THE CONVERGENCE RATE AND ASYMPTOTIC DISTRIBU- TION …stat.columbia.edu/~jcliu/paper/BootQuantile2APT.pdf · THE CONVERGENCE RATE AND ASYMPTOTIC DISTRIBU-TION OF BOOTSTRAP QUANTILE

Applied Probability Trust (8 February 2011)

THE CONVERGENCE RATE AND ASYMPTOTIC DISTRIBU-

TION OF BOOTSTRAP QUANTILE VARIANCE ESTIMATOR

FOR IMPORTANCE SAMPLING

JINGCHEN LIU AND XUAN YANG,∗ Columbia University

Abstract

Importance sampling is a widely used variance reduction technique to compute

sample quantiles such as value-at-risk. The variance of the weight sample

quantile estimator is usually a difficult quantity to compute. In this paper,

we present the exact convergence rate and asymptotic distributions of the

bootstrap variance estimators for quantiles of weighted empirical distributions.

Under regularity conditions, we show that the bootstrap variance estimator is

asymptotically normal and has relative standard deviation of order O(n−1/4).

1. Introduction

In this paper, we derive the asymptotic distributions of the bootstrap quantile

variance estimators for weighted samples. Let F be a cumulative distribution

function (c.d.f.), f be its density function, and αp = infx : F (x) ≥ p be

its p-th quantile. It is well known that the asymptotic variance of the p-th

sample quantile is inversely proportional to f(αp) (c.f. [6]). When f(αp) is

close to zero (e.g. p is close to zero or one), the sample quantile becomes very

unstable since the “effective samples” size is small. In the scenario of Monte

Carlo, one solution is using importance sampling for variance reduction by

distributing more samples around the neighborhood of the interesting quantile

αp. Such a technique has been widely employed in multiple disciplines. In

portfolio risk management, the p-th quantile of a portfolio’s total asset price

is an important risk measure. This quantile is also known as the value-at-risk.

∗ Postal address: 1255 Amsterdam Ave, Department of Statistics, New York, NY 10027, U.S.A.

1

2 Jingchen Liu and Xuan Yang

Typically, the probability p in this context is very close to zero (or one). A

partial list of literature of using importance sampling to compute the value-at-

risk includes [25, 26, 28, 42, 43, 44, 14, 23]. A recent work by [30] discussed

efficient importance sampling for risk measure computation for heavy-tailed

distributions. In the system stability assessment of engineering, the extreme

quantile evaluation is of interest. In this context, the interesting probabilities

are typically of a smaller order than those of the portfolio risk analysis.

Upon considering p being close to zero or one, the computation of αp can

be viewed as the inverse problem of rare-event simulation. The task of the

latter topic is computing the tail probabilities 1−F (b) when b tends to infinity.

Similar to the usage in the quantile estimation, importance sampling is also a

standard variance reduction technique for rare-event simulation. The first work

on this topic is given by [41], which not only presents an efficient importance

sampling estimator but also defines a second-moment-based efficiency measure.

We will later see that such a measure is also closely related to the asymptotic

variance of the weighted quantiles. Such a connection allows people to adapt

the efficient algorithms designed for rare-event simulations to the computation

of quantiles (c.f. [24, 30]). More recent works of rare-event simulations for

light-tailed distributions include [19, 39, 17] and for heavy-tailed distributions

include [2, 4, 18, 31, 7, 8, 10, 11]. There are also standard textbooks such as

[13, 3].

Another related field of this line of work is survey sampling where unequal

probability sampling and weighted samples are prevailing (c.f. [32, 36]). The

weights are typically defined as the inverse of the inclusion probabilities.

The estimation of distribution quantile is a classic topic. The almost sure

result of sample quantile is established by [6]. The asymptotic distribution of

(unweighted) sample quantile can be found in standard textbook such as [15].

Estimation of the (unweighted) sample quantile variance via bootstrap was

proposed by [37, 38, 40, 5, 22]. There are also other kernel based estimators

Bootstrap for Weighted Quantile Variances 3

(to estimate f(αp)) for such variances (c.f. [21]).

There are several pieces of works immediately related to the current one.

The first one is [29], which derived the asymptotic distribution of the bootstrap

quantile variance estimator for unweighted i.i.d. samples. Another one is given

by [27] who derived the asymptotic distribution of weighted quantile estimators;

see also [14] for a confidence interval construction. A more detailed discussion

of these results is given in Section 2.2.

The asymptotic variance of weighted sample quantile, as reported in [27],

contains the density function f(αp), whose evaluation typically consists of

computation of high dimensional convolutions and therefore is usually not

straightforward. In this paper, we propose to use bootstrap method to com-

pute/estimate the variance of such a weighted quantile. Bootstrap is a generic

method that is easy to implement and does not consist of tuning parameters in

contrast to the kernel based methods for estimating f(αp). This paper derives

the convergence rate and asymptotic distribution of the bootstrap variance

estimator for weighted quantiles. More specifically, the main contributions are

to first provide conditions under which the quantiles of weighted samples have

finite variances and develop their asymptotic approximations. Second, we de-

rive the asymptotic distribution of the bootstrap estimators for such variances.

Let n denote the sample size. Under regularity conditions (for instance, moment

conditions and continuity conditions for the density functions), we show that

the bootstrap variance estimator is asymptotically normal with a convergence

rate of order O(n−5/4). Given that the quantile variance decays at the rate of

O(n−1), the relative standard deviation of a bootstrap estimator is O(n−1/4).

The technical challenge lies in that many classic results of order statistics

are not applicable. This is mainly caused by the variations introduced by the

weights, which in the current context is the Radon-Nikodym derivative, and

the weighted sample quantile does not map directly to the ordered statistics.

In this paper, we employed Edgeworth expansion combined with the strong


approximation of empirical processes ([33]) to derive the results.

This paper is organized as follows. In Section 2, we present our main results

and summarize the related results in literature. A numerical implementation

is given in Section 3 to illustrate the performance of the bootstrap estimator.

The proofs of the theorems are provided in Sections 4 and 5.

2. Main results

2.1. Problem setting

Consider a probability space (Ω,F , P ) and a random variable X admitting

cumulative distribution function (c.d.f.) F (x) = P (X ≤ x) and density func-

tion

f(x) = F ′(x)

for all x ∈ R. Let αp be its p-th quantile, that is,

αp = infx : F (x) ≥ p.

