 0 (recalling that Aj

4,2(⇡(4,2),1,⌦3

) is defined in equation (70)).

Thus altogether we have for j = (j1

, j2

, j3

, j4

, j1

, j6

) and the partition ⇡(4,2),1, that |A

j

4,2(⇡(4,2),1)| =O( 1�). However, it is important to note for all other j 2 D

5

and partitions in P4,2 the same method

will lead to a similar decomposition given in (70) and the rank one matrix given in (71). The rank

one matrix means one ‘free’ m in the sinc functions and this |Aj

4,2(⇡4,2)| = O( 1�) for all j 2 D5

and

⇡4,2 2 P

4,2. Thus, since |D5

| = O(n5) we have

1

n6

X

j2D5

X

⇡4,22P4,2

Aj

4,2(⇡2) = O

✓1

�n+

log2 a

�2n

◆= O

✓log2 a

�2

◆

if n >> �/ log2(a) (ie. �n log

2 a! 0). For j 2 D

4

and j 2 D3

we use the same argument, noting

that the number of free m’s in the sinc functions goes down but to compensate, |D4

| = O(n4) and

|D3

| = O(n3). Therefore, if n >> log2(a)/�, then

B4,2 =

1

n6

3X

i=1

X

j2D6�i

X

⇡4,22P4,2

Aj

4,2(⇡2) = O

✓log2 a

�2

◆.

Bound for B6

Finally, we bound B6

. By using Lemma D.2(ii) we observe that Aj

6

(k1

, k2

, k3

) = 0 if more than

four elements of j are di↵erent. Thus

B6

=1

n6

3X

i=2

X

j2D6�i

Aj

6

.

68

We start by considering the case that j = (j1

, j2

, j1

, j4

, j1

, j6

) (three elements in j are the same),

then by using conditional cumulants we have

Aj

6

=aX

k1,k2,k3=�a

cum[Z(s1

)eis1!k1 , Z(s2

)e�is2!k1 , Z(s1

)eis1!k2 , Z(s4

)e�is4!k2 , Z(s1

)eis1!k3 , Z(s6

)e�is6!k3 ]

=aX

k1,k2,k3=�a

X

⌦2RA

j

6

(⌦),

where R is the set of all pairwise partitions of {1, 2, 1, 4, 1, 6}, for example

Aj

6

(⌦1

) = cum⇥c(s

1

� s2

)ei!k1(s1�s2), c(s

1

� s4

)ei!k2(s1�s4), c(s

1

� s6

)ei!k3(s1�s6)

⇤.

We will first bound the above and then explain how this generalises to the other ⌦ 2 R and j 2 D4

.

Expanding the above third order cumulant in terms of expectations gives

Aj

6

(⌦1

)

=aX

k1,k2,k3=�a

⇢E⇥c(s

1

� s2

)ei!k1(s1�s2)c(s

1

� s4

)ei!k2(s1�s4)c(s

1

� s6

)ei!k3(s1�s6)

⇤�

E⇥c(s

1

� s2

)ei!k1(s1�s2)

⇤E⇥c(s

1

� s4

)ei!k2(s1�s4)c(s

1

� s6

)ei!k3(s1�s6)

⇤�

E⇥c(s

1

� s2

)ei!k1(s1�s2)c(s

1

� s4

)ei!k2(s1�s4)

�E⇥c(s

1

� s6

)ei!k3(s1�s6)

⇤�

E⇥c(s

1

� s2

)ei!k1(s1�s2)c(s

1

� s6

)ei!k3(s1�s6)

⇤E⇥c(s

1

� s4

)ei!k2(s1�s4)

⇤+

2E⇥c(s

1

� s2

)ei!k1(s1�s2)

⇤E⇥c(s

1

� s4

)ei!k2(s1�s4)

⇤E⇥c(s

1

� s6

)ei!k3(s1�s6)

⇤�

=5X

`=1

Aj

6

(⌦1

,⇧`).

We observe that for 2 ` 5, Aj

6

(⌦1

,⇧`) resembles Aj

4,2(⇡(4,2),1,⌦1

,⇧1

) defined in (71), thus the

same proof used to bound the terms in (71) can be use to show that for 2 ` 5Aj

6

(⌦1

,⇧`) = O( 1�).

However, the first term Aj

6

(⌦1

,⇧1

) involves just one expectation, and is not included in the previous

cases. By using the spectral representation theorem we have

Aj

6

(⌦1

,⇧1

)

=aX

k1,k2,k3=�a

Ehc(s

1

� s2

)ei!k1(s1�s2)c(s

1

� s4

)ei!k2(s1�s4)c(s

1

� s6

)ei!k3(s1�s6)

i

=1

(2⇡)3

aX

k1,k2,k3=�a

Z Z Zf(x)f(y)f(z)sinc

✓�(x+ y + z)

2+ (k

1

+ k2

+ k3

)⇡

◆sinc

✓�x

2+ k

1

⇡

◆

sinc

✓�y

2+ k

2

⇡

◆sinc

✓�z

2+ k

3

⇡

◆dxdydz

=23

(2⇡)3�3

aX

k1,k2,k3=�a

Z Z Zf(

2u

�� !k1)f(

2v

�� !k2)f(

2w

�� !k3)⇥

sinc(u+ v + w)sinc(u)sinc(v)sinc(w)dudvdw.

69

It is obvious that the ks within the sinc function correspond to a rank zero matrix, and thus

A6

(⌦1

,⇧1

) = O(1). Therefore, Aj

6

(⌦1

) = O(1). A similar bound holds for all j 2 D4

, this we have

1

n6

X

j2D4

Aj

6

(⌦1

) = O

✓1

n2

◆,

since |D4

| = O(n4). Indeed, the same argument applies to the other partitions ⌦ and j 2 D3

, thus

altogether we have

B6

=1

n6

X

⌦2R

3X

i=2

X

j2D6�i

Aj

6

(⌦) = O

✓1

n2

◆.

Altogether, using the bounds derived for B2,2,2, B4,2 and B

6

we have

cum4

( eQa,�(1, 0)) = O

✓log2(a)

�2

+1

n�+

log2(a)

�2n+

1

n2

◆= O

✓log2(a)

�2

◆,

where the last bound is due to the conditions on a, n and �. This gives the result. ⇤

We now generalize the above results to higher order cumulants.

