Mathematical Notes, VoL 68, No. 3, ~000

Normal Approximation for Linear Stochastic Processes and Random Fields in Hilbert Space

A. N. Nazarova UDC 519.214

ABSTRACT. The central limit theorem is proved for linear random fields defined on an integer-valued lattice of arbitrary dimension and taking values in Hilbert space. I t is shown that the conditions in the central limit theorem are optimal.

KEY WOl~S: central llmlt theorem, Linear random field, second moment.

The study of the limiting behavior of snm.q of random variables with values in Banach or Hilbert spaces is the traditional domain of probability theory having a number of statistical applications. In the present paper we study sum.q generated by linear random fields with values in Hilbert space. Such sum.q arise in estimation and prediction problems for some classes of random processes and fields (see, for example, [1]). The goal of the present paper is to prove the central limit theorem (CLT) for linear fields defined on an integer-valued lattice Z d , where is d _> 1 arbitrary. This extends the results of the recent paper [2]. It is also proved that the conditions given in this paper are optimal.

We need a number of definitions. Suppose that H is a separable real Hilbert space with norm ]] �9 ]]H induced by the inner product (- , - )

and E(H) is the space of linear bounded operators f r o m / - / t o H . Suppose that {~k, k E Z ~} is the field of random variables defined on a probability space (fl, ~ ' , P) and taking values in H , while {ak, k E Zd} is the family of linear operators from L:(H).

By a linear process (or field) we mean a set of random variables of the following form:

Xk = E a j (~k- j ) , k E Z d. (1) jEZ ~

Here and further, we assume that the random variables ~k are independent, identically distributed, with mean zero, EH~0[[ 2 < oo, and ~ [[aj[I 2 < co. Then it is readily seen (see [3, Chap. 3.2]) that the series (1) is convergent almost surely in the space s (fl) consisting of functions mapping fl into H and possessing the integrable square of the norm.

Let us define the partial sums

S,,= ~ xh, n > l . l~/e_<n

For the vectors k = ( k l , . . . , ks) and n = ( n l , . . . , n~) the inequality k ~ n means that k~ < n~ for all i = l , . . . , d .

For the simplest case in which the random variables ~ are real-valued, the numbers ak are real, and the index k E Z, i.e., d = 1, the behavior of Sn is well studied. (For such random variables instead of aj(~k_j) in (1) we write a j~k- j .) For example, the following theorem is well known.

T h e o r e m 1 (see [4, Theorem 18.6.5]). Suppose that {~k}keZ are independent, identically distributed, zero-mean random variables with a finite second moment, {ak}kez is a sequence of real numbers such that

O 0

< (2) k---~--oo

Translated from Matematicheskie Zamethi, Vol. 68, No. 3, pp. 421-428, September, 2000. Original article submitted December 27, 1999.

0001-4346/2000/6834-0363525.00 (~)2000 Kluwer Academic/Plenum Publishers 363

and Xk , Sn are as defined above. Then i f s 2 -+ oo , n -+ c r then

s~ -~ x(o, ~).

For the case in which the ~k assume values in an infinite-dimensional space, the situation is essentially different. So in Theorem 3 from [2] it was shown that condition (2) (with lakl replaced by HakHr(H)) does

not ensure even the density of the sequence of distributions of S n / ~ 2 . For the CLT with normalization by v/n to be valid, it turns out that the following stronger condition

is sufficient (see [2, Theorem 2]) CO

Ila~ll,~c_~) < ~ - (3) k=-co

In the present paper, the CLT is proved for random variables taking values in Hilbert space and indexed by the multivariate index k 6 Z d (see Theorem 2). The sufficient condition turns out to be similar to (3). Using the ideas of [2], it is shown that the obtained condition is sharp (see Theorem 3).

For any vector n = (nl , . . . , rtd) 6 N d, set In[ = n l ' ' ' r i d .