Consider a change of measure Q, under which X admits a cumulative distribu-

tion function G(x) = Q(X ≤ x) and density

g(x) = G′(x).

Let

L(x) =f(x)

g(x),

and X1,...,Xn be i.i.d. copies of X under Q. Assume that P and Q are

absolutely continuous with respect to each other, then EQL(Xi) = 1. The

corresponding weighted empirical c.d.f. is

FX(x) =

∑ni=1 L(Xi)I(Xi ≤ x)∑n

i=1 L(Xi). (1)

A natural estimator of αp is

αp(X) = infx ∈ R : FX(x) ≥ p. (2)


Of interest in this paper is the variance of αp(X) under the sampling distribution

of Xi, that is

σ2n = V arQ(αp(X)). (3)

The notations EQ(·) and V arQ(·) are used to denote the expectation and

variance under measure Q.

Let Y1, ..., Yn be i.i.d. bootstrap samples from the empirical distribution

G(x) =1

n

n∑i=1

I(Xi ≤ x).

The bootstrap estimator for σ2n in (3) is defined as

σ2n =

n∑i=1

Q (αp(Y) = Xi) (Xi − αp(X))2, (4)

where Y = (Y1, ..., Yn) and Q is the measure induced by G, that is, under Q

Y1, ..., Yn are i.i.d. following empirical distribution G. Note that both G and Q

depend on X. To simplify the notations, we do not include the index of X in

the notations of Q and G.

Remark 1. There are multiple ways to form an estimate of F . One alternative

to (1) is

FX(x) =1

n

n∑i=1

L(Xi)I(Xi ≤ x). (5)

The analysis of FX is analogous to and simpler than that of (1). This is because

the denominator is a constant. We will briefly mention the corresponding results

for (5) without a rigorous proof.

The weighted sample c.d.f.’s in (1) and (5) are useful under different situa-

tions. If one can only compute L(x) up to an unknown normalizing constant,

then it is necessary to consider (1) for the quantile computation. On the other

hand, for some situations such as rare-event simulation, the measure P is

sometimes not absolutely continuous with respect to Q. In that case, (1) is

a consistent estimator of the conditional c.d.f. and αp(X) is the conditional


quantile estimator given event A, where A = ∩B : Q(B) = 1. In order to

compute the quantile of F under this situation, (5) is appropriate.

Remark 2. The bootstrap method is a generic estimation procedure and only

requires that the samples are i.i.d.. In addition, this method does not involve

choosing appropriate tuning parameters. From the computation point of view,

one typically uses Monte Carlo to compute the bootstrap estimator in (4).

More precisely, let Y(k)1 , ..., Y

(k)n be i.i.d. samples with replacement from the set

X1, ..., Xn. Define Y(k) = (Y(k)1 , ..., Y

(k)n ) for k = 1, ...,m, where Y(k)’s are

independent. A Monte Carlo estimator of σ2n is

1

m

m∑k=1

(α(Y(k))− α(X))2.

In this paper, we do not pursue the analysis of Monte Carlo error for computing

σ2n.

Another advantage of bootstrap is that one does not have to regenerate the

samples in contrast to evaluating σ2p via Monte Carlo directly by regenerating

samples from Q. Sometimes, generating a single X from Q requires nonig-

norable computational cost, for instance, X =∑m

i=1 Zi with m large. A nice

feature of bootstrap is that the computation only consists of X (not the Zi’s)

and therefore is free of the complexity of the system.

2.2. Related results

In this section, we present two related results in the literature. First, [29]

established asymptotic distribution of the the bootstrap variance estimators for

the (unweighted) sample quantiles. In particular, it showed that if the density

function f(x) is Holder continuous with index 12 + δ0 then

n5/4(σ2n − σ2n)⇒ N(0, 2π−1/2[p(1− p)]3/2f(αp)−4)

as n→∞. This is consistent with the results in Theorem 2 by setting L(x) ≡ 1.

This paper can be viewed as a natural extension of [29], though the proof

techniques are different.


In the context of importance sampling, as shown by [27], if EQ|L(x)|3 <∞,

the asymptotic distribution of a weighted quantile is

√n(αp(X)− αp)⇒ N

(0,V arQ(Wp)

f(αp)2

)(6)

as n → ∞, where Wp = L(X)(I(X < αp) − p). More general results in terms

of weighted empirical processes are given by [30]

Remark 3. An alternative variance estimator to the bootstrap method is the

approximation V arQ(Wp)/f2(αp). Such an approximation requires the evalua-

tion of V arQ(Wp) and f(αp). The computation of f(αp) is usually difficult for

complex systems.

We illustrate this issue by one example (chapter 4 in [13]). Consider n

i.i.d. random variables W1, ...,Wm. Let Sm =∑m

i=1Wi. We are interested

in computing the p-th quantile of Sm. Note that the density of Sm is the

convolution of m density functions. The computation overhead for evaluating

this density function with certain relative error is substantial especially when it

is of a small value. For more complicated systems, the computation of marginal

density is even harder.

Remark 4. Note that the weak convergence in (6) requires weaker conditions

than those in Theorems 1 and 2 in the following subsection. The weak conver-

gence does not require αp(X) to have a finite variance. In contrast, in order to

apply the bootstrap variance estimator, one needs to have the estimand well

defined, that is, V arQ(αp(X)) <∞.

We now provide a brief discussion on the efficient quantile computation via

importance sampling. The sample quantile admits a large variance when f(αp)

is small. One typical situation is that p is very close to zero or one. To fix

ideas, we consider the case where p tends to zero. The asymptotic variance of

the p-th quantile of n i.i.d. samples is

1− pnp

p2

f(αp)2.


Then, in order to obtain an estimate of an ε error with at least 1−δ probability,

the necessary number of i.i.d. samples is proportional to p−1 p2

f2(αp)which

grows to infinity as p → 0. Typically, the inverse of the hazard function,

p/f(αp), varies slowly as p tends to zero. For instance, p/f(αp) is bounded if

X is a light-tailed random variable and grows at the most linearly in αp for

most heavy-tailed distributions (e.g. regularly varying distribution, log-normal

distribution).

The asymptotic variance of the quantiles of FX defined in (5) is

V arQ(L(X)I(X ≤ αp))np2

p2

f(αp)2.

