Non-central limit theorem for non-linear vector functions of Gaussian vector processes

(1)

NON-CENTRAL LIMIT THEOREM FOR NON-LINEAR VECTOR FUNCTIONS OF GAUSSIAN VECTOR

PROCESSES

TAE IL JEON

ABSTRACT. We formulate a non-central limit theorem for non-linear functionals of stationary Gaussian vector processes with dependence.

1.

Introduction

Let X

t

= (Xl, xl) be a stationary Gaussian vector process such that EXI = EX; = 0, E(XI)2 = E(Xl)2 =

1.

Let

G = (Gll _G ^G12)

21 G ₂₂

be the corresponding random spectral matrix. Suppose that G ll ^and

G22 are absolutely continuous. Then so are G12 ^and G21 . ^Let 9a/3()..)

=

d)" Ga/3()..) ,

d

where a,

f3

= 1 or 2. Then we can write the relations between variances and spectral density functions

r

a/3(t)

⁼

EXg xf

⁼

1:

^eitA

^g a/3()")d>-',

where a,

f3

= 1 or 2. Assume the random spectral matrix dG is strictly positive definite. Let Z = ^{(Zl, Z2)} be the corresponding random vector

Received September 1, 1997. Revised July 16, 1999.

1991 Mathematics Subject Classification: 60F05.

Key words and phrases: non-central limit theorem.

This paper was supported by the Basic Science Research Institute Program,Min- istry of Education, Korea, 1996, Project No. BSRI-96-1439.

2 Tae

n

Jean

measure such that

xl = 1: ^e

^it>'dZ1

^{(,X), x; =} 1: ^e

^it>.dZ2

^('x).

Let T = [-1r,1r]. Let B(T) be its Borel subsets. Assume that the JR2-valued noise Z

= (Z1, Z2)

in

T

satisfies the following condition:

For any

A

E B(T),

Z(A)

=

(Zl(A), Z2(A))

is defined on a probabil- ity space (n, F, P) such that for any nonintersecting sets

A1, ... ,An E

B(T), n 2': 1,

Z(A1), ••• ,Z(An)

are independent and

Z(A_{1 )}

+ ... +

Z(An)

=

^Z(Uk=lAk).

Introduce the Hilbert space L

²

(T) of functions

f : T ~

C

²

such that

1

11 f 11=

[l(dG(t)f(t),f(t))r

<

^00.

Since

dG

is strictly positive definite,

L²(T)

is complete if we identify as usual functions which are equal a.e. with respect to the Lebesgue measure. It can be shown that n-tuple tensor product (0L

²

(T)t can be identified with the Hilbert space L

²

(Tn) consisting of all functions

f :

rn - (0C

²

t,

^f

=

(fil,... ,iJil,...,4,=1,2

with finite norm

1

11 f ^II

n

⁼

[L ^L Jj(n)(t(n»)Jj,(n}(t(n»)dGjlj~ ^(t

^{1 ) •••}

dGjnj~ ^(tn)]

²

^<

^00,

j(n) j,(n}=1,2

where

t

^{en) -}

- 1,···, t t

^n,

^J

- ( n ) '^{=J1, ... ,In,}.

in) ⁼ j~,

^...

,j~,

^j(n)

= 1,2 means each

jk

= 1,2.

Symmetric tensor product [0L2(T)]n can be identified with the subspace

-

V(Tn)

C

L

²

(Tn) consisting of symmetric functions: f =

^symf,

where, for fiXed

^j(n)

.. (symf)j(n).(

ten») =~! E !::iO"(i)".·jq(n) (to-(l) , .. " to-(n»)

0-

and the sum is taken over all permutations

^(j

on {1, . .. ,n}.

DEFINITION

1. A simple function f(

t1, ... ,tn)

is called special if f

vanishes except for the case that

t1, ... ,tn

are all different. We shall

denote L;(Tn) the set of all special functions.

Non-CLT for Gaussian vector processes 3

We state the following theorem without proof (see [6]).