T h e o r e m 2. Suppose that {~k, k E Z a} is the random field described above. Let

Ila~ll < ~. (4) j E Z a

T h e n

1 -~ Af(o, AC~oA* ) n ~ co, (5) V S" where A = ~'~jez d at , A* is the adjoint operator for A , C~o is the covariance operator of ~o , and n -~ co means that [hi -+ co w/th all the coordinates nondecreasing.

To prove this, we need the following auxiliary result.

L e m m a 1. Suppose that {bj}jeza are elements of some normalized space satisfying

IIb~ll < oo , (6) j E Z ~

b~ = o. (~) j E Z a

Then

n -+co. (8)

By the notation [-x, x] for any vector x = (zl, ..., Xd) E R d we mean the rectangle in Ra:

[-x, x] = [-xl, xl] • • [-~, ~d]-

Likewise,

(--X, X) ---- (--ZI~ Zl) X--- • (--Xd, Zd).

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P r o o f o f L e m m a 1. Denote D~ =

By (6) we have D. -~ 0 as n -+ c~. Now we note that

IIb~ll-

1 II ( ) , I,~ E E b, < ~ E E IIb, II j ~ ( - - 2 n , 2 n ) 1--j<_i<_n--j j~ ( - -2n ,2n ) 1--j<_i<n--~

<-lnl ~ Ilbdl llbdl

<_ D . - ~ x- j_ _ , , - j j~zd n - + oo; (9)

here we have again taken into account inequality (6). Consider the f tm~ion

2

~ b,

for all z E [ - 2 , 2]. Here [-] stands for the integral part . For a vector from ]R d , the integral part is calculated coordinate-wise. Note that it follows from (6) and (7) that for all z ~ 0 h,,(~v) -+ 0 as n --+ oo and ,,)2

Ih.(a:)l <_ Ilb~-[t.,=] _< IIb~ < ~ .

I_ _ n IEZ d

Then, by Lebesgue's theorem on majorized convergence, we have

je [ -2n ,2n- l j 1-j_i_n-j

~q{--2.,2~t--11 v - , ~E[-- n - x ] /I, '-I, C i+ I ) / In . I ]

= f[_ hn ( z )dz<_f [_ hn(z)dz- -+O, n - + o o . 2-/I-1 ,n/l-l] 2,2]

(10)

Now from (9) and (10) we obtain (8). The lemma is proved. []

P r o o f o f T h e o r e m 2. First, let us show that under the assumptions of this theorem the following relation is satisfied:

i_<~__n

Indeed,

l<_k<n l<k<_n vr~EZ ct ~EZ I_ _n

Therefore,

i_<k_<, i_<~_<n i_<k_<, j~[i,.] i_ _. jez i<k<n

where bo = ao - A and bl = ai for all i # 0.

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Obviously, the operators {bk}k~Zd satisfy the assumptions of Lemma 1. Then, using (12), we have

l<__~_<n �9 - ' _ ' _ -" 3~Z d 1 3 < z < n 3

n --+ CO.

Further, from the GLT for independent, identically distributed, zero-mean random variables with values in H having a finite second moment (see for example, [5]), we obtain

1

l<~<n n -+ c0.

Since A is a linear operator, we have

1 ~ ~ ~ -~ Jr ~c~oA*), ~-+ o~. (13)

l<k<r,

Now from (11) and (13), using Theorem 4.1 from [6], we obtain (5). The theorem is proved. []

The following theorem shows that condition (4) of Theorem 2 cannot be improved, and hence in the case of an infinite-dimensional space there is no analog of Theorem 1.

Let us agree to call a family of random variables dense if the family of their distributions is dense.