There is a wealth of literature on the design of importance sampling algo-

rithms particularly adapted to the context in which p is close to zero. A

well accepted efficiency measure is precisely based on the relative variance

p−2V arQ(L(X)I(X ≤ αp)) as p→ 0. More precisely, the change of measure is

called strongly efficient, if p−2V arQ(L(X)I(X ≤ αp)) is bounded for arbitrarily

small p. A partial list of recent developments of importance sampling algorithms

in the rare event setting includes [1, 9, 10, 12, 19]. Therefore, the change of

measure designed to estimate p can be adapted without much additional effort

to the quantile estimation problem. For a more thorough discussion, see [30, 14].

The current paper is presented based on such efficient algorithms. Whereas,

we take a simplified approach by considering a fixed p and sending the sample

size n to infinity.

2.3. The main results

In this subsection, we provide an asymptotic approximation of σ2n and the

asymptotic distribution of σ2n. We first list a set of conditions which we will

refer to in the statements of our theorems.

A1 There exists an α > 4 such that

EQ|L(X)|α <∞.


A2 There exists a β > 3 such that

EQ|X|β <∞.

A3 Assume thatα

3>β + 2

β − 3.

A4 There exists a δ0 > 0 such that the density functions f(x) and g(x) are

Holder continuous with index 12 +δ0 in a domain of αp, that is, there exists

a constant c such that

|f(x)− f(y)| ≤ c|x− y|1

2+δ0 , |g(x)− g(y)| ≤ c|x− y|

1

2+δ0 ,

for all x and y in a domain of αp.

A5 The measures P and Q are absolutely continuous with respect to each

other. The likelihood ratio L(x) ∈ (0,∞) is Lipschitz continuous in a

domain of αp.

A6 Assume f(αp) > 0.

Theorem 1. Let F and G be the cumulative distribution functions of a random

variable X under probability measures P and Q respectively. The distributions

F and G have density functions f(x) = F ′(x) and g(x) = G′(x). We assume

that conditions A1 - A6 hold. Let

Wp = L(X)I(X ≤ αp)− pL(X),

and αp(X) be as defined in (2). Then,

σ2n , V arQ(αp(X)) =V arQ(Wp)

nf(αp)2+ o(n−5/4), EQ(αp(X)) = αp + o(n−3/4)

as n→∞.

Theorem 2. Suppose that the conditions in Theorem 1 hold and L(X) has

density under Q. Let σ2n be defined as in (4). Then, under Q

n5/4(σ2n − σ2n)⇒ N(0, τ2p ) (7)


as n→∞, where “⇒” denotes weak convergence and

τ2p = 2π−1/2L(αp)f(αp)−4(V arQ(Wp))

3/2.

Remark 5. Conditions A1-3 are imposed to insure that αp(X) has a finite

variance under Q. In the context of light-tailed systems, such moment con-

ditions are typically satisfied. However, for very heavy-tailed simulations, the

moment conditions may not be in place, e.g. regularly varying distributions

with very heavy tails (c.f. [18, 10, 11]).

The continuity assumptions on the density function f and the likelihood ratio

function L (conditions A4 and A5) are typically satisfied in practice. Condition

A6 is necessary for the quantile to have a variance of order O(n−1).

Remark 6. Once a consistent estimate of σ2n has been obtained, one can use

the weak convergence result in (6) to construct approximate confidence intervals

for αp.

If one considers the empirical distribution FX defined as in (5) and quantile

estimator

αp(X) = infx ∈ R : FX(x) ≥ p.

The corresponding result to those in Theorem 1 is that

σ2∗n = V arQ(αp(X)) =V arQ(Wp)

nf(αp)2+ o(n−5/4), (8)

where Wp = L(X)I(X ≤ αp). Under Q, the corresponding bootstrap variance

estimator has asymptotic distribution

n5/4(σ2∗n − σ2∗n)⇒ N(0, τ2p ) (9)

as n→∞, where

τ2p = 2π−1/2L(αp)f(αp)−4(V arQ(Wp))

3/2.

The proofs of (8) and (9) are similar to those of Theorems 1 and 2 and therefore

are omitted.


p α1−p α1−p σ2n σ2

n σ2n

0.05 15.70 15.67 0.0032 0.0031 0.0027

0.04 16.16 16.13 0.0034 0.0033 0.0029

0.03 16.73 16.70 0.0037 0.0036 0.0032

0.02 17.51 17.47 0.0042 0.0041 0.0037

0.01 18.78 18.74 0.0054 0.0052 0.0047

σ2n: the quantile variance computed by crude Monte Carlo.

σ2n: the asymptotic approximation of σ2

n in Theorem 1.

σ2n: the bootstrap estimate of σ2

n.

Table 1: Comparison of variance estimators. The standard error of Monte Carlo errors are

reported in the parentheses, n = 10000.

3. A numerical example

In this section, we provide one numerical example to illustrate the perfor-

mance of the bootstrap variance estimator. In order to compare the boot-

strap estimator with the asymptotic approximation in Theorem 1, we choose

an example for which the marginal density f(x) is in a closed form and αp

can be computed numerically. This example is simply for comparison and

illustration purpose. The proposed bootstrap estimator is applicable to much

more complicated situations. Consider a partial sum

X =

10∑i=1

Zi

where Zi’s are i.i.d. exponential random variables with rate one. Then, the

density function of X is

f(x) =x9

9!e−x.

We are interested in computing X’s (1− p)-th quantile via exponential change

of measure that is

dQθdP

=

10∏i=1

eθZi−φ(θ), (10)


where φ(θ) = − log(1 − θ) for θ < 1. This family of change of measure is

equivalent to the exponential family generated by the Gamma distribution

itself.

We further choose θ = θ∗ so that θ∗ = arg supθ(θαp − 10φ(θ)). Note that

such a choice of change of measure is the optimal change of measure, in terms

of minimizing EQ(L2(X);X ≥ αp), among all the change of measures under

which Zi’s are i.i.d..

We generate n i.i.d. replicates of (Z1, ..., Z10) fromQθ∗ , that is, (Z(k)1 , ..., Z

(k)10 )

for k = 1, ..., n; then, use Xk =∑10

i=1 Z(k)i , k = 1, ..., n and the associated

weights to form an empirical distribution and further α1−p(X). Let σ2n =

V arQθ∗ (α1−p(X)), σ2n be the approximation of σ2n in Theorem 1, and σ2n be

the bootstrap estimator of σ2n in Theorem 2. We use Monte Carlo to compute

both σ2n and σ2n by generating independent replicates of αp(X) under Q and

bootstrap samples under Q respectively. In particular, they are computed via

1000 independent Monte Carlo simulations with which the Monte Carlo error

is small enough to be ignored. The variance estimates are reported in Table 1.

