A central limit theorem for functionals of level overshoot by a Gaussian field with dependence

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A CENTRAL LIMIT THEOREM FOR FUNCTIONALS OF LEVEL OVERSHOOT BY A GAUSSIAN FIELD

WITH DEPENDENCE

TAE IL JEON

ABSTRACT. Inthis article we prove a limit theorem for a functional generated by intersections of realization of stationary Gaussian fields with moving level. A sequence of random variables is defined in terms of the functional and the sequencehasa normal limiting distribution.

1. Introduction

Many papers [1), (2), (4), [6j, [7j, (9), (10), [121 have considered the asymptotical distribution of non-linear functionals of Gaussian fields. It has been shown that the central limit theorems hold for some non-linear functionals of stationary Gaussian fields if the correlation function of the underlying field tends fast enough to zero. In this paper we con- sider a functional generated by intersections of realizations of stationary Gaussian fields with moving level. Leonenko and Kh.-el'-Bassionll [8]

has considered a geometric-type functional of a homogeneous isotropic Gaussian random fields with strong dependence. Basically we will con- sider the same functionals which are of interest in the theory of rough- ness of surfaces. But the sequence of random variables in what we are interested is different from the one considered in [81. The functionals presented in this paper come within the scope of the general non-linear functionals dealt in [1]. But the random variables of a sequence of functionals in the present paper are dependent on the level increase.

Let ZV denote the integer lattice in the v-dimensional Euclidean space Received September 11, 1996.

1991 Mathematics Subject Classification: 60F05.

Key words and phrases: central limit theoremm, level overshoot.

This workis supported by NON DIRECTED RESEARCH FUND, Korea Re- search Foundation, 1995.

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(1)

RV. Let X(n), n E ZV, be a stationary Gaussian field with zero mean and unit variance. Letr(n) = EX(m)X(m+n),n,m E zv. ^{We define}

the sets, f<;>r nE ZV,N = 1,2,3, ... , B(n, N) ={m = (m(I), ... ,m~v»)

E ZVln(j)N <mU) ~ (nU) +l)N,1_~j _~v}

and the random fields, for nE zv,^N ⁼ ^{1,2, ... ,}

Z: =

L

^fN(X(m)),

meB(n,N)

wherefN(X) =max{O,x-a(N)},anda(N) is a slowly varying sequence.

We use the standard notation </>(x) = exp(-x²/2)/..;2ir,x E R, and

<I?(a) = J~oo</>(x)dx. Let j(x) be a real-valued function with .

J

j(x)</>(x)dx= 0,

J

^f²(x)</>(x)dx <00.

Thenj (x) can be expandedin the form

00 00

f(x) =LcjHj(x), LcJj! < 00,

j=1 j=1

whereHj isthe j-th Hermite polynomial with leading coefficient 1. Itis well known that these polynomials {Hk(x)lk = 0,1,2, ... } form an or- thogonal systeminthe space L2(R,</>(x)dx). Indeed, for k =0, 1,2, ... ,

x² d x²

Hk(x) =(-1)kexp(_)(-.)kexp(__ )

2 dx 2

and it satisfies

(~)k</>(X)⁼ (-ll Hk(x)</>(x).

LetX(m), m EZ^v^,v~ 2, be a stationary Gaussian field withEX(m) =

0, EX²(m) = 1, andrem) = EX(O)X(m) = rl(lml)for some decreasing functionrl. Note thatrem) depends only on the lengthImlof the vector m and not its direction. So if no confusion arises, we simply use the notationsr(lml) insted ofrl(lml). Since .

EjN(X(O)) = 100 (x - a(N))</>(x)dx= </>(a(N)) - a(N)[1- <I?(a(N))]

a(N)

(3)

Ef~(X(O))⁼ 100 (x - a(N))2<j>(x)dx < 00, a(N)

fN has Hermite expansion (2)

00

fN(X(m)) = L CNkHk(X(m)), k=O

where

(3) CNk =

:! f

fN (y)Hk (y)<j>(y)dy, k=0, 1,2, ....

By a simple computation we have (_1)k

f

^{d k} ¹

CNk= _-~ (dy) - <j>(y)dy, ^k =1,2, ....

The first two Hermite coefficients offN are computed by integration by parts and we obtain

CNO =</>(a(N)) - a(N)[1 - ~(a(N))], ^CNl =^{1 -} ~(a(N)).

By assumptions on X(m),

(4) EHk(X(m»=0,EHk(X(ml»Hz(X(m2»=8kZk!,.k(lml - m21),

whereOkl is the Kronecker's delta. These are immediate consequence of the orthogonality of Hermite polynomials of Gaussian random variables.

For more computations of product of Hermite polynomials of radom field we refer to the diagram formula [12].