Lemma D.4 Suppose Assumptions 2.1, 2.2, 2.4 and 2.5(b) are satisfied, and d = 1. Then for

q � 3 we have

cumq[ eQa,�(g, r)] = O

✓log2(q�2)(a)

�q�1

◆, (72)

cumq[ eQa,�(g, r1), . . . , eQa,�(g, rq)] = O

✓log2(q�2)(a)

�q�1

◆(73)

and in the case d > 1, we have

cumq[ eQa,�(g, r1), . . . , eQa,�(g, rq)] = O

✓log2d(q�2)(a)

�d(q�1)

◆(74)

with �d/(log2(a)n) ! 0 as � ! 1, n ! 1 and a ! 1.

PROOF. The proof essentially follows the same method used to bound the second and third cu-

mulants. We first prove (72). To simplify the notation we prove the result for g = 1 and r = 0,

noting that the proof in the general case is identical. Expanding out cumq[ eQa,�(1, 0)] and using

indecomposable partitions gives

cumq[ eQa,�(1, 0)]

=1

n2q

X

j1,...,j2q2D

aX

k1,...,kq=�a

cumhZ(sj1)Z(sj2)e

i!k1(sj1�sj2 ), . . . , Z(sj2q�1)Z(sj2q)e

i!kq (sj2q�1�sj2q )i

=1

n2q

X

j2D

X

b2Bq

X

⇡b2P2b

Aj

2b(⇡2b)

70

where Bq corresponds to the set of integer partitions of q (more precisely, each partition is a se-

quence of positive integers which sums to q). The notation used here is simply a generalization

of the notation used in Lemma D.3. Let b = (b1

, . . . , bm) (notingP

j bj = q) denote one of these

partitions, then P2b is the set of all indecomposable partitions of {(1, 2), (3, 4), . . . , (2q � 1, 2q)}

where the size of each partition is 2b1

, 2b2

, . . . , 2bm. For example, if q = 3, then one example of an

element of B3

is b = (1, 1, 1) and P2b = P

(2,2,2) corresponds to all pairwise indecomposable partitions

of {(1, 2), (3, 4), (5, 6)}. Finally, Aj

2b(⇡2b) corresponds to the product of one indecomposable par-

tition of the cumulant cum⇥Z(sj1)Z(sj2)e

i!k1(sj1�sj2 ), . . . , Z(sj2q�1)Z(sj2q)e

i!kq (sj2q�1�sj2q )⇤, where

the cumulants are of order 2b1

, 2b2

, . . . , 2bm (examples, in the case q = 3 are given in equation

(62)-(64)). Let

B2b =

1

n2q

X

j2D

X

⇡2b2P2b

Aj

2b(⇡2b),

therefore cumq[ eQa,�(1, 0)] =P

b2BqB

2b.

Just as in the proof of Lemma D.3, we will show that under the condition n >> �/ log2(a), the

pairwise decomposition B2,...,2 is the denominating term. We start with a ‘typical’ decomposition

⇡(2,...,2),1 2 P

2,...,2,

Aj

2,...,2(⇡(2,...,2),1) =aX

k1,...,kq=�a

cum[Z(s1

)eis1!k1 , Z(s2q)e

�is2q!kq ]q�1Y

c=1

cum[Z(s2c)e

�is2c!kc , Z(s2c+1

)eis2c+1!kc+1 ].

and

1

n2q

X

j2DA

j

2,...,2(⇡(2,...,2),1) =1

n2q

q�1X

i=0

X

j2D2q�i

Aj

2,...,2(⇡(2,...,2),1),

where D2q denotes the set where all elements of j are di↵erent and D

2q�i denotes the set that

(2q � i) elements in j are di↵erent. We first consider the case that j = (1, 2, . . . , 2q) 2 D2q. Using

identical arguments to those used for var[ eQa,�(1, 0)] and cum3

[ eQa,�(1, 0)] we can show that

Aj

2,...,2(⇡(2,...,2),1) =aX

k1,...,kq=�a

Z

Rqf(xq)sinc

✓2xq�

+ k1

⇡

◆sinc

✓2xq�

+ kq⇡

◆⇥

q�1Y

c=1

f(xc)sinc

✓2xc�

� kc⇡

◆sinc

✓2xc�

� kc+1

◆ qY

c=1

dxc. (75)

By a change of variables we get

Aj

2,...,2(⇡(2,...,2),1) =1

�q

aX

k1,...,kq=�a

Z

Rq

q�1Y

c=1

f

✓�uc2

+ !c

◆sinc(uc)sinc(uc � (kc+1

� kc)⇡)

⇥f

✓�uq2

� !1

◆sinc(uq)sinc(uq + (kq � k

1

)⇡)qY

c=1

dxc.

71

As in the proof of the third order cumulant we can rewrite the ks in the above as a matrix equation

0

BBBB@

�1 1 0 . . . 0 0

0 �1 1 . . . 0 0...

......

... 0 0

�1 0 . . . . . . 0 1

1

CCCCA

0

BBBB@

k1

k2

...

kq

1

CCCCA,

noting that that above is a (q� 1)-rank matrix. Therefore applying the same arguments that were

used in the proof of cum3

[ eQa,�(1, 0)] and also Lemma C.1(iii) we can show that Aj

2,...,2(⇡(2,...,2),1) =

O( log2(q�2)

(a)�q�1 ). Thus for j 2 D

2q we have 1

n2q

Pj2D2q

Aj

2,...,2(⇡(2,...,2),1) = O( log2(q�2)

(a)�q�1 ).

In the case that j 2 D2q�1

((2q�1)-terms in j are di↵erent) by using the same arguments as those

used to bound A2,2,2 (in the proof of cum[ eQa,�(1, 0)]) we have A

j

2,...,2(⇡(2,...,2),1) = O( log2(q�3)

(a)�q�2 ),

similarly if (2q � 2)-terms in j are di↵erent, then Aj

2,...,2(⇡(2,...,2),1) = O( log2(q�4)

(a)�q�3 ) and so forth.

Therefore, since |D2q�i| = O(n2q�i) we have

1

n2q

X

j2D|Aj

2,...,2(⇡(2,...,2),1)| CqX

i=0

log2(q�2�i)(a)

�q�1�ini.

Now by using that n >> �/ log2(a) we have

1

n2q

X

j2D|Aj

2,...,2(⇡(2,...,2),1)| = O

log2(q�2)(a)

�q�1

!.

The same argument holds for every other second order cumulant indecomposable partition, because

the corresponding matrix will always have rank (q � 1) in the case that j 2 D2q or for j 2

D2q�i and the dependent sj ’s lie in di↵erent cumulants (see Deo and Chen (2000)), thus B

2,...,2 =

O( log2(q�1)

(a)�q�1 ).