With sample size n = 10000, the bootstrap variance estimator is very closed to

the (estimated) true variance and its asymptotic approximation. Nonetheless,

when computable, the asymptotic approximation σ2n provides a more accurate

estimate of σ2n.

4. Proof of Theorem 1

Throughout our discussion we use the following notations for asymptotic

behavior. We say that 0 ≤ g(b) = O(h(b)) if g(b) ≤ ch(b) for some constant

c ∈ (0,∞) and all b ≥ b0 > 0. Similarly, g(b) = Ω(h(b)) if g(b) ≥ ch(b) for all

b ≥ b0 > 0. We also write g(b) = Θ(h(b)) if g(b) = O(h(b)) and g(b) = Ω(h(b)).

Finally, g(b) = o(h(b)) as b→∞ if g(b)/h(b)→ 0 as b→∞.

Before the proof of Theorem 1, we first present a few useful lemmas.


Lemma 1. X is random variable with finite second moment, then

EX =

∫x>0

P (X > x)dx−∫x<0

P (X < x)dx

EX2 =

∫x>0

2xP (X > x)dx−∫x<0

2xP (X < x)dx

Proof of Lemma 1. For each nonnegative random variable X,

EX =

∫x>0

xP (dx) =

∫x>0

∫0<u<x

duP (dx) =

∫u>0

P (X > u)du,

and

EX2 =

∫x>0

x2P (dx) =

∫x>0

∫0<u<x

2uduP (dx) =

∫u>0

2uP (X > u)du.

For a general random variable, let X = max(X, 0)−max(−X, 0) and apply the

above derivation to the positive part and negative part separately. Thereby, we

conclude the proof.

Lemma 2. Let X1, ..., Xn be i.i.d. random variables with EXi = 0 and E|Xi|α <

∞ for some α > 2. For each ε > 0, there exists a constant κ depending on ε,

EX2i and E|Xi|α such that

E

∣∣∣∣∣n∑i=1

Xi/√n

∣∣∣∣∣α−ε

≤ κ

for all n > 0.

Proof of Lemma 2. Let σ2 = EX21 . According to Theorem 2.18 in [16], we

obtain that for all δ > 0, x > 0, and n > 1

P

(n∑i=1

Xi >√nx

)≤ P ( max

1≤i≤nXi > δ

√nx) +

(3

1 + δσ−2x2

)1/δ

.

One can first choose δ small enough such that the second term decays fast

enough. For the first term, by Chebyshev inequality and the fact that α > 2,

for x > 1 and n > 1

P

(max1≤i≤n

Xi > δ√nx

)= 1− P (X1 ≤ δ

√nx)n

≤ 1− (1− κ0n−α/2x−α)n

≤ κ1x−α.


Therefore, the tail probability has a bound

P

(n∑i=1

Xi >√nx

)≤ κ1x−α +

(3

1 + δσ−2x2

)1/δ

,

which is free of n. For the cases that P (Sn < −√nx), the development is

completely analogous. Then, the conclusion follows.

Lemma 3. Let h(x) be a non-negative function. There exists ζ0 > 0 such that

h(x) ≤ xζ0 for all x sufficiently large. Then, for all ζ1, ζ2, λ > 0 such that

λ(ζ1 − 1) < ζ2, we obtain∫ nλ

0h(x)Φ(−x+ o(xζ1n−ζ2))dx =

∫ nλ

0h(x)Φ(−x)dx+ o(n−ζ2),

as n→∞, where Φ is the c.d.f. of a standard Gaussian distribution.

Proof of Lemma 3. We first split the integral into∫ nλ

0h(x)Φ(−x+ o(xζ1n−ζ2))dx

=

∫ (logn)2

0h(x)Φ(−x+ o(xζ1n−ζ2))dx+

∫ nλ

(logn)2h(x)Φ(−x+ o(xζ1n−ζ2))dx.

Note that the second term∫ nλ

(logn)2h(x)Φ(−x+ o(xζ1n−ζ2))dx ≤ nλ(ζ0+1)Φ(− (log n)2 /2) = o(n−ζ2).

For the first term, note that for all 0 ≤ x ≤ (log n)2,

Φ(−x+ o(xζ1n−ζ2)) = (1 + o(xζ1+1n−ζ2))Φ(−x).

Then∫ (logn)2

0h(x)Φ(−x+ o(xζ1n−ζ2))dx = (1 + o(n−ζ2))

∫ (logn)2

0h(x)Φ(−x)dx.

Therefore, the conclusion follows immediately.


Proof of Theorem 1. Let αp(X) be defined in (2). To simplify the notation,

we omit the index X and write αp(X) as αp. We use Lemma 1 to compute the

moments. In particular, we need to approximate the following probability,

Q(n1/2(αp − αp) > x

)= Q(FX(αp + xn−1/2) < p) (11)

= Q

(n∑i=1

L(Xi)(I(Xi ≤ αp + xn−1/2)− p

)< 0

).

For some λ ∈ ( 14(α−2) ,

18), we provide approximations for (11) of the following

three cases: 0 < x ≤ nλ, nλ ≤ x ≤ c√n, and x >

√n. The development for

Q(n1/2(αp − αp) < x)

on the region that x ≤ 0 is the same as that of the positive side.

Case 1: 0 < x ≤ nλ.

Let

Wx,n,i = L(Xi)(I(Xi ≤ αp + xn−1/2)− p

)− F (αp + xn−1/2) + p.

According to Berry-Esseen bound (c.f. [20]),

Q

(n∑i=1

L(Xi)(I(Xi ≤ αp + xn−1/2)− p

)< 0

)

= Q

1n

∑ni=1Wx,n,i√

1nV ar

QWx,n,1

< −F (αp + xn−1/2)− p√1nV ar

QWx,n,1

= Φ

− (F (αp + xn−1/2)− p)√

1nV ar

QWx,n,1

+D1(x). (12)

There exists a constant κ1 such that

|D1(x)| ≤ κ1

(V arQWx,n,1)3/2

n−1/2.


Case 2: nλ ≤ x ≤ c√n.