Using (4) and some change of variables, we have

[

V 0) v I (i) \

E(Z:)2 = N^V {c}.,0mE~N)JJ(l_ln;" 1)+C~<lLBJJ(l-

:r

^)r(lml)}

00 v I 011 ]

+ £;^c}.,kk!mE~N)

D

^{(1 -} ^mN ),.k(lml) ,

where

B(o,N) = {m = (m(l), ... ,m(v))I_N +1_~ m(j) _~ N -1,j = 1, ... ,v}.

Let

(4)

where

v C)

D(N) = L IT(1 - Im; I)r(lml).

mEB(O,N)j=l

(5)

2. Theorem

THEOREM 1. LetX(m), mE zv ^be^astationary Gaussian field with EX(m) = 0,EX(m)2 = 1,r(lml) = EX(O)X(m) andr(lml) converges monotonically to zeroas Iml ~00. Suppose that asN ~ 00

(i) L ^r²^(lml) ^<⁰⁰

mEZ"

(ii) a2(N) '" In(lnN)

(iii) (D(N))-l =o((lnN)-l).

Then U;: ~ n(O, 1), where~ means convergenceindistribution and n(O, 1) is the standard normal random variableas N - 00.

Proof According to (1), (2) and (3) U;: can be expressed as a sum of two independent random variables

00

CNl L H1(X(m)) L L CNkHk(X(m))

N mEB(O,N) mEB(O,N) k=2

Un = NV/2cN1(D(N))l/2 + NV/2_CNl(D(N))l/2

From the orthogonality condition of Hermite polynomials it follows that

00

(6) Var(U::) = 1+Var

L LCNkHk(X(m))

mEB(n,N) k=2

(7)

Ifthe variance on the right side in (6) converges to zero, then variance of

U::

converges to 1. This implies that

U::

has a limiting distribution identical with that of

CNl L H1(X(m))

mEB(n,N)

(5)

But (7) has a standard normal distribution as the limiting distribution.

Therefore it suffices to prove that the last term in (6) converges to zero.

Using the orthogonality of Hermite polynomial of stationary Gaussian process with mean zero, variance 1 and correlation function r(lml), we have the following

Var

L L00 CNkHk(X(m))

meB(n,N) k=2

=

Since the correlation function converges to zero as Iml goes to infinity, we may assume that, for some 0< f3 < 1/2 andM > 0,

(8)

Therefore we have

(9)

Taking into account the fact L>2(lml) < 00 and n;=I(l - Im:)I) < 1, and by Schwartz inequality, we have

11 (j)

L IT_{_ .} ^{(1 -} E:..J_N _)~2(lml)

meB(O,N) ;=1

(6)

Therefore we have, for someK > 0,

Lk!4k,8k~2;,00 ^.

(10) I: :s; K k=2 ] 1/2"

v (j)

eJv1 [

~

^U(l- ^Im_N ^l)r(lml)

meB(O,N) 3=1 ,

Using integration by parts , for k ~ 2, we have

1 [100 ^d ]2

k!c?Nk = k' (dx)k-l(j>(x)dx

• a(N) ,

(11) .!- [.(.!!:-.)k-2(j>(a(N))]²

k!, dx

_ ~! ^[(_1)k-2 Hk- 2(a(N))(j>(a(N))]²

1 2

- k! [Hk_2(a(N)W[(j>(a(N))] "

Applying (11) to (10) and using change of variable, we have

00 ^,8k

(j>2(a(N)) L k! H;(a(N)) _

IN < K k=O '

n - . 'eJv1(D(N))!

Considering the asymptotic behaViour of function (j>,'we may regard (j>(a(N)) ^rv exp{-a2(N)/2}/V2ii ^rv a(N)(l - q;(a(N))). Consequently we have a constant K1 and, for sufficiently largeN,

(12)

Using Mehler's formula, (22) on page 194 of [5J, we abserve that

(13) ~ ,8k H 2 aN))= 1 e {2,8a2

~ ^k! ^{k( (} ^Jl-4,82 ^xp ^1+2,8(N)}^.

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Replace the appropriate term in (12) by (13). By the assumption (5) the right side in (12) is asymptotically equal to, for some K2 > 0,

In(lnN)exp{ (2/3/(1+2/3)) In(lnN)} In(ln N)(lnN)2^f3/(1+2f3)

K2 (D(N))1/2 =K2 (D(N))1/2

Since (D(N))-l = o((lnNt1) as N approachs infinity we have (14) I~ :::; K1ln(lnN)(lnN)2f3/(l+2f3)-1/2.

Since 2/3/(1+2/3) < 1/2, the last term in (14) converges to zero as N approaches infinity and this completes the proof. 0

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Department of Mathematics Taejon University

Taejon 300-716, Korea