Now, we bound the other extreme B2q. Using the conditional cumulant expansion (56) and not-

ing that cum2q[Z(sj1)e

isj1!k1 , . . . , Z(sj2q)e�isj2q!kq ] is non-zero, only when at most (q+1) elements

of j are di↵erent we have

B2q =

1

n2q

X

j2D

aX

k1,...,kq=�a

cum2q[Z(sj1)e

isj1!k1 , . . . , Z(sj2q)e�isj2q!kq ]

=1

n2q

X

j2Dq+1

X

⌦2R2q

Aj

2q(⌦).

whereR2q is the set of all pairwise partitions of {1, 2, 1, 3, . . . , 1, q}. We consider a ‘typical’ partition

⌦1

2 R2q

Aj

2q(⌦1

) =aX

k1,...,kq=�a

cum⇥c(s

1

� s2

)ei(s1�s2)!k1 , . . . , c(s1

� sq+1

)ei(s1�sq+1)!kq⇤. (76)

72

By expanding the above the cumulant as the sum of the product of expectations we have

Aj

2q(⌦1

) =1

n2q

X

j2Dq+1

X

⌦2R2q

X

⇧2Sq

Aj

2q(⌦1

,⇧),

where Sq is the set of all partitions of {1, . . . , q}. As we have seen in both the var[ eQa,�(1, 0)] and

cum3

[ eQa,�(1, 0)] calculations, the leading term in the cumulant expansion is the expectation over

all the covariance terms. The same result holds true for higher order cumulants, the expectation

over all the covariances in that cumulant is the leading term because the it gives the linear equation

of the ks in the sinc function with the lowest order rank (we recall the lower the rank the less ‘free’

�s). Based on this we will only derive bounds for the expectation over all the covariances. Let

⇧1

2 Sq, where

Aj

2q(⌦1

,⇧1

) =aX

k1,...,kq=�a

E⇥ qY

c=1

c(s1

� sc+1

)ei(s1�sc+1)!kc⇤.

Representing the above expectation as an integral and using the spectral representation theorem

and a change of variables gives

Aj

2q(⌦1

,⇧1

) =aX

k1,...,kq=�a

E⇥ qY

c=1

c(s1

� sc+1

)ei(s1�sc+1)!kc⇤

=1

(2⇡)q

aX

k1,k2,k3=�a

Z

Rqsinc

✓�(Pq

c=1

xc)

2+ ⇡

qX

c=1

kc

◆ qY

c=1

f(xc)sinc(xc + kc⇡)qY

c=1

dxc

=2q

(2⇡)q�q

aX

k1,...,kq=�a

Z

Rqsinc(

qX

c=1

uc)qY

c=1

f

✓2uc�

� !kc

◆sinc(uc)duc = O(1),

where the last line follows from Lemma C.1, equation (44). Therefore, Aj

2q(⌦1

) = O(1). By using

the same method on every partition ⌦ 2 Rq+1

and j 2 Dq+1

and |Dq+1

| = O(nq+1), we have

B2q =

1

n2q

X

j2Dq+1

X

⌦2R2q

X

⇧2S2q

Aj(⌦1

,⇧1

) = O(1

nq�1

).

Finally, we briefly discuss the terms B2b which lie between the two extremes B

2,...,2 and B2q.

Since B2b is the product of 2b1, . . . , 2bm cumulants, by Lemma D.2(ii) at most

Pmj=1

(bj+1) = q+m

elements of j can be di↵erent. Thus

B2b =

1

n2q

q+mX

i=q

X

j2Di

X

⇡2b2P2b

Aj

2b(⇡2b).

By expanding the cumulants in terms of the cumulants of covariances conditioned on the location

(which is due to Gaussianity of the random field, see for example, (76)) we have

B2b =

1

n2q

q+mX

i=q

X

j2Di

X

⇡2b2P2b

X

⌦2R2b

Aj

2b(⇡2b,⌦),

73

where R2b is the set of all paired partitions of {(1, . . . , 2b

1

), . . . , (2bm�1

+1, . . . , 2bm)}. The leadingterms are the highest order expectations. This term leads to a matrix equation for the k’s within

the sinc functions, where the rank of the corresponding matrix is at least (m� 1) (we do not give

a formal proof of this). Therefore, B2b = O( log

2(m�1)(a)

nq�m�m�1 ) = O( log2(q�2)

(a)�q�1 ) (since n >> �/ log2(a)).

This concludes the proof of (72).

The proof of (73) is identical and we omit the details.

To prove the result for d > 1, (74) we use the same method, the main di↵erence is that the

spectral density function in (75) is a multivariate function of dimension d, there are 2dp sinc

functions and the integral is over Rdp, however the analysis is identical. ⇤

PROOF of Theorem 3.7 By using the well known identities

cov(<A,<B) =1

2

�<cov(A,B) + <cov(A, B)

�

cov(=A,=B) =1

2

�<cov(A,B)�<cov(A, B)

�,

cov(<A,=B) =�1

2

�=cov(A,B)�=cov(A, B)

�, (77)

and equation (22), we immediately obtain

�dvar[< eQa,�(g; 0)] =1

2(C

1

(0) + <C2

(0)) +O(`�,a,n),


i=

8><

>:

<2

C1

(!r) +O(`�,a,n) r1

= r2

(= r)<2

C2

(!r) +O(`�,a,n) r1

= �r2

(= r)

O(`�,a,n) otherwise

�dcovh= eQa,�(g; r1),= eQa,�(g; r2)

i=

8><

>:

<2

C1

(!r) +O(`�,a,n) r1

= r2

(= r)�<2

C2

(!r) +O(`�,a,n)) r1

= �r2

(= r)

O(`�,a,n) otherwise

and

�dcovh< eQa,�(g; r1),= eQa,�(g; r2)

i=

(O(`�,a,n) r

1

6= �r2

=2

C2

(!r) +O(`�,a,n) r1

= �r2

(= r)

Similar expressions for the covariances of <Qa,�(g; r) and =Qa,�(g; r) can also be derived.

Finally, asymptotic normality of < eQa,�(g; r) and = eQa,�(g; r) follows from Lemma D.4. Thus

giving (23). ⇤

PROOF of Theorem 3.4 The proof is identical to the proof Lemma D.4, we omit the details.

⇤

PROOF of Theorem 3.5 The proof is similar to the proof of Theorem 3.7, we omit the

details.

74

E Additional proofs

In this section we prove the remaining results required in this paper.

For example, Theorems 3.1, 3.2, 3.3(i,iii), 3.6 and A.2, involve replacing sums with integrals.

In the case that the frequency domain is increasing stronger assumptions are required than in the

case of fixed frequency domain. We state the required result in the following lemma.