T h e o r e m 3. Condition (4) of Theorem 2 is sharp, i.e., for any se~ of positive numbers tk ( k E 7. d) such that

t~ = oo, ~_~ t 2 < oo, k E Z a k E Z ~t

there exists a field of independent, identically distribuSed, zero-mean random variables {~i, J E Z d} with values in H for which 0 < EII~olI~- < ~ , ~ d there ~-i~ts ~ /~may ol o~rator, {a,~ e Z.(H), k e Z '~} with n o , n Ila~ll = tk ; m o r e o ~ ,

1) the family {S,~/]X/~},~es~ is not dense; 2) there ~ t s a ~ r ~ ' e n ~ { = ~ } ~ N c Z d ,~ch that ~k -+ oo and ellS.~ll~/l=kl -+ ~ as k -* ~ ; 3) the famil~ { S = / ~ } . ~ N " is not dense.

P r o o f . To prove this theorem, we construct the sets of random variables and operators described above. Set

T . = ~ t~, ~ e N ~, z~ = (1 + T<[2~/~] ..... [,~/~D) -~, k e N,

gp = ~/zp - zp+~, p e N .

Suppose that {Np,k}~N, hez~ is an array-of independent, identically distributed, real random variables having the standard normal distribution and {ek}~_z is an orthonormal basis in the space H . Let us define the action of the linear operators {ah} on the basis vectors:

ak(ep) = tke-[Ikll2/Pep = ak~)ep, k E Z d, p >_ 1,

where Ilkll - v/k~ + - - - + k~. Obviously, Ilaklls ---- tk. Let us now define the random variables

CO

F = I

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It is readily seen that F_.~ = 0 and 0 < EIIZ~oll 2 < oo, because

~ 1 1 I E 2 2 E I l ~ o l l = a F - ~ ; = = zl = - - ~ = ~ . = ,o I + T[z/3I 1 + To 1 + to

p = l p = l

T h e n

s.: E E E E E (-) l<k<_n l<k_<n ~EZ d l~k<r* j~Z d p:l p=l

where s,,~) = g,, ~ ~ _,v,,,,,_j,,~). . - .

l < k < n j E Z d

Note tha t for any fixed p >_ 1 the operators {aj(p)}jeza satisfy the assumpt ions of T h e o r e m 2 in the

one-dimensional space H (p) = {Aep}xeR. Indeed,

Ilai@)llctH~,)) = ~ t i e -(~++~='~/~ < 00. . iEZ d j E Z a

Therefore, by Theorem 2 we have

s~(p) _~ ~r(0, ~) , ,-, ~ oo, (15)

where the c~ are some positive numbers. Consider the set

{ EHSn"2 n > l } rn = in----f--, _ �9

Let us choose a sequence nk E N d , nk = ( k , . . . , k), k E l~l, and prove t h a t

(16)

r n u ---). (x) a s k --+ o o . (17)

Indeed,

2

" ~ - In~,l I,',~,--T E ES2k ( p ) = ~ ~ E aj (p)Np, ,_ j ~=ln1,12 Inkl p=ln l= l 2 1 _ _ ~=

p=lm, I 2 ./eZd 1_ _ ~,

= Inkl ~ tle-IMIt2/p ~=l,-,kl = jez,~ x+.i<z<n~=+j

(18)

The cube Aj = {l E Z d : 1 + j < l < nk + j } contains the smaller cube

A = {m ~ Zd: --Ink/3] < m < [n~/3]}

if the following inequalities hold:

(19)

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In coordinate representation this means that

i = l , . . . , d .

Thus the vector ] can assume exactly

values along each coordinate. Since k - 2[k/3] >_ k/3, it follows that the number of different j satisfying system (19) is at least (k/a) ~ = Inkl/3 d. Taking this into account, we continue inequality (18) as follows:

" " -> I,~t ~ g~" " t~e-"="'/" > " " Z l n " l " " T[n"/3] p=lnkl 2 --[.~/3]<_l<[nk/3] - [nkl

1 d T= T= _ _ ~ - 2 / 9 . ([k/a] ... . . [k/s]) = l e _ 2 / 9 ~ ([k/a] ... . . [k/a1) -> 3 d - 1 + T([ 2~/1~kl2/3] ..... [ =d InVfF~/3] ) 3 d " 1 + T([k/3] . . . . . [h/a]) --~ c o ' k - + c o .