THEOREM

1. L;(P) is a dense linear subspace in L

²

(Tn).

Define the multiple stochastic integral of the function f

E

L

²

(Tn) with respect to L2 noise Z = (Zl, Z2) in T, following Surgailis [9]. See [9] for the proof of the following.

LEMMA

1. Hf

E

L;(Tn), then there exists r. v. ](n)(f) called multiple stochastic integral of

f

with respect to Z such that

(i) I(n)(f) = I(n)(symf)

E

L2(0) (ii) E[I(n)(f)] =

0

(iii) E[I(n)(f)I(k)(g)] = 8 nk , (symf, g) 1 n.

for any k

~

1 and g

E

L;(Tk), where 8nk is the Kronecker delta and (".) is the inner product in L

²

(Tn).

For arbitrary f

E

L2(Tn) let I(n)(f) = limI(n)(f(k), where {f(k)} is a

k-->oo

sequence in L;(Tn) which converges to f in L2(Tn). By (iii) such a limit exists and the limit is independent of choice of {f(k)}. The limit also o satisfies (i) - (iii)-. We also denote it by

I(n)(f) = ~ n' 1 '"

^L..J

^f-

^{)1,-", ) n ) 1}

^. (t(n)z· (dt l )··· z- (dt )

^{In n '}

. Tn jl,." ,jn=1,2

For simplicity of subscripts we use the following notation:

.(n) . . . .

Jk )1,'"

,)k-l,)k+l,'" ,)n t k

(n)

= t l ,'" ,tk-l,tk+l,'" ,tn J(k=i)

.(n) =

jl,'" ,jk-l,i,jk+l,'" ,jn'

Define two functions generated from f(t(n) = (fj(n)(t(n))j(n)=1,2

E

L2(Tn) and g(t)

=

(gl(t), g2(t»

E

L2(T):

(f

X(k)

g)ji

n )

(tin)) = h. L ^~~~~i) (t(n))gj(tk)dGij(tk),

Z,)=1,2

and

(f 0

g)j(n~l)

(t(n+l))

=

fj(n) (t(n))gjn_l (tn+d.

Then (f 0 g) is an element in L2(Tn+l) and its norm satisfies the in-

equality

11

(f0g) Iln+l ::;

11

f IIn·1I g

11·

Let f(l}, ... f(nl,g

E

L

²

(T). Let

4 Tae Il Jeon

f = ®£=If(e). Then f

E

L

²

(Tn). Moreover (f

^XCk)

g), k = 1,2, ... ,n, ^are

in L

²

(Tn-1) and 11 (f

X Ck)

g) II n-

¹

S 11 f Iln ·11 g 11·

DEFINITION

2. f(l) = (f?), fJ1»), ... ,f(m) = (f}m) , ft») E L2(T) are said to be orthonormal if, for any

i =1=

j, (f(i) , f ⁰

»

= 0, that is,

i ^L f~i)(t)fy)(t)dGa{3(t) ^{= 0}

a,{3=1,2 and

The following theorem is called Ito's formula (see [6J for the proof).

Ito's formula for the case of one dimensional process

X_t

is developed in [8J. In this sense we may call the following 2-dimensional version of Ito's formula.

THEOREM 2.

L

²

(T). Then

Suppose

f(l) , ..• ,f(m) E

L

²

(T) are orthonormal in

H

_n1

[I(1)(f(1»)JH

_n2

[I(1)(f(2»)] ... H

_nm

[I(l)(f(m»)]

=

(n1 + ... +n

_m

)!·

I(nl+-+nm)

[(0f(l)t1 ® (0f(2»)n

²

® ... ® (®f(m»)nmj, where Hn(x) is the Hermite polynomial of leading coefficient 1 defined by

Let H(x, y)

=

Hk(x)He(Y),

k

+.e

⁼

m. Consider a process

N-1 N-1

( )

2

YHN

=

A ..