Thanks to Lemma 2, for each ε > 0, the (α−ε)-th moment of 1√n

∑ni=1Wx,n,i

is bounded. By Chebyshev’s inequality, we obtain that

Q

(n∑i=1

L(Xi)(I(Xi ≤ αp + xn−1/2)− p) < 0

)= Q

(1√n

n∑i=1

Wx,n,i ≤√n(p− F (αp + x/

√n)))

≤ κ1

(1

√n(F (αp + xn−1/2)− p)

)α−ε≤ κ2x−α+ε.

Since λ > 14(α−2) , we choose ε small enough such that

∫ c√n

nλxQ(αp − αp > xn−1/2)dx = O(n−λ(α−2−ε)) = o(n−1/4). (13)

Case 3: x > c√n.

Note that

Q

(n∑i=1

L(Xi)(I(Xi ≤ αp + xn−1/2)− p) < 0

)

is a non-increasing function of x. Therefore, for all x > c√n, from Case 2, we

obtain that

Q

(n∑i=1

L(Xi)(I(Xi ≤ αp + xn−1/2)− p) < 0

)≤ κ2(c

√n)−α+ε = κ3n

−α/2+ε/2.

For c√n < x ≤ nα/6−ε/6, we have that

Q

(n∑i=1

L(Xi)(I(Xi ≤ αp + xn−1/2)− p) < 0

)≤ κ3n−α/2+ε/2 ≤ κ3x−3.

In addition, note that for all xβ−3 > n1+β/2,

Q(αp > αp + x/

√n)≤ Q(sup

iXi > αp + xn−1/2) = 1−Gn(αp + xn−1/2)

≤ O(1)n1+β/2x−β = O(x−3).

Therefore, Q (αp > αp + x/√n) = O(x−3) on the region c

√n < x ≤ nα/6−ε/3∪

x > nβ+2

2(β−3) . Since α3 > β+2

β−3 , one can choose ε small enough such that


x > nα/6−ε/6 implies xβ−3 > n1+β/2. Therefore, for all x > c√n, we obtain

that

Q(αp > αp + x/

√n)≤ x−3,

and ∫ ∞c√nxQ(αp > αp + x/

√n)

= O(n−1/2). (14)

A summary of Cases 1, 2, and 3.

Summarizing the Cases 2 and 3, more specifically (13) and (14), we obtain

that ∫ ∞nλ

xQ(αp > αp + x/

√n)dx = o(n−1/4).

Using the result in (12), we obtain that∫ ∞0

xQ(αp > αp + x/

√n)dx

=

∫ nλ

0x

Φ

− (F (αp + xn−1/2)− p)√

1nV ar

QWx,n,1

+O(n−1/2)

dx+ o(n−1/4)

=

∫ nλ

0xΦ

− (F (αp + xn−1/2)− p)√

1nV ar

QWx,n,1

dx+O(n2λ−1/2) + o(n−1/4).(15)

Given that λ < 18 , we have that O(n2λ−1/2) = o(n−1/4). Thanks to condition

A4 and the fact that V arQ(Wx,n,1) = (1 +O(xn−1/2))V arQ(W0,n,1), we have

−(F (αp + xn−1/2)− p

)√1nV ar

QWx,n,1

= − xf(αp)√V arQW0,n,1

+O(x3

2+δ0n−1/4−δ0/2).

Insert this approximation to (15). Together with the results from Lemma 3, we

obtain that∫ nλ

0xQ(αp > αp + x/

√n)dx =

∫ nλ

0xΦ

(− xf(αp)√

V arQW0,n,1

)dx+ o(n−1/4).


Therefore∫ ∞0

xQ (αp − αp > x) dx =1

n

∫ ∞0

xQ(αp > αp + x/

√n)dx

=1

n

∫ ∞0

xΦ

(− xf(αp)√

V arQW0,n,1

)dx+ o(n−5/4)

=V arQW0,n,1

nf2(αp)

∫ ∞0

xΦ(−x)dx+ o(n−5/4)

=V arQW0,n,1

2nf2(αp)+ o(n−5/4).

Similarly,∫ ∞0

Q (αp > αp + x) dx =1√n

∫ ∞0

Q(αp > αp + x/

√n)dx

=1√n

∫ ∞0

Φ

(− xf(αp)√

V arQW0,n,1

)dx+ o(n−3/4).

For Q(αp < αp − x) and x > 0, the approximations are completely the same

and therefore are omitted. We summarize the results of x > 0 and x ≤ 0 and

obtain that

EQ (αp − αp)2 =

∫ ∞0

xQ (αp > αp + x) dx+

∫ ∞0

xQ (αp < αp − x) dx

= n−1[V arQW0,n,1

f2(αp)+ o(n−1/4)

]EQ (αp − αp) =

∫ ∞0

Q (αp > αp + x) dx−∫ ∞0

Q (αp < αp − x) dx

= o(n−3/4).

5. Proof of Theorem 2

We first present a lemma that localizes the event. This lemma can be proven

straightforwardly by standard results of empirical processes (c.f. [33, 34, 35])

along with the strong law of large numbers and the central limit theorem.

Therefore, we omit it. Let Y1, ..., Yn be i.i.d. bootstrap samples and Y be a

generic random variable equal in distribution to Yi. Let Q be the probability

measure associated with the empirical distribution G(x) = 1n

∑ni=1 I(Xi ≤ x).


Lemma 4. Let Cn be the set in which the following events occur

E1 EQ|L(Y )|ζ < 2EQ|L(X)|ζ for ζ = 2, 3, α; EQ|L(Y )|2 > 12E

Q|L(X)|2;

EQ|X|β ≤ 2EQ|X|β.

E2 Suppose that αp = X(r). Then, assume that |r/n−G(αp)| < n−1/2 log n

and |αp − αp| < n−1/2 log n.

E3 There exists δ ∈ (0, 1) such that for all 1 < x <√n

δ ≤∑n

i=1 I(X(i) ∈ (αp, αp + xn−1/2])

nQ(αp < X ≤ αp + n−1/2x))≤ δ−1,

and

δ ≤∑n

i=1 I(X(i) ∈ (αp − xn−1/2, αp])nQ(αp − n−1/2x < X ≤ αp)

≤ δ−1.

Then,

limn→∞

Q(Cn) = 1.

Lemma 5. Under conditions A1 and A5, let Y be a random variable following

c.d.f. G. Then, for each λ ∈ (0, 1/2)

sup|x|≤cnλ−1/2

∣∣∣V arQ[L(Y )(I(Y ≤ αp + x)− p)]− V arQ[L(X)(I(αp + x)− p)]∣∣∣ = Op(n

−1/2+λ).