Thus we have proved (17), and hence assertion 2) of Theorem 3. Obviously,

s.~ _ s.~ / I~1 (20) ~/EI IS- . I I 2 - Iv~h-~" VEI IS"~I I2"

By the symbol O we denote the identically zero random variable O: ~2 -+ /-/. Recall that by finite- dimensional distributions of a random element ~ : f~ =+ H we mean the fA.mily of distributions of the projections of ff on finite-dimensional subspaces //k C//. Then relations (14)-(17) and (20) imply the following assertion:

A: all finite-dimensional distributions of the sequence {SnJ%/EHS,, ~ 1[2 }hen converge to the ilnite- dimensional distributions of 8.

Suppose that the following assumption is also valid:

B: the sequence {Sn~/~/EHSnkH 2 }ken is dense.

Then, by A and B, this sequence converges in distribution to zero in the same way as to an element of

the Hilbert space /-/ (see [5, Chap. 2.1]). Let us consider the sequence Y~ = l[S=k H=/E[IS,,, ]]2 and prove that it is ,miformly integrable. Indeed,

l ira sup EYk-T{Y~ > c} = l im sup 1 E l lS . . II=Z{llS.. II 2 > cEIIS,~ II 2} r k r ~ EIIS.~II =

< Jim sup 1 . !El lS.~l l4" -o - ,oo k (EIIS.~II2) 2 c

(21)

Since the quantities

IIS~,ll 2 ~ = = = vj~ 9 , j E Z '~ x N

are independent and v~ > 0, we have

r/j ,-~ N'(O, 1),

j E Z d x N (, = ~ ' " 3 ~ 3 (E I IS . . I I= ) = v ] E r / j + 2 Z 2 2 2 2 vi v] Er/~ Er/J < =

j E Z d x N i ~ j EZ d x N

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and we can continue inequality (21). We obtain

3 Jim sup EYkI{Yk > c} _< l lra sup - --- O.

Since Yk -~ 0, k -+ c~, and the random variables {Yk} are uniformly integrable, by Theo rem 5.4 f rom [6] we obtain EYk -+ E0 = 0. But EYk = 1 -~ 0. We have come to a contradiction. Hence the assnmption B is false and the sequence (S,~k/v/EIISnk]] 2 }~N is not dense; it follows tha t t he family ( S ~ / ~ } n ~ Z d is not dense. This proves assertion 3) of Theorem 3.

Further, it is readily seen from (20) that the sequence {S,~/~/-~}k~N is also not dense, which implies assertion 1). Theorem 3 is thereby proved. []

The author wishes to thank Professor A. V. B,,lln~kii for setting the problem and a t t en t ion ~to the au thor ' s work on the paper.

R e f e r e n c e s

1. F. MerlevL~de, "Sur l'inversibilit6 des processus lin6aires ~ valeurs dans un ~pace de Hilbert," Uompt. Rend. Acad. Sci. Paris. Sdr. 1, 321, 477-480 (1995).

2. F. MerlevL~de, M. Peligrad, and S. Utev, "Sharp conditions for the CLT of linear processes in a HUbert space," J. Theor. Probab., 10, No. 3, 681-693 (1997).

3. A. Araujo and E. Gin~, The Centra/Lindt Theorem for Real and Banach-Valued Random Variables, J. Wiley, New York (1980).

4. I. A. Ibragimov and Yu. V. Linnilr Independent and Stationary Dependent Random Variables [in Russian] Naulm, Moscow (1965).

5. M. Ledoux and M. Talagrand, Probability in Banach Space, Springer, Berlin (1991). 6. P. Billlngsley, Conwrgence os Probability Measures, J. Wiley, New York-London-Sydney-Toronto (1968).

M. V. LOMONOSOV MOSCOW STATE UNIVERSITY E-mail: uspenska~issp.ac.ru

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