A central limit theorem for the stationary linear process generated by an associated process

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J.Korean Math. Soc. 37 (2000), No. 3, pp. 463-471

A CENTRAL LIMIT THEOREM FOR THE STATIONARY LINEAR PROCESS GENERATED BY AN ASSOCIATED PROCESS

TAE-SUNG KIM, JONG-IL BAEK AND JAE-HAK LIM

ABSTRACT. A central limit theoremisobtained for stationary linear process Xt= 2::i=oajEt-j,where{Et}isa strictly stationary associated sequence withEEt= 0, EE; <00and{aj}isa sequence of real numbers with2::i=olajl <00. A functional central limit theoremis also derived.

1. Introduction

In the last years there has been growing interest inconcepts of positive dependence for families of random variables. Such concepts are of considerable use in deriving inequalities in probability and statistics.

Lehmann (1966) introduced a simple and natural definition of positive dependence: Two random variables X and Y are said to be positive quadrant dependent (PQn) if for any real x, y

P(X >x, Y >y) ~P(X >x)P(Y > y).

A much stronger concept than positive quadrant dependence was con- sidered by Esary, Proschan and Walkup (1967).

A collection of random variables{El,'" ,Em} is said to be associated if for any two coordinatewise nondecreasing functions !I,h on lR^m such that

h

⁼ h(El"" ,Em) has finite variance for j = 1,2,

COV(11 ,12) _~o.

Received September 5, 1999.

1991 Mathematics Subject Classification: 60G10.

Key words and phrases: central limit. theorem, associated, linear process, stationary, maximal inequality for associated sequence.

This work was supported by Korea Foundation Grant (KRF-99-042-D00022- D1201).

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An infinite collection {Et, t E Z}, Z = {O, ±1, ±2, "'}, is said to be associated ifevery finite subcollection is associated.

A large amount of papers has been concerned with limit theorems for associated processes (see, for example, Newman (1984)).

Newman (1980) established a central limit theorem for strictly stationary associated process {Et} with E(Et) =0 and E(E;) < 00; more precisely, if

(1)

00

o<^0"2 ^=; Ef.i+2L E(El Et) < ⁰⁰

t=2

(3) then

(2) n-!(El+ ... +f._{n )} _~ N(O, 0"2).

Newman and Wright (1981) also improved this central limit theorem, that is they derived a functional central limit theorem: Let {Et, t E Z}

be strictly stationary associated process with E(Et) =0 andE(f.;) < 00.

If(1) holds, then

[nt]

Wn(t) =n-!O"-l

L

^Ei, ^0:::;^{t :::; 1}

i=l

converges weakly to the Wiener measureW on D[O, 1] the space of all functions on [0, 1] which have left limits and are continuous from the right.

Let{Xt , t E Z+}, Z+ = {1,2, "'}, be a stationary linear process defined on a probability space (Q, F, P) of the form

00

Xt=LajEt-j,

j=O

where {aj} is the sequence of coefficients satisfying a certain summability condition, that is, {aj} is a sequence of real numbers withL:~olajI< ^00, and{Et} is a strictly stationary sequence of associated random variables such that

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Here we suppose that the coefficients aj are geometrically bounded, Le.

there exist constants A > 0 and 0 < ^p < 1 such that lajl :::; Api for all j ~ o.

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Linear process generated by an associated process 465

o

This class of processes is a basic object in time series analysis.

A vast amount of literature