1 "" b H Xt,)(t (

^{1 2)}

= ALJHk(Xt )He(Xt ),N 1 ""

^{1 2}

= 1,2, ...

N t=O N t=O

with an appropriate norming factor

AN.

Let fp)(A) = (eitA-, 0), ~(2)(A) =

(0, eit >.). Then (fp), ri

^2»

^{= 0 in L}

²

^(T). Applying Ito's formula we have

Since

(4)

and

(5)

Hk(I(l) (fp)) _)He(I(1) (f?)))

=

H

^k

(1:

^eitA

^Z

¹

^{(dA)) He} (1:

^eitA

^Z

²

^(dA))

=

H

k

(xl)H e

^(Xt^2),

we can rewrite (2) using (3), (4) and (5) as

PROPOSITION

1. Assume the stationary Gaussian vector process X

_t

satisfies the conditions stated at the beginning of the section and have the correlation functions

and

where f3i > 0 for

i

= 1,2,3 and 4. Since !nl-.84

^rv

^T21(n) = T12( -n)

^{r v}

\nl-.8a we may assume \nl-.8a

^rv

Inl-,84. The notation

^rv

means that the

two terms are asymptotically the same. Let

G

be the random spectral

6 Tae Il Jeon

N= 1,2, ...

A

E

B([-7r,7r]) matrix corresponding to X

t

and define

Gf;(A) =N{31G n (~), G~(A) ^{= NP2G}

²²

(~)

^,

G{';(A) = Nf3aG

I2

(~

^{) ,}

G~(A) ^{= N{34G}

^2I

(~)

^.

Then there exist locally finite measures

_G~I'

Gg2'

G~2'

and GgI ^{such that} lim G~{3 = _G~{3' a, (3 = 1 or 2,

N->oo

in the sense of locally weak convergence.

This result follows from [1] and [4]. Let G

^N

and GO be the random spectral matrices with entries G;:{3 and

_~{3'

a, {3 = 1,2. Now we state the main result.

THEOREM 3. Suppose the stationary Gaussian vector process X

_t

sat- isfies the conditions stated at the beginning of the section and the con- ditions in Proposition 1. Assume {3I < {34, {32 < (33 and k{34 + .e{33 < 1.

With the choice of

AN =

N -I~2 ,

the distribution of the random variables defined in (2) tends to that of random variable Y H, ^{given by}

Y H ⁼ J Ko(y)Z{(dYl)'" Z{(dYk)Zf(dYk+I)'" Zf'(dYm),

where

y

= (YI, ... , Ym)

E

IRm. The last formula is a multiple stochastic integral with respect to the random spectral matrix determined by the

measures

_~{3'

a, (3 = 1,2 and .

ei(Yl+'+Ym) -

1

Ko(Y) = . .

Z(Yl + ... + Ym)

Replacing AN in (6) by the one in Theorem 3 and using the change

of variables

in

multiple stochastic integrals, we can see that the random

variables (6) have the same joint distribution for fixed

N

as the following one, which we identify with them for the sake of simplicity.

(9) Yf = 1

[-1I"N,1I"N]m^KN(Y)

ZfN (dYl)'" z r (dYk)zf N

(dYk+l)'" ZfN (dYm) , where ZGN = (ZfN, ZfN) is the random measure corresponding to the random spectral matrix G

^N

and

N-l

1

^.1

1

eⁱ(Y1+'+Ym) -

1 (10)

KN(y)

= "

_ezNt(YI+"+Ym)

=

^{.1 •}

L....J

N N e^ZN(Y1+"+Ym) -

1

t=O

Note that the variance of Yf is different from that of [1] because it involves cross-correlation functions. The following lemmas hold true under the assumptions of Theorem 3.

LEMMA

2.