Lemma E.1 Let us suppose the function g1

, g2

are bounded (sup!2Rd |g1

(!)| < 1 and sup!2Rd |g2

(!)| <1) and for all 1 j d, sup!2Rd |@g1(!)

@!j| < 1 and sup!2Rd |@g2(!)

@!j| < 1.

(i) Suppose a/� = C (where C is a fixed finite constant) and h is a bounded function whose first

partial derivative 1 j d, sup!2Rd |@h(!)

@!j| < 1. Then we have

��1

�d

aX

k=�a

g1

(!k)h(!k)�1

(2⇡)d

Z

2⇡[�C,C]

dg1

(!)h(!)!

�� K��1,

where K is a finite constant independent of �.

(ii) Suppose a/� ! 1 as � ! 1. Furthermore, h(!) ��(!) and for all 1 j d the partial

derivatives satisfy |@h(!)

@!j| ��(!). Then uniformly over a we have

��1

�d

aX

k=�a

g1

(!k)h (!k)�1

(2⇡)d

Z

2⇡[�a/�,a/�]dg1

(!)h (!) d!

�� K��1

(iii) Suppose a/� ! 1 as � ! 1. Furthermore, f4

(!1

,!2

,!3

) ��(!1

)��(!2

)��(!3

) and for

all 1 j 3d the partial derivatives satisfy |@f4(!1,!2,!3)

@!j| ��(!).

��1

�2d

aX

k1,k2=�a

g1

(!k1)g2(!k2)f4 (!k1+r1 ,!k2 ,!k2+r2)�

1

(2⇡)d

Z

2⇡[�a/�,a/�]d

Z

2⇡[�a/�,a/�]dg1

(!1

)g2

(!2

)f4

(!1

+ !r1 ,!2

,!2

+ !r2)

�� K��1.

PROOF. We first prove the result in the univariate case. We expand the di↵erence between sum

and integral

1

�

aX

k=�a

g1

(!k)h(!k)�1

2⇡

Z2⇡a/�

�2⇡a/�g1

(!)h(!)d!

=1

�

aX

k=�a

g1

(!k)h(!k)�1

2⇡

a�1X

k=�a

Z2⇡/�

0

g1

(!k + !)h(!k + !)d!.

By applying the mean value theorem for integrals to the integral above we have

=1

�

aX

k=�a

g1

(!k)h(!k)�1

2⇡

a�1X

k=�a

Z2⇡/�

0

g1

(!k + !)h(!k + !)d!

=1

�

aX

k=�a

[g1

(!k)h(!k)� g1

(!k + !k)h(!k + !k)]

75

where !k 2 [0, 2⇡� ]. Next, by applying the mean value theorem to the di↵erence above we have��1

�

aX

k=�a

[g1

(!k)h(!k)� g1

(!k + !k)h(!k + !k)]

�� 1

�2

aX

k=�a

⇥g01

(e!k)h(e!k) + g1

(e!k)h0(e!k)

⇤(78)

where e!k 2 [!k,!k+!] (note this is analogous to the expression given in Brillinger (1981), Exercise

1.7.14).

Under the condition that a = C�, g1

(!) and h(!) are bounded and using (78) it is clear that��1

�

aX

k=�a

g1

(!k)h(!k)�1

2⇡

Z2⇡a/�

�2⇡a/�g1

(!)h(!)d!

��

sup! |h0(!)g1

(!)|+ sup! |h(!)g01

(!)|�2

aX

k=�a

1 = C��1.

For d = 1, this proves (i).

In the case that a/� ! 1 as � ! 1, we use that h and h0 are dominated by a montonic

function and that g1

is bounded. Thus by using (78) we have��1

�

aX

k=�a

g1

(!k)h(!k)�1

2⇡

Z2⇡a/�

�2⇡a/�g1

(!)h(!)d!

�� 1

�2

aX

k=�a

sup!k!!k+1

(|g01

(!)h(!)|+ |g1

(!)h0(!)|)

2(sup! |g1

(!)|+ sup! |g01

(!)|)�2

aX

k=0

��(!k) C

�

Z 1

0

��(!)d! = O(��1).

For d = 1, this proves (ii).

To prove the result for d = 2 we take di↵erences, that is

1

�2

aX

k1=�a

aX

k2=�a

g(!k1 ,!k2)h(!k1 ,!k2)�1

(2⇡)2

Z a/�

�a/�

Z a/�

�a/�g(!

1

,!2

)h(!1

,!2

)

=1

�

aX

k1=�a

0

@ 1

�

aX

k2=�a

g(!k1 ,!k2)h(!k1 ,!k2)�1

(2⇡)

Z a/�

�a/�g(!k1 ,!2

)h(!k1 ,!2

)

1

A

+

Z a/�

�a/�

0

@ 1

�

aX

k1=�a

g(!k1 ,!2

)h(!k1 ,!2

)�Z a/�

�a/�g(!

1

,!2

)h(!1

,!2

)

1

A .

For each of the terms above we apply the method described for the case d = 1; for the first term

we take the partial derivative over !2

and the for the second term we take the partial derivative

over !1

. This method can easily be generalized to the case d > 2. The proof of (iii) is identical to

the proof of (ii).

We mention that the assumptions on the derivatives (used replace sum with integral) can be

relaxed to that of bounded variation of the function. However, since we require the bounded

derivatives to decay at certain rates (to prove other results) we do not relax the assumption here.

⇤

We now prove the claims at the start of Appendix A.2.

76

Lemma E.2 Suppose Assumptions 2.1, 2.2 and

(i) Assumptions 2.4(i) and 2.4(a,c) hold. Then we have


i=

(C1

(!r) +O( 1� + �d

n ) r1

= r2

(= r)

O( 1� + �d

n ) r1

6= r2

and


i=

(C2

(!r) +O( 1� + �d

n ) r1

= �r2

(= r)

O( 1� + �d

n ) r1

6= �r2

where

C1

(!r) = C1,1(!r) + C

1,2(!r) and C2

(!r) = C2,1(!r) + C

2,2(!r), (79)

with

C1,1(!r) =

1

(2⇡)d

Z

2⇡[�a/�,a/�]df(!)f(! + !r)|g(!)|2d!,

C1,2(!r) =

1

(2⇡)d

Z

Dr

f(!)f(! + !r)g(!)g(�! � !r)d!,

C2,1(!r) =

1

(2⇡)d

Z

2⇡[�a/�,a/�]df(!)f(! + !r)g(!)g(�!)d!,

C2,2(!r) =

1

(2⇡)d

Z

Dr

f(!)f(! + !r)g(!)g(! + !r)d!, (80)

where the integral is defined asRDr

=R2⇡min(a,a�r1)/�2⇡max(�a,�a�r1)/�

. . .R2⇡min(a,a�rd)/�2⇡max(�a,�a�rd)/�

(note that C1,1(!r) and C

1,2(!r) are real).