Proof of Lemma 5. Note that

EQL2(Y )(I(Y ≤ αp + x)− p)2 =1

n

n∑i=1

L2(Xi)(I(Xi ≤ αp + x)− p)2

=1

n

n∑i=1

L2(Xi)(I(Xi ≤ αp + x)− p)2

+L2 (αp)Op

(1

n

n∑i=1

I(min(αp, αp) ≤ Xi − x ≤ max(αp, αp))

).

For the first term, by central limit theorem, continuity of L(x), and Taylor’s

expansion, we obtain that

sup|x|≤cn−1/2+λ

∣∣∣∣∣ 1nn∑i=1

L2(Xi)(I(Xi ≤ αp + x)− p)2 − EQ(L2(X)(I(X ≤ αp + x)− p)2

)∣∣∣∣∣ = Op(n−1/2+λ).


Thanks to the weak convergence of empirical measure and αp−αp = O(n−1/2),

we have that the second term

L2 (αp)Op

(1

n

n∑i=1

I(min(αp, αp) ≤ Xi − x ≤ max(αp, αp))

)= Op(n

−1/2).

Therefore,

sup|x|≤cn−1/2+λ

∣∣∣EQL2(Y )(I (Y ≤ αp + x)− p)2 − EQ(L2(X)(I(X ≤ αp + x)− p)2)∣∣∣ = Op(n

−1/2+λ).

With a very similar argument, we have that

sup−cn−1/2≤x≤cn−1/2

∣∣∣EQL(Y )(I (Y ≤ αp + x)− p)− EQ(L(X)(I(X ≤ αp + x)− p))∣∣∣ = Op(n

−1/2+λ).

Thereby, we conclude the proof.

Proof of Theorem 2. Let X(1), ..., X(n) be the order statistics of X1, ..., Xn in

an ascending order. Since we aim at proving weak convergence, it is sufficient

to consider the case that X ∈ Cn as in Lemma 4. Throughout the proof, we

assume that X ∈ Cn.

Similar to the notations in the proof of Theorem 1, we write αp(X) as αp and

keep the notation αp(Y) to differentiate them. We use Lemma 1 to compute

the second moment of αp(Y)− αp under Q, that is,

σ2n =

∫ ∞0

xQ(αp(Y) > αp + x)dx+

∫ ∞0

xQ(αp(Y) < αp − x)dx.

We first consider the case that x > 0 and proceed to a similar derivation as

that of Theorem 1. Choose λ ∈ ( 14(α−2) ,

18).

Case 1: 0 < x ≤ nλ.

Similar to the proof of Theorem 1 by Berry-Esseen bound, for all x ∈ R

Q(n1/2(αp(Y)− αp) > x

)= Φ

−∑ni=1 L(Xi)

(I(Xi ≤ αp + xn−1/2)− p

)√nV arQWx,n

+D2,


where

Wx,n = L(Y )(I(Y ≤ αp + xn−1/2)− p

)− 1

n

n∑i=1

L(Xi)(I(Xi ≤ αp + x/

√n)− p

),

and (thanks to E1 in Lemma 4)

|D2| ≤3EQ

∣∣∣Wx,n

∣∣∣3√n(V arQWx,n

)3/2 = O(n−1/2).

In what follows, we further consider the cases that x > nλ. We will essentially

follow the Cases 2 and 3 in the proof of Theorem 1.

Case 2: nλ ≤ x ≤ c√n.

Note thatn∑i=1

L(Xi) (I(Xi ≤ αp)− p) = O(1).

With exactly the same argument as in Case 2 of Theorem 1 and thanks to E1

in Lemma 4, we obtain that for each ε > 0

Q(αp(Y)− αp > xn−1/2

)≤ κ

(1√n

n∑i=1

L(Xi)(I(Xi ≤ αp + x/

√n)− p

))−α+ε

= κ

(1√n

n∑i=1

L(Xi)I(αp < Xi ≤ αp + x/√n) +O(1/

√n)

)−α+εFurther, thanks to E3 in Lemma 4, we have

Q(αp(Y)− αp > xn−1/2

)= O(x−α+ε).

With ε sufficiently small, we have∫ √nnλ

xQ(αp(Y)− αp > xn−1/2

)dx = O(n−λ(α−ε−2)) = o(n−1/4).

Case 3: x > c√n.

Note that

Q

(n∑i=1

L(Yi)(I(Yi ≤ αp + xn−1/2)− p

)< 0

),


is a monotone non-increasing function of x. Therefore, for all x > c√n, from

Case 2, we obtain that

Q

(n∑i=1

L(Yi)(I(Yi ≤ αp + xn−1/2)− p

)< 0

)≤ κ3n−α/2+ε/2.

For x ≤ nα/6−ε/6, we obtain that

Q

(n∑i=1

L(Yi)(I(Yi ≤ αp + xn−1/2)− p

)< 0

)≤ κ3n−α/2+ε/2 ≤ κ3x−3.

Thanks to condition A3, with ε sufficiently small, we have that x > nα/6−ε/6

implies that xβ−3 > n1+β/2. Therefore, because of E1 in Lemma 4, for all

x > nα/6−ε/6 (therefore xβ−3 > n1+β/2)

Q(α(Y) > αp+xn−1/2) ≤ Q(sup

iYi > αp+xn

−1/2) = O(1)n1+β/2x−β = O(x−3)

Therefore, we have that

∫ ∞c√nxQ(αp(Y)− αp > xn−1/2

)dx = O(n−1/2).

Summary of Cases 2 and 3.

From the results of Cases 2 and 3, we obtain that for X ∈ Cn

∫ ∞nλ

xQ(αp(Y) > αp + x/√n)dx = o(n−1/4). (16)

With exactly the same proof, we can show that

∫ ∞nλ

xQ(αp(Y) < αp − x/√n)dx = o(n−1/4). (17)


Case 1 revisit.