EIYfl

²

= kW 1 ^L ~b

^IKN(Y)12

[-1I"N,1I"N]mj(m)EC

G~l (dYl)'" G~l (dYk)G~-H2(dYk+l)" . Gfm2 (dYm) ,

where

k m

b

=

b(j(m))

=

L

^(3(j(j;)

⁺ L

^{(3-y(j;) -}

k{31 - f{32,

i=l i=k+l

8(ji)

= {I ₄ ^{~f ~i} _{If Ji} ⁼

₌

1 _2, (.) {2 ^if

^ji

⁼ 2

'"Y

Ji = 3 if ji = 1, and

(11) C

= {j(m) =

jl,"

.jm

I k of

jIs

are 1 and

f

of

jIs

are 2}.

Let us introduce the following piecewise constant modification of the Fourier transform: .

(12)

'l/JN(X)

= kW 1 .

^m

^L ~beik(VJ.YI+"+lImYm)IKN(Y)12

[-1I"N,1I"N] j(m)EC

G~l(dYl)'" G~1(dYk)G~-;-12(dYk+1)'" Gfm2(dYm) ,

where x =

(XI, . . . ,Xm ),

l/p = [xpN],

p

= 1,2, ... ,m.

8 Tae

n

Joon

LEMMA

3. Assume

(31

<

^(34,

fh < (3a and k(34 < 1, l(3a < 1. For fixed

j(m) E

C let

hN(x) = (

ei:k(II1Yl+O+VmYm)

IKN(y)

¹²

1

1^-1rN,1rN}m

G~1(dY1)··· G~1(dYk)G~+l2(dYk+1)··· Gf.,,2(dYm), where vp = [xpNJ. Then hN(x)

^-+

h(x) uniformly on every bounded region, where

1

I

^{X m}

+

^{ZI1-'7(im)f.l}

^dz.

LEMMA

4. lim'l/JN(x) = g(x) uniformly

in

every bounded region,

. N-.oo

where

g(x) = kW { (1 - Izl)

1 1

^-1,1]

1 1 1 1

IX

_m

+ zlih dz.

LEMMA

5. Let

J.L1, J.L2, •••

be a sequence of finite measures on lR

^m

such that

Let

where vp

⁼

[xpN], p

=

1, ... ,m. If, for every x

=

(Xl,' .. ,X

m ) E

lR

^m,

the sequence </>N(X) tends to </>(x) , which is continuous at the origin, then

J.LN

converges weakly to a finite measure 110, where </>(x) is the Fourier

transform of /LO.

LEMMA

6. Suppose

G~{3

converges locally weakly to

~{3

for each

0:,

(3

=

1,2 and KN(y) converges to a continuous function Ko(Y) uni- formly in any [-A, A]m. Moreover, let the functions K N satisfy the re- lation

uniformly for N

=

1,2, ... , and Ko satisfies the relation

Then the multiple stochastic integral

(15) J Ko(y)Zf(dYl)··· z f (dYk)Zf (dYk+l) ... Zf(dYm) exists and the sequence of multiple stochastic integrals

tends in distribution to the integral (15) as N

---+00.

We will apply Lemma 6 to show the distribution of the random vari- able YI! tends to that of the random variable YR' We have to check the validity of the conditions of Lemma 6. The convergences of

G~{3^---+ ~{3

is stated in Proposition 1. The uniform convergence of K N

^---+

Ko in [-A, A]m is easy to show. Let

J.lN(A)

=

1 ^L ~bIKN(Y)12

Aj(m)EC

G~l (dYl)··· G~l(dYk)G~+12(dYk+l)·· . Gfn2(dYm) , where N

=

1,2, ... and

J.lo(A) = 1 ^{L IK o (y)1}

²

Aj(m)EC

GJll (dYl) ... GJk1 (dYk)GJk+12( dYk+l) ... Gt2(dYm) ,

10 Tae

n

Jeon

where A

E

BClRm). The measures

J1.I, J1.2,'"

are finite and concentrated on the rectangle [-7rN, 7rN]m. Then we have

'l/JN(X) =

k!£! r ^ei-b

(VJ.Y1+"+V^mYm)J1.N

(dy), JIRm

and therefore

'l/JN ~9

by Lemma 4. Since

9

is continuous at the origin,

J1.N ~

J.Lo weakly by Lemma 5, where

9

is the Fourier transform of J1.0' Weakly convergence of

J1.N ~

J.Lo is equivalent to

J1.N ~ J1.o

locally weB.kly and lim

sUPJ1.N(lxl

> A) = O. Therefore condition (13) of Lemma 6 is

A--+oo A

satisfied. Thus Lemma 6 implies Theorem 3.