(ii) Assumptions 2.4(ii) and 2.5(b) hold. Then


i= A

1

(r1

, r2

) +A2

(r1

, r2

) +O(�d

n),


i= A

3

(r1

, r2

) +A4

(r1

, r2

) +O(�d

n),

where

A1

(r1

, r2

) =1

⇡2d�d

2aX

m=�2a

min(a,a+m)X

k=max(�a,�a+m)

Z

R2df(

2u

�� !k)f(

2v

�+ !k + !r1)

g(!k)g(!k � !m)Sinc(u�m⇡)Sinc(v + (m+ r1

� r2


A2

(r1

, r2

) =1

⇡2d�d

2aX

m=�2a

min(a,a+m)X

k=max(�a,�a+m)

Z

R2df(

2u

�� !k)f(

2v

�+ !k + !r1)

g(!k)g(!m � !k)Sinc(u� (m+ r2

)⇡)Sinc(v + (m+ r1


77

A3

(r1

, r2

) =1

⇡2d�d

2aX

m=�2a

min(a,a+m)X

k=max(�a,�a+m)

Z

R2df(

2u

�� !k)f(

2v

�+ !k + !r1)

g(!k)g(!m � !k)Sinc(u+m⇡)Sinc(v + (m+ r2

+ r1


A4

(r1

, r2

) =1

⇡2d�d

2aX

m=�2a

min(a,a+m)X

k=max(�a,�a+m)

Z

R2df(

2u

�� !k)f(

2v

�+ !k + !r1)

g(!k)g(!k � !m)Sinc(u� (m� r2

)⇡)Sinc(v + (m+ r1


where k = max(�a,�a � m) = {k1

= max(�a,�a � m1

), . . . , kd = max(�a,�a � md)},k = min(�a+m) = {k

1

= max(�a,�a+m1

), . . . , kd = min(�a+md)}.

(iii) Assumptions 2.4(ii) and 2.5(b) hold. Then we have

�d supa


i�� < 1 and �d supa


i�� < 1,

if �d/n ! c (0 c < 1) as � ! 1 and n ! 1.

PROOF We prove the result in the case d = 1 (the proof for d > 1 is identical). We first prove (i).

By using indecomposable partitions, Theorem 2.1 and Lemma D.1 in Subba Rao (2015b), noting

that the fourth order cumulant is of order O(1/n), it is straightforward to show that


i

=1

�

aX

k1,k2=�a

g(!k1)g(!k2)

cov

✓Jn(!k1)Jn(!k1+r1)�

�d

n

nX

j=1

Z(sj)2e�isj!r ,

Jn(!k2)Jn(!k2+r2)��d

n

nX

j=1

Z(sj)2e�isj!r

◆�

=

8><

>:

1

�

Pak=�a f(!k)f(!k + !r)g(!k)g(!k)

+ 1

�

Pmin(a,a�r)k=max(�a,�a�r) f(!k)f(!k + !r)g(!k)g(�!k+r) +O( a

�2 + �n) r

1

= r2

O( a�2 + �

n) r1

6= r2

Since a = C� we have O( a�2 ) = O( 1�) and by replacing sum with integral (using Lemma E.1(i)) we

have


i=

(C1

(!r) +O( 1� + �n) r

1

= r2

(= r)

O( 1� + �n) r

1

6= r2

where

C1

(!r) = C11

(!r) + C12

(!r) +O

✓1

�

◆,

with

C11

(!r) =1

2⇡

Z2⇡a/�

�2⇡a/�f(!)f(! + !r)g(!)g(!)d!

C12

(!r) =1

2⇡

Z2⇡min(a,a�r)/�

2⇡max(�a,�a�r)/�f(!)f(! + !r)g(!)g(�! � !r)d!.

78

We now show that C1

(!r) = C11

(!r) + C12

(!r). It is clear that C11

(!r) is real. Thus we focus

on C12

(!r). To do this, we write g(!) = g1

(!) + ig2

(!), therefore

=g(!)g(�! � !r) = [g2

(!)g1

(�! � !r)� g1

(!)g2

(�! � !r)].

Substituting the above into =C12

(!r) gives us =C12

(r) = [C121

(r) + C122

(r)] where

C121

(r) =1

2⇡


2⇡max(�a,�a�r)/�f(!)f(! + !r)g2(!)g1(�! � !r)d!

C122

(r) = � 1

2⇡


2⇡max(�a,�a�r)/�f(!)f(! + !r)g1(!)g2(�! � !r)d!

We will show that C122

(r) = �C121

(r). Focusing on C122

and making the change of variables

u = �! � !r gives us

C122

=1

2⇡

Z2⇡max(�a,�a�r)/�

2⇡min(a,a�r)/�f(u+ !r)f(�u)g

1

(�u� !r)g2(u)du,

noting that the spectral density function is symmetric with f(�u) = f(u), and thatR2⇡max(�a,�a�r)/�2⇡min(a,a�r)/� =

�R2⇡min(a,a�r)/�2⇡max(�a,�a�r)/�. Thus we have =C

12

(r) = 0, which shows that C1

(!r) is real. The proof of

�dcov[ eQa,�(g; r1), eQa,�(g; r2)] is the same. Thus, we have proven (i).

To prove (ii) we first expand cov[ eQa,�(g; r1), eQa,�(g; r2)] to give

�cov[ eQa,�(g; r1), eQa,�(g; r2)]

= �aX

k1,k2=�a

g(!k1)g(!k2)nX

j1,...,j4=1

✓E⇥c(sj1 � sj3)e

isj1!k1�isj3!k2

⇤E⇥c(sj2 � sj4)e

�isj2!k1+r1+isj4!k2+r2

⇤+

E⇥c(sj1 � sj4)e

isj1!k1+isj4!k2+r2

⇤E⇥c(sj2 � sj3)e

�isj2!k1+r1�isj3!k2

⇤+

cum⇥Z(sj1)e

i!k1sj1 , Z(sj2)e

�isj2 (!k1+!r2 ), Z(sj3)e

�i!k2sj3 , Z(sj4)e

isj4 (!k2+!r2 )

⇤◆.