Cases 2 and 3 imply that the integral in the region where |x| > nλ can be

ignored. In the region 0 ≤ x ≤ nλ, on the set Cn, for λ < 1/8, we obtain that∫ nλ

0xQ(αp(Y) > αp + x/

√n)dx

=

∫ nλ

0x

Φ

−∑ni=1 L(Xi)

(I(Xi ≤ αp + xn−1/2)− p

)√nV arQWx,n

+D2

dx=

∫ nλ

0xΦ

−∑ni=1 L(Xi)

(I(Xi ≤ αp + xn−1/2)− p

)√nV arQWx,n

dx

+o(n−1/4). (18)

We now take a closer look at the integrand. Note that

n∑i=1

L(Xi)(I(Xi ≤ αp + xn−1/2)− p

)=

n∑i=1

L(Xi) (I(Xi ≤ αp)− p) +

n∑i=1

L(Xi)I(αp < Xi ≤ αp + xn−1/2).(19)

Suppose that αp = X(r). Then,

r∑i=1

L(X(i)) ≥ pn∑i=1

L(Xi), and

r−1∑i=1

L(X(i)) < p

n∑i=1

L(Xi).

Therefore,

p

n∑i=1

L(Xi) ≤n∑i=1

L(Xi)I(Xi ≤ αp) < L(αp) + p

n∑i=1

L(Xi).

We plug this back to (19) and obtain that

n∑i=1

L(Xi)(I(Xi ≤ αp + xn−1/2)− p

)= O(L(αp)) +

n∑i=1

L(Xi)I(αp < Xi ≤ αp + xn−1/2). (20)

In what follows, we study the dominating term in (18) via (20). For all


x ∈ (0, nλ), thanks to (20), we obtain that

Φ

−∑ni=1 L(Xi)I(Xi ≤ αp + xn−1/2)− np√

nV arQWx,n

= Φ

−∑ni=1 L(Xi)I(αp < Xi ≤ αp + xn−1/2)√

nV arQWx,n

+O(n−1/2)

. (21)

Note that the above display is a functional of (X1, ..., Xn) and is also a stochastic

process indexed by x. In what follows we show that it is asymptotically a

Gaussian process. The distribution of (21) is not straightforward to obtain.

The strategy is to first consider a slightly different quantity and then connect it

to (21). For each (x(r), r) such that |x(r)−αp| ≤ n−1/2 log n and |r/n−G(αp)| ≤

n−1/2 log n, conditional on X(r) = x(r), X(r+1), ..., X(n) are equal in distribution

to the order statistics of (n−r) i.i.d. samples from Q(X ∈ ·|X > x(r)). Thanks

to the fact that L(x) is locally Lipschitz continuous and E3 in Lemma 4, we

obtain

Φ

−∑ni=r+1 L(X(i))I(x(r) < X(i) ≤ x(r) + xn−1/2)√

nV arQWx,n

+O(n−1/2)

= Φ

− L(x(r))√nV arQWx,n

n∑i=r+1

I(X(i) ∈ (x(r), x(r) + xn−1/2]) +O(x2n−1/2)

.

Note that the above display equals to (21) if αp = X(r) = x(r). For the time

being, we proceed by conditioning only on X(r) = x(r) and then further derive

the conditional distribution of (21) given αp = X(r) = x(r). Due to Lemma 5,

we further simplify the denominator and the above display equals to

Φ

(−

L(x(r))√nV arQW0,n

n∑i=r+1

I(X(i) ∈ (x(r), x(r) + xn−1/2]) +O(x2n−1/2+λ)

).

(22)

Let

Gx(r)(x) =

G(x(r) + x)−G(x(r))

1−G(x(r))= Q(X ≤ x(r) + x|X > x(r)).


Thanks to the result of strong approximation ([33, 34, 35]), given X(r) = x(r),

there exists a Brownian bridge B(t) : t ∈ [0, 1], such that

n∑i=r+1

I(X(i) ∈ (x(r), x(r) + xn−1/2])

= (n− r)Gx(r)(xn−1/2) +

√n− rB(Gx(r)

(xn−1/2)) +Op(log(n− r)),

where the Op(log(n − r)) is uniform in x. Again, we can localize the event by

considering a set in which the error term in the above display is O(log(n−r))2.

We plug this strong approximation back to (22) and obtain

Φ

(−


(n− r)Gx(r)(xn−1/2) +Op(x

2n−1/2+λ(log n)2)

)(23)

−ϕ

(−


(n− r)Gx(r)(xn−1/2) +Op(x

2n−1/4(log n)2)

)

×(n− r)1/2L(x(r))√

nV arQW0,n

B(Gx(r)(xn−1/2)).

In addition, thanks to condition A4,


(n− r)Gx(r)(xn−1/2) =

f(x(r))√V arQW0,n

x+O(xδ0+3/2n−1/4−δ0/2).

(24)

Let

ξ(x) =(n− r)1/2L(x(r))√

nV arQW0,n

B(Gx(r)(xn−1/2)) (25)

which is a Gaussian process with mean zero and covariance function

Cov(ξ(x), ξ(y)) =(n− r)L2(x(r))

nV arQW0,nGx(r)

(x/√n)(1−Gx(r)

(y/√n))

= (1 +O(n−1/4+λ/2))L(x(r))f(x(r))

V arQW0,n

x√n

(26)

for 0 ≤ x ≤ y ≤ nλ. Insert (24) and (25) back to (23) and obtain that given


X(r) = x(r)

∫ nλ

0Φ


nV arQWx,n

+O(n−1/2)

dx

=

∫ nλ

02xΦ

(−

f(x(r))√V arQW0,n

x+ o(x2n−1/4)

)dx (27)

−∫ nλ

02xϕ

(−

f(x(r))√V arQW0,n

x+ o(x2n−1/4)

)ξ(x)dx+ o(n−1/4),

where ϕ(x) is the standard Gaussian density function. Due to Lemma 3 and

|x(r) − αp| ≤ n−1/2 log n, the first term on the right side of (27) is

∫ nλ

02xΦ

(−

f(x(r))√V arQW0,n

x+ o(x2n−1/4)

)dx (28)

= (1 + o(n−1/4))

∫ ∞0

2xΦ

(− f(αp)√

V arQW0,n

x

)dx+ op(n

−1/4).