2. Proof of the Lemmas

Proof of Lemma 2. The integrations'in the rest of this paper are over the range [-7rN,7rN]m unless we specify it. By (9)

EIYfl

²

= E 11 ^K

^N

(y)ZrN(dY1)'" ZrN(dYk)

[-1l"N,1l"Nlm

zf N(dYk+1)'" ZfN (dYm) r.

Note that

(16) KN(y) ~ (~~[(

",ril)' '"

«N?')']) ^(y)

- 1,... ,1,2,... ,2

with ri

^1)(y)

^{= (ei-bty,O) and} ~(2)(y) = (0, eikty ) in L

²

([-7rN,7rN}). The subscript in (16) is 1, ... ,1,2, ... ,2. For simplicity, let

---

^{k l.}

{

KN(Y) if

j(m)

= 1, .. ". ,1,2, .. ". ,2.

(fN

)j<~)(Y)

⁼ - - ; ; - -

~

o ^otherwise.

To avoid complicated subscript we use F instead of F

N

for the time being.

If

we need the precise notation in any context we will specify it.

By (ill) in Lemma 1,

EIYfl

² =

(m!?E[I(m)(F)I(m)(F)]

=

m! (symF, F)

P([-1l"N,1l"Nlm) "

Now compute

symF.

For fixed

^j(m),

by the definition of

symF,

{

0 if

^j(m)

(j. C

(symF)'(m)(Y)

= 1 " F ( )

'f ^-(m) E

C

J

m!

~ ja(l)"··ja(m)

Ya(l),'" ,Ya(m)

1

J , where C is the set in (11) and

^(J"

is a permutation on {I, 2, ... , m}.

Examining F, for

^j(m) E

C, we have (17)

Thus

(s

y

mF).m ( ) J' )

Y = k!f!F

_m.

, 1 ,... ,1,2, ... ,2( )Y = k!f!K ( )

_m. ,N

^{Y .}

m

where

T

=

T(j(m)~j/(m))

= Lp(ji, jD - k{31 -

ff32,

i=l

if

ji = j~ =

1 if

ji = j~ =

2

if

ji =

1 and

j~ =

2 if

ji

= 2 and

j~

=

1.

Using (17) we have

EIY:1 ²

⁼

^kW 1

[-'JrN,'JrN]mj(m)EC

^{L ~bIKN(Y)12}

Gfr1(dY1)'" G~1(dYk)G~+12(dYk+1)'" GS":n2(dYm)' This completes the proof of Lemma 2.

Proofs of Lemmas 3,4 and 5. First note that

N-1

IK

_N

(y)1

²

= ~2 L ^(N _lu\)ei~u(Yl+··+Ym).

u=-N+1

12

Therefore

Tae

n

Jeon

N-l

hN(x) = 1 ^{~2 L} (N - lul)eik [Cu+III)Y1+...Cu+v

^m

)Ym]

[-1I"N,1I"N]m u=-N+l

Gfrl (dYl) ... G~l(dYk)G~+l2(dYk+l) ... Gf,2(dYm)

N-l

I I

1 ^{~ L (1 -} ~)ei~lCU+Vq)AqN'E~l.8ci(jp)+~k+l.8-y(jp)

[-1I",1I"jm

N u=-N+l N

Gitl(dAl) ... Gjkl(dAk)Gjk+l2(dAk+l)··· Gjm2 (dAm)

N-l I I

::;0::

-!. "" ^{(1 -} ~)N'E~l.8ci(jp)+'E';k+l.8-y(ip)