By applying Lemma D.1, (59) in Subba Rao (2015b), to the fourth order cumulant we can show

that

�covheQa,�(r1), eQa,�(g; r2)

i= A

1

(r1

, r2

) +A2

(r1

, r2

) +O

✓�

n

◆(81)

where

A1

(r1

, r2

) = �aX

k1,k2=�a

g(!k1)g(!k2)E⇥c(s

1

� s3

) exp(is1

!k1 � is3

!k2)⇤⇥

E⇥c(s

2

� s4

) exp(�is2

!k1+r1 + is4

!k2+r2)⇤

A2

(r1

, r2

) = �aX

k1,k2=�a

g(!k1)g(!k2)E⇥c(s

1

� s4

) exp(is1

!k1 + is4

!k2+r2)⇤⇥

E⇥c(s

2

� s3

) exp(�is2

!k1+r1 � is3

!k2)⇤.

79

Note that the O(�/n) term include the error n�1[A1

(r1

, r2

)] + A2

(r1

, r2

)] (we show below that

A1

(r1

, r2

) and A2

(r1

, r2

) are both bounded over � and a). To write A1

(r1

, r2

) in the form stated in

the lemma we integrate over s1

, s2

, s3

and s4

to give

A1

(r1

, r2

)

= �aX

k1,k2=�a

g(!k1)g(!k2)1

�4

Z

[��/2,�/2]4c(s

1

� s3

)c(s2

� s4

)eis1!k1�is3!k2e�is2!k1+r1

+is4!k2+r2ds1

. . . ds4

.

By using the spectral representation theorem and integrating out s1

, . . . , s4

we can write the above

as

A1

(r1

, r2

)

=�

(2⇡)2

aX

k1,k2=�a

g(!k1)g(!k2)

Z 1

�1

Z 1

�1f(x)f(y)sinc

✓�x

2+ k

1

⇡

◆

sinc

✓�y

2� (r

1

+ k1

)⇡

◆sinc

✓�x

2+ k

2

⇡

◆sinc

✓�y

2� (r

2

+ k2

)⇡

◆dxdy

=1

⇡2�

aX

k1,k2=�a

g(!k1)g(!k2)

Z 1

�1

Z 1

�1f(

2u

�� !k1)f(

2v

�+ !k1 + !r1)

⇥sinc(u)sinc(u+ (k2

� k1

)⇡)sinc(v)sinc(v + (k1

� k2

+ r1

� r2

)⇡)dudv,

where the second equality is due to the change of variables u = �x2

+ k1

⇡ and v = �y2

� (r1

+ k1

)⇡.

Finally, by making a change of variables k = k1

and m = k1

� k2

(k1

= �k2

+m) we obtain the

expression for A1

(r1

, r2

) given in Lemma E.2.

A similar method can be used to obtain the expression for A2

(r1

, r2

)

A2

(r1

, r2

)

=�

(2⇡)2

aX

k1,k2=�a

g(!k1)g(!k2)

Z 1

�1

Z 1

�1f(x)f(y)sinc

✓�x

2+ k

1

⇡

◆

sinc

✓�y

2� (r

1

+ k1

)⇡

◆sinc

✓�y

2+ k

2

⇡

◆sinc

✓�x

2� (r

2

+ k2

)⇡

◆dxdy

=1

⇡2�

aX

k1,k2=�a

g(!k1)g(!k2)

Z 1

�1

Z 1

�1f(

2u

�� !k1)f(

2v

�+ !k1 + !r1)

⇥sinc(u)sinc(u� (k2

+ r2

+ k1

)⇡)sinc(v)sinc(v + (k2

+ k1

+ r1

)⇡)dudv.

By making the change of variables k = k1

and m = k1

+ k2

(k1

= �k2

+m) we obtain the stated

expression for A2

(r1

, r2

).

Finally following the same steps as those above we obtain

�cov( eQa,�(g; r1), eQa,�(g; r2)) = A3

(r1

, r2

) +A4

(r1

, r2

) +O(�

n),

80

where

A3

(r1

, r2

) =1

�3

aX

k1,k2=�a

g(!k1)g(!k2)

Z

[��/2,�/2]4ei(s1!k1

+s3!k2)e�is2!k1+r1

�is4!k2+r2

⇥c(s1

� s3

)c(s2

� s4

)ds1

ds2

ds3

ds4

A4

(r1

, r2

) =1

�3

aX

k1,k2=�a

g(!k1)g(!k2)

Z

[��/2,�/2]4ei(s1!k1

+s3!k2)e�is2!k1+r1

�is4!k2+r2

⇥c(s1

� s4

)c(s2

� s3

)ds1

ds2

ds3

ds4

.

Again by replacing the covariances in A3

(r1

, r2

) and A4

(r1

, r4

) with their spectral representation

gives (ii) for d = 1. The result for d > 1 is identical.

It is clear that (iii) is true under Assumption 2.4(a,b). To prove (iii) under Assumption 2.5(a,b)

we will show that for 1 j 4, supa |Aj(r1, r2)| < 1. To do this, we first note that by the Cauchy

Schwarz inequality we have

supa,�

1

⇡2d

��1

�d

min(�a+m)X

k=max(�a,�a+m)

g(!k)g(!k � !m)f(2u

�� !k)f(

2v

�+ !k + !r1)

��

C sup!

|g(!)|2kfk22

,

where kfk2

is the L2

norm of the spectral density function and C is a finite constant. Thus by

taking absolutes of A1

(r1

, r2

) we have

|A1

(r1

, r2

)|

C sup!

|g(!)|2kfk22

1X

m=�1

Z

R2d|Sinc(u�m⇡)Sinc(v + (m+ r

1

� r2

)⇡)Sinc(u)Sinc(v)|dudv.

Finally, by using Lemma C.1(iii), Subba Rao (2015b), we have that supa |A1

(r1

, r2

)| < 1. By using

the same method we can show that supa |A1

(r1

, r2

)|, . . . , supa |A4

(r1

, r4

)| < 1. This completes

the proof. ⇤

PROOF of Theorem 5.1 Making the classical variance-bias decomposition we have

E⇣eV � �dvar[ eQa,�(g; 0)]

⌘2

= var[eV ] +⇣E[eV ]� �dvar[ eQa,�(g; 0)]

⌘2

.

We first analysis the bias term, in particular E[eV ]. We note that by using the expectation and

variance result in Theorem 3.1 and equation (22), respectively, we have

E[eV ] =�d

|S|X

r2Svar[ eQa,�(g; r)] +

�d

|S|X

r2S

��E[ eQa,�(g; r)]��2

| {z }=O(��2d

Qdj=1(log �+log |rj |)2)

=1

|S|X

r2SC1

(!r) +O

✓`�,a,n +

[log �+ logM ]d

�d

◆

= C1

+O

✓`a,�,n +

[log �+ logM ]d

�d+

|M |�

◆.