The second term on the right side of (27) converges weakly to a Gaussian

distribution with mean zero and variance

∫ nλ

0

∫ nλ

0xyϕ

(−

f(x(r))√V arQW0,n

x+ o(x2n−1/4)

)

ϕ

(−

f(x(r))√V arQW0,n

y + o(x2n−1/4)

)Cov(ξ(x), ξ(y))dxdy

=1 + o(1)√

n

∫ nλ

0

∫ nλ

04xyϕ

(−

f(x(r))√V arQW0,n

x

)ϕ

(−

f(x(r))√V arQW0,n

y

)

×L(x(r))f(x(r))

V arQW0,nmin(x, y)dxdy

= (1 + o(1))n−1/2L(αp)

(V arQW0,n

)3/2f4(αp)

√π

. (29)

We insert the estimates in (28) and (29) back to (27) and obtain that conditional


on X(r) = x(r),

n1/4∫ nλ

0Φ


nV arQWx,n

+O(n−1/2)

dx

−∫ ∞0

2xΦ

(− f(αp)√

V arQW0,n

x

)dx

=⇒ N(0, τ2p /2) (30)

as n − r, r → ∞ subject to the constraint that |r/n − G−1(αp)| ≤ n−1/2 log n,

where τ2p is defined in the statement of the theorem. One may consider that the

left-hand-side of (30) is indexed by r and n− r. The limit is in the sense that

both r and n−r tend to infinity in the region that |r/n−G−1(αp)| ≤ n−1/2 log n.

The limiting distribution of (30) conditional on αp = X(r) = x(r).

We now consider the limiting distribution of the left-hand-side of (30) further

conditional on αp = X(r) = x(r). To simplify the notation, let

Vn = n1/4∫ nλ

02xϕ

(− f(αp)√

nV arQW0,n

x+ o(x2n−1/4)

)

L(αp)

∑ni=r+1 I(x(r) < X(i) ≤ x(r) + xn−1/2)− (n− r)Gx(r)

(xn−1/2)√nV arQW0,n

dx.

Then,∫ nλ

0Φ


nV arQWx,n

+O(n−1/2)

dx

−∫ ∞0

2xΦ

(− f(αp)√

V arQW0,n

x

)dx

= n−1/4Vn + o(n−1/4).

The weak convergence result in (30) says that for each compact set A,

Q(Vn ∈ A|X(r) = x(r))→ P (Z ∈ A),

as n − r, r → ∞ subject to the constraint that |r/n − G−1(αp)| ≤ n−1/2 log n,

where Z is a Gaussian random variable with mean zero and variance τ2p /2. Note


that αp = X(r) = x(r) is equivalent to

0 ≤ H =

r∑i=1

L(X(i))(1− p)− pn∑

i=r+1

L(X(i)) ≤ L(x(r)).

Let

Un =

n∑i=r+1

I(x(r) < X(i) ≤ x(r) + nλ−1/2)− (n− r)Gx(r)(nλ−1/2)

and

Bn =|Un| ≤ nλ/2+1/4 log n

.

Note that, given the partial sum Un, H is independent of the Xi’s in the interval

(x(r), x(r) + nλ−1/2) and therefore is independent of Vn. For each compact set

A and An = Vn ∈ A ∩Bn, we have

Q(An|αp = X(r) = x(r)

)(31)

=Q(0 ≤ H ≤ L(x(r))|X(r) = x(r), An

)Q(0 ≤ H ≤ L(x(r))|X(r) = x(r)

) Q(An|X(r) = x)

= EQ

[Q(0 ≤ H ≤ L(x(r))|X(r) = x(r), Un

)Q(0 ≤ H ≤ L(x(r))|X(r) = x(r)

) ∣∣∣∣∣X(r) = x(r), An

]Q(An|X(r) = x(r))

The second step of the above equation uses the fact that on the set Bn

Q(0 ≤ H ≤ L(x(r))|X(r) = x(r), Un

)= Q

(0 ≤ H ≤ L(x(r))|X(r) = x(r), Un, An

).

Note that Un only depends on the Xi’s in (x(r), x(r) + nλ−1/2), while H is the

weighted sum of all the samples. Therefore, on the setBn =|Un| ≤ nλ/2+1/4 log n

Q(0 ≤ H ≤ L(x(r))|X(r) = x(r), Un

)Q(0 ≤ H ≤ L(x(r))|X(r) = x(r)

) = 1 + o(1), (32)

and the o(1) is uniform in Bn. The rigorous proof of the above approximation

can be developed using the Edgeworth expansion of density functions straight-

forwardly, but is tedious. Therefore, we omit it. We plug (32) back to (31).

Note that Q(Bn|X(r) = x(r))→ 1 and we obtain that for each A

Q(Vn ∈ A|αp = X(r) = x(r)

)−Q(Vn ∈ A|X(r) = x(r))→ 0.


Therefore, we obtain that conditional on αp = X(r), |αp − αp| ≤ n−1/2 log n,

and |r/n−G−1(αp)| ≤ n−1/2 log n, as n→∞

n1/4

[∫ nλ

0Q(αp(Y) > αp + x/

√n)dx−

∫ ∞0

2xΦ

(− f(αp)√

V arQW0,n

x

)dx

]

= n1/4∫ nλ

0Φ

−∑ni=r+1 L(X(i))I(αp < X(i) ≤ αp + xn−1/2)√

nV arQWx,n

+O(n−1/2)

dx

−∫ ∞0

2xΦ

(− f(αp)√

V arQW0,n

x

)dx

+ op(1)

= Vn + op(1) =⇒ N(0, τ2p /2).

Together with E2 in Lemma 4, this convergence indicates that asymptotically

the bootstrap variance estimator is independent of αp. Therefore, the uncon-

ditional asymptotic distribution is

n1/4

[∫ nλ

0Q(αp(Y) > αp + x/

√n)dx−

∫ ∞0

2xΦ

(− f(αp)√

V arQW0,n

x

)dx

]=⇒ N(0, τ2p /2).

(33)

With exactly the same argument, we have the asymptotic distribution of the

negative part of the integral

n1/4

[∫ nλ

0Q(αp(Y) < αp − x/

√n)dx−

∫ ∞0

2xΦ

(− f(αp)√

V arQW0,n

x

)dx

]=⇒ N(0, τ2p /2).

(34)

Using a conditional independence argument, we obtain that the negative part

and the positive part of the integral are asymptotically independent. Putting

together the results in Theorem 1, (16), (17), (33), (34), and the moment

calculations of Gaussian distributions, we conclude that

σ2n =

∫ ∞0

2x[Q (αp(Y) < αp − x) + Q (αp(Y) > αp + x)

]dx

=1

n

∫ ∞0

2x[Q(αp(Y) < αp − x/

√n)

+ Q(αp(Y) > αp + x/

√n)]dx

d=

V arQ(Wp)

nf(αp)2+ Zn−5/4 + o(n−5/4)

= σ2n + Zn−5/4 + o(n−5/4).


where Z ∼ N(0, τ2p ).

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