N L N

-N+l

IT 111" eiCu+vp)Gjpl(dAp) IT ^111" eiCu+vp)Gjp2(dAp)

p=l

-11" p=k+l

^-11"

By (1) and assumptions of Proposition 1

k m

.rr ^{lu +} lIpl-.8ci(jp) rr ^{lu +} lIpl-.8-y(jp)

p=l

p=k+l

= ~ ~ ^{(1 _} M) rrk ^{[Iu +}

^lip

^I

^r.8ci(ip)

rr

^m

^{[Iu +}

^lip

^I

^r.8-y(jp)

N L N N N ·

-N+l

p=l

p=k+l

Let

fN(Xl, ... ,xm,.Z)

=c(l~/[~:JI)fIrjpl~~J:lIP) IT rjp2~~L:vp)

p=l p=k+l

and

where z

^E

[-1,1]. We can show that

hN(xl, ... ,x

m )

= 1

¹

IN(X1,'" ,X

m ,

z)dz.

-1

Let

Ac:(x) = {z

E

[-1,1] I Ixp+z\ < c for somep = 1, ... ,m}, and

Then

IhN - hi

=

11: ^{UN - f)dZ\ ~} 1: ^{I/N - Ildz}

< 1

[-1,1J-Ae(x)

^{IIN - Ildz + {}

}A£(x)

^{I/Nldz + (}

} A£(x)

^I/ldz

<

l-1,1J-A£(X)

I/N - Ildz + t, JAe(Xi) I/Nldz + t, ^{[(Xi) I/ldz.}

Let IXpl

~

K,

p

= 1, ... ,m, ^and c ^> O. Then it is not hard to show lim sup 1 ^I/N(X, z) - I(x, z)ldz = O.

N-oo Ixpl~K. [-1,l]-A£(x)

p=1,...

^,m

In order to complete the proof of Lemma 3 it is sufficient to show that (18) 1 _Ae(Xq) ^{I/N(X, z)ldz < M(c)}

and

(19) 1 _Ae(xq) ^{I/(x, z)ldz < M(c),}

for every q

=

1, ... ,m, if Ixpl ^< K,p = 1, ... ,m, ^{where M(c)}

^-+

^{0 as}

c

-+

O. Let us show (19) first. By Holder's inequality

1

{ I/(x, z)ldz ~ ^Mo IT {1- ^{xp +c: 1}

^k(3/j(j)

^dZ}

¹ⁱ

} Ae(xq) p=1 -xp-c: Ixp + zl p

14

For a fixed

p

Therefore

Tae

n

Jeon

[ If(x, z)ldz

JAe(xq )

k 1 m 1

<

Mo

IT

{Mpcl-kf35Up)}k

IT

{Mpcl-lf3liUp)}1

p=l p=k+l

<

^M

c

2-

2:;"1

^f35Up)-I:~k+l^f3;(jp)

<

M e2-kf34-l/33 .

This completes the proof of (19). Therefore, for large enough

N,

[ IfN(X, z)ldz

JAe(xp )

1

^k

¹

^m

¹

r v

(1 - Izl) .. dz.

Ae(x_p)

!! ^{Iz +}

^Xplf35Up)

^{p=\t Iz +}

^{x p}

^l

^f3;(jp)

An estimation similar to the argument above implies (18) and it com- pletes the proof of Lemma 3.

The following argument and Lemma 3 imply Lemma 4. Examining (12) we can interchange the order of sum and integration because each term has finite integral. Therefore

'l/JN(X)

= kW 2: -; 1

ei;{r(VJ.Y1+OO-+VmYm)IKN(Y)12

j(m)EC

N

[-1l"N,1l"Nlm

Gfl(dYl)'" GZ

¹

(dYk) GZ.H2(dYk+l) ... Gf".2(dYm).