81

Next we consider var[eV ], by using the classical cumulant decomposition we have

var[eV ] =�2d

|S|2X

r1,r22S

✓ ��covheQa,�(g; r1), eQa,�(g; r2)

i��2

+��cov

heQa,�(g; r1), eQa,�(g; r2)

i��2

+�2d

|S|2X

r1,r22Scum

⇣eQa,�(g; r1), eQa,�(g; r1), eQa,�(g; r2), eQa,�(g; r2)

⌘

+O

0

@ �2d

|S|2X

r1,r22S

��cum⇣eQa,�(g; r1), eQa,�(g; r1), eQa,�(g; r2)

⌘E⇣eQa,�(g; r2)

⌘��

1

A .

By substituting the variance/covariance results for eQa,�(·) based on uniformly sampled locations

in (22) and the cumulant bounds in Lemma D.4 into the above we have

var[eV ] =�2d

|S|2X

r2S

��varheQa,�(g; r)

i��2

+O

`�,a,n +

log4d(a)

�d

!= O

1

|S| + `�,a,n +log4d(a)

�d

!.

Thus altogether we have the result. ⇤

F Sample properties of Qa,�(g; r)

In this section we summarize the result for Qa,�(g; r). To simplify notation we state the results

only for the case that the locations are uniformly distributed.

We recall that

Qa,�(g; r) = eQa,�(g; r) +G�Vr (82)

where Vr = 1

n

Pnj=1

Z(sj) exp(�i!rsj).

Theorem F.1 Suppose Assumptions 2.1(i), 2.2, b = b(r) denotes the number of zero elements in

the vector r 2 Zd and

(i) Assumptions 2.4(i) and 2.5(a,c) hold. Then we have

E [Qa,�(g; r)]

=

(O( 1

�d�b ) r 2 Zd/{0}1

(2⇡)d

R!22⇡[�C,C]

d f(!)g(!)d! +O( 1� + �d

n ) r = 0

(ii) Suppose the Assumptions 2.4(ii) and Assumption 2.5(b,c) hold and {m1

, . . . ,md�b} is the

subset of non-zero values in r = (r1

, . . . , rd), then we have

E [Qa,�(g; r)]

=

8<

:O⇣

1

�d�b

Qd�bj=1

(log �+ log |mj |)⌘

r 2 Zd/{0}1

(2⇡)d

R!2Rd f(!)g(!)d! + c(0)

n

Pak=�a g(!k) +O

�log �+log krk1

� + 1

n

�r = 0

.

82

PROOF The proof of (i) immediately follows from Theorem 2.1.

The proof of (ii) follows from writing Qa,�(g; r) as a quadratic form and taking expectations

E [Qa,�(g; r)]

= c2

aX

k=�a

g(!k)1

�d

Z

[��/2,�/2]dc(s

1

� s2

) exp(i!0k(s1 � s

2

)� is02

!r)ds1ds2 +Wr,

where c2

= n(n�1)/n2 and Wr = c(0)I(r=0)

n

Pak=�a g(!k) (I(r = 0) denotes the indicator variable).

We then follow the same proof used to prove Theorem 3.1. ⇤

Theorem F.2 [Asymptotic expression for variance] Suppose Assumptions 2.1, 2.2, 2.4 2.5(b,c)

hold. Then we have

�dcov [Qa,�(g; r1), Qa,�(g; r2)] =

(C1

(!r) + 2<[C3

(!r)G�] + 2f2

(!r)|G�|2 +O(`�,a,n) r1

= r2

(= r)

O(`�,a,n) r1

6= r2

�dcovhQa,�(g; r1), Qa,�(g; r2)

i=

(C2

(!r) + 2G�C3

(!r) + 2f2

(!r)G2

� +O(`�,a,n) r1

= �r2

(= r)

O(`�,a,n) r1

6= �r2

,

where C1

(!r) and C2

(!r) are defined in (79), G� = 1

n

Pak=�a g(!k), ad = O(n),

C3

(!r) =2

(2⇡)d

Z

2⇡[�a/�,a/�]dg(!)f(�!)f(! + !r)d! and f

2

(!r) =

Z

Rdf(�)f(!r � �)d�.(83)

PROOF To prove the result we use (82). Using this expansion we rewrite cov[Qa,�(g; r1), Qa,�(g; r2)]

in terms of eQa,�(g; r) and Vr

cov [Qa,�(g; r1), Qa,�(g; r2)] =

covheQa,�(g; r1), eQa,�(g; r2)

i+ |G�|2cov [Vr1 , Vr2 ] +G�cov

heQa,�(g; r1), Vr2

i+G�cov

heQa,�(g; r2), Vr1

i.

In Theorem A.1 we have evaluated an asymptotic expression for covheQa,�(g; r1), eQa,�(g; r2)

i. We

now evaluate similar expressions for cov [Vr1 , Vr2 ] and covheQa,�(g; r1), Vr2

i. It is straightforward

to show that

�dcov[Vr1 , Vr2 ] =

(O( 1�) r

1

6= r2

2f2

(!r1) +O⇣

1

� + �d

n

⌘r1

= r2

(84)

To evaluate an expression for covheQa,�(g; r1), Vr2

iwe consider the case d = 1. Using similar

83

arguments to those in the proof of Lemma E.2 we can show that

�covheQa,�(g; r1), Vr2

i

= 2�c3

aX

k=�a

g(!k)Ehc(s

1

� s3

)c(s2

� s3

)eis1!keis2(y�!k�!r)eis3(�x�y�!r)i+O

✓�

n2

◆

=2c

3

�

(2⇡)2

aX

k=�a

g(!k)

Z

R3f(x)f(y)sinc

✓�x

2+ k⇡

◆sinc

✓�x

2� (k + r

1

)⇡

◆sinc

✓�(x+ y)

2� r

2

⇡

◆dxdy

+O

✓�

n2

◆

=2c

3

�⇡2

aX

k=�a

g(!k)

Z

R3f

✓2u

�� !k

◆f

✓2v

�+ !k+r

◆sinc (u) sinc (v) sinc (u+ v + (r

1

� r2

)⇡) dudv

+O

✓�

n2

◆,

where c3

= n(n � 1)(n � 2)/n3. Finally by replacing sum with an integral, using Lemma E.1 and

replacing f�2u� � !

�f�2v� + ! + !r

�with f (�!) f (! + !r) and by using (54) we have

�covheQa,�(g; r1), Vr2

i

=2c

3

2⇡⇡2

Z2⇡a/�

�2⇡a/�g(!)f (�!) f (! + !r1) d!

Z

R3sinc (u) sinc (v) sinc (u+ v + (r

1

� r2

)⇡) dudv

+O

✓�

n2

+log2 �

�

◆

=

8<

:C3

(!r1) +O⇣

�n2 + log

2 �� + 1

n

⌘r1

= r2

O⇣�d

n2 + log

2 �� + 1

n

⌘r1

6= r2

(85)

Altogether, by using Theorem A.1, (85) and (84) we obtain the result. ⇤

In the following lemma we show that we further simplify the expression for the asymptotic

variance if we keep r fixed.

Corollary F.1 Suppose Assumption 2.4, 2.5(a,c) or 2.5(b,c) holds, and r is fixed. Let C3

(!) and

f2

(!) be defined in (83). Then we have

C3

(!r) = C3

+O

✓krk

1

�

◆,

and f2

(!r) = f2

+O⇣krk1�

⌘, where

C3

=2

(2⇡)d

Z

2⇡[�a/�,a/�]dg(!)f(!)2d!

and f2

= f2

(0) (note, if g(!) = g(�!), then C1

= C2

).

PROOF The proof is the same as the proof of Corollary 3.1. ⇤

84

Theorem F.3 [CLT on real and imaginary parts] Suppose Assumptions 2.1, 2.2, 2.4 and 2.5(b,c)

hold. Let C1

, C2

, C3

and f2

be defined as in Corollary F.1. We define the m-dimension complex

random vectors Qm = (Qa,�(g, r1), . . . , Qa,�(g, rm)), where r1

, . . . , rm are such that ri 6= �rj and

ri 6= 0. Under these conditions we have

2�d/2

E1

✓E

1

E1

+ <E2

<Qa,�(g, 0),<Qm,=Qm

◆P! N

�0, I

2m+1

�, (86)

where E1

= C1

+ 2<[G�C3

] + 2f2

|G�|2 and E2

= C2

+ 2G�C3

+ 2f2

G2

� with �d

n log

2d(a)

! 0 and

log

2(a)

�1/2 ! 0 as � ! 1, n ! 1 and a ! 1.

PROOF Using the same method used to prove Lemma D.4, and analogous results can be derived

for the cumulants of Qa,�(g; r). Asymptotic normality follows from this. We omit the details. ⇤

Application to nonparametric covariance estimator without bias correction

In this section we apply the results to the nonparametric estimator considered in Example 2.2(ii);

the estimator without bias correction and is a non-negative definite sequence. We recall that

bcn(v) = T

✓2v

�

◆ecn(v) where ecn(v) =

1

�d

aX

k=�a

|Jn(!k)|2 exp(iv0!k)

!.

and T is the d-dimensional triangle kernel. It is clear the asymptotic sampling properties of bcn(v)are determined by ecn(v). Therefore, we first derive the asymptotic sampling properties of ecn(v).We observe that ecn(v) = Qa,�(eiv

0·; 0), thus we use the results in Section 3 to derive the asymptotic

sampling properties of ecn(v). By using Theorem 3.1(ii)(a,b) and Assumption 2.4(ii) we have

E[ecn(v)] =1

(2⇡)d

Z

2⇡[�a/�,a/�]df(!) exp(iv0)d! +O

✓log �

�

◆

=1

(2⇡)d

Z

Rdf(!) exp(iv0)d! +O

✓�

a

◆�

+log �

�

!, (87)

where � is such that it satisfies Assumption 2.4(ii)(a). Therefore, if ��+d/2/a� ! 0 as a ! 1 and

� ! 1, then by using Theorem F.3 we have

�d/2 (ecn(v)� c(v))D! N (0,⌃) (88)

where

⌃ =1

(2⇡)d

Z

2⇡[�a/�,a/�]df(!)2

�1 + exp(i2v0!)

�d! +

2G�

(2⇡)d

Z

2⇡[�a/�,a/�]df(!)2 exp(iv0!)d! + 2G2

�f2,

and G� = 1

n

Pak=�a exp(iv

0!k), which is real. We now derive the sampling properties of bcn(v). Byusing (87), (88) and properties of the triangle kernel we have

E[bcn(v)] = c(v) +O

✓��

a�+

log �

�

�✓1� min

1id |vi|�

◆+

max1id |vi|�

◆

85

and

�d/2

✓bcn(v)� T

✓2v

�

◆c(v)

◆D! N

0, T

✓2v

�

◆2

⌃

!,

with ��+d/2/a� ! 0, �d

n log

2d(a)

! 0 and log

2(a)

�1/2 ! 0 as a ! 1 and � ! 1.

Application to parameter estimation using an L2

criterion

In this section we consider the asymptotic sampling properties of b✓n = argmin✓2⇥ Ln(✓), where

Ln(·) is defined in (10) and ⇥ is a compact set. We will assume that there exists a ✓0

2 ⇥, such

that for all ! 2 Rd, f✓0(!) = f(!) and there does not exist another ✓ 2 ⇥ such that for all ! 2 Rd

f✓0(!) = f✓(!) and in additionRRd kr✓f(!; ✓

0

)k21

d! < 1. Furthermore, we will assume thatb✓n ! ✓

0

.

Making the usual Taylor expansion we have �d/2(b✓n � ✓0

) = A�1�d/2rLn(✓0) + op(1), where

A =2

(2⇡)d

Z

Rd[r✓f(!; ✓

0

)][r✓f(!; ✓0

)]0d!, (89)

and it is clear the asymptotic sampling properties of b✓n are determined by r✓Ln(✓0), which we see

from (11) can be written as r✓Ln(✓0) = Qa,�(�2r✓f✓0(·); 0) + 1

�d

Pak=�ar✓f✓0(!k)2.

Thus by using Theorem F.3 we have �d/2r✓Ln(✓0)D! N (0, B), where

B =4

(2⇡)d

Z

Rdf(!)2 [r✓f✓(!)] [r✓f✓(!)]0c✓=✓0d!

�4<G�

(2⇡)d

Z

Rdf(!)2r✓f✓(!)c✓=✓0d! + 2|G�|2

Z

Rdf(!)2d!

and G� = 2

n

Pak=�ar✓f✓(!k)c✓=✓0 . Therefore, by using the above we have

�d/2(b✓n � ✓0

)P! N (0, A�1BA�1)

with �d

n ! 0 and log

2(a)

�1/2 ! 0 as a ! 1 and � ! 1.

86

Documents

Fourier based statistics for irregular spaced spatial …suhasini/papers/fourier_based...Fourier based statistics for irregular spaced spatial data Suhasini Subba Rao Department of