For

^{j(m) E}

C whose b = b(j(m») is positive, the term converges to 0 in any bounded region by Lemma 3. The only term which converges to non- zero is the one with

^j(m}

= 1, ... ,12, ... ,2. Indeed this subscript yields

--...---...-

k l

b(I, ... ,1,2, ... ,2)

=

O. Therefore the proof of Lemma 4 is completed.

Lemma 5 is an analogue of the theorem about the equivalence of the weak convergence of measures and the convergence of their Fourier transforms.

See [IJ for the proof.

Proof of Lemma 6.

If

hE L;([-nN, 1rN]m), then

(20) lim

I(m)

(h) =

^I(m)

(h)

N-+00 zeN zCO'

in the sense of convergence in distribution, where ZeN

=

(ZfN, ZfN) is the random measure corresponding to the random spectral matrix

G^N

and ZCO = (Zfo, Zfo) is the random measure corresponding to the random spectral matrix GO. Indeed the multiple stochastic integral of (20) is a sum of terms like the constant times of the following forms:

(21) ZfN _(~d'" ZfN (~k)ZfN(~k+l)'" ZfN _(~m),

where

~i

is in a regular system of rectangles

~(M),

for some positive integer M. Since the joint distribution of the variables in (21) tends to the joint distribution of the variables

Zr(~l)" .Zr(~k)Zr(~k+d··.Zr(~m),

(20) is true. Let c > O. Then by (13), there exists A o ^{such that}

1 ^{L ~b} IKN(yWCfl(dYl)'" C.fm2(dYm) <

^C

JRm_[-Ao,Ao]mj(m)EC

uniformly for N

=

1,2,. " . Therefore, for sufficiently large N > A

_O

/1r, we have

for sufficiently large

^N,

which implies the following (22) El 1-1rN,1rNJIn (KN(y) - Ko(Y))

Zr (dYl)'" ZfN (dYk)Zf N

(dYk+l)'" ZfN (dYm)

1

2

< c

16

Tae

n

Jeon

(26)

for large N. Since Ko can be approximated by h

E L~[-7rN,

1rN]m, by (20), we have the following

(23) Ell ^Ko(y) ZfN (dYl)··· z t (dYk)Zf N

(dYk+l)··· ZfN (dYm)

[-1rN,trNlm

-1 ^{Ko (y)} z f (dyd ... z f (dYk) z f (dYk+l) ... Zf(dYm)

¹²

< c

[-trN,trN]m

for large

N.

By (14) we can show that, for large N,

(24)

EI'Lm_[_trN,trNlm

Ko(Y)

z f (dYl)'" zfo (dYk)Zf (dYk+l)··· z f (dYm)j2 < c.

Therefore, by (23) and (24), we have

(25) 1 Ko(y)zfN(dYl)···zfN(dYk)zfN(dYk+l)···zfN(dYm)

[-trN,trNlm

~ J Ko (y)Zf (dYl) .. . Zf(dYk)Zf(dYk+l)··· Zf(dYm) Consequently (22) and (25) complete the proof of Lemma 6.

Suppose that

H(x, y) = L c(k,eyHk(x)He(y).

k+e=m Then we have

N-l

N 1 ~ ~ ^{1 2}

YH

= AN L....J L....J c(k,e)Hk(X

t

)He(X

t )·

t=O

k+e;=m Using similar technique we can have

. Y/!= L C(k,e)l ... KN(y) k+e=m

[-trN,trN]m

eN ( eN eN ) eN ( )

Zl dYI)'" Zl (dYk)Z2 (dYk+l ... Z2 dYm,

where KN(y) is the same to ( 10). Considering (26) we can formulate

the more general result. The norming constant AN should be uniform

for all terms in (26).

THEOREM 4.

Suppose the stationary Gaussian vector process X

t

sat- isfies the conditions on section

1

and the conditions in Proposition

l.

Assume (31 < (34,(32 < (33 and

_~m=

min{k(31 +f(32!k+f

=

m} <

1.

With the choice of

A

_N-

- N

^1-Sm.₂

the distribution of the random variables defined in (26) tends to that of the random variable Y H, given by the formula

Y; ⁼ L

^C(k,e)

J ^Ko(Y)

(k,e)EDm

Zf(dY1)··· z f (dYk)Zfo (dYk+l) ... Zf(dYm), where Dm = {(k, f) Ik + f = m, k(31 + ff32 =

~m}

and

y

=

(yr, . .. ,

Ym)

E Rm.

The last formula is a multiple stochastic integral with respect to the random spectral matrix determined by the measures

~,6'

af3

=

1,2 and

Since the proof of Theorem

4

is basically the same to that of Theorem 3 we will skip it. Consider the case of general H(x, y) which has the

00

Hermite expansion H(x, y)

⁼

L L

^cm.Hml

^(x)H

^m2

^{(y) with}

j=mlml=j

(27)

N-1 00

Z;

⁼

L L L

^cm

^H

^m1

^(Xl)H

^ffi2

^(X;)

t=O

j=m+1Iml=j

N= 1,2, ....

18 Tae

n

Jeon

Then we have (see [7] for the detail)

E(Z~)2

00

min(ml,lr) [ , Il Il , ]

'""' '""' '""' m1· m 2·

1· 2·

=.L- L-.

CmCI

L- S!(l1 - s)!(m1 - S)!(l2 -

m1

+ s)!

J=m+1Iml=llj=J

s=max(O,ml-h)

. [~ I: I: ^{rf1 (t1 -} t2)r~1-S(t1 ^- t2)r~rml+S(t1 ^- t2)r~rS(t1 ^{- t2)] .}

tl=O t2=O

It is

easy to check with the help of (27), (7) and (8) that

E(Z~)2

= o ^{(N 2}

^{-E.m+l )}

^{+ O(N) as N --}

^00.

Therefore

A~l

zIf -- ⁰

ⁱⁿ

probability as N --

00

and this implies that H(x, y) can be replaced with L ^emH m1 (x)H _m2 (y)

in

Th~orem 4.

Iml=m

References

[1] Dobrushin,R.L. and Major, P., Non-central limit theorems for non-linear func- tionals of Gaussianfields, Z. Wahrsch. Verw. Gebiete 50 (1979), 27-52.

[2] Fox,R.,and Taqqu, M.,Multiple stochastic integrals with dependent integrators, J. Multivariate Analysis 21 (1987), 105-127.

[3] Giraitis, L. and Surgailis, D., CLT and other limit theorems for functionals of Gaussian processes,Z. Wahrsch. Verw. Gebiete 70 (1985), 191-212.

[4] Ho, H. C., and Sun, T. C., Limiting distributions of non-linear vector functions of stationary Gaussian process,Ann. Probability 18 (1990), No. 3, 1159-1173.

[5] Hsiao, C. T., Central limit theorem for stationary random vectors, unpublished note (1983).

[6J

Jeon, T. I., £2 multiple stochastic integral and Ita formula, J. Chungcheong Math. Soc. 6 (1989), 71-87.

[7] Jeon, T. 1. and Sun, T. C., A Central limit theorem for non-linear vector Gauss- ian processes,Proc. 1990 Taipei SymposiuminStat. Inst. of Stat. SeL, Academia Sinica (1990).

[8J Major, .P.,Multiple .Wiener-Ito. Integrals, Lecture notes in Math..849, Springer- Verlag, New York/Berlin (1981).

[9J

Surgailis, D., On L² and non-L² multiple stochastic integration.In Stochastic Differential Systems. Lecture Notes in Control and Information Sciences, Vol.

36, Springer-Verlag, New York, 1981, pp. 212-226.

[10] Taqqu, M., Laws of the iterated logarithm for sums of non-linear functions of Gaussian variables that exhibit a long-range dependence,Z. Wahrsch. Verw. Ge- biete 40 (1977), 103-238.

[11J Zygmund, A., Trigonometric Series, Cambridge: Cambridge Univ. Press, 1959.

Department of Mathematics Taejon University

Taejon, 300-716, Korea

E-mail: [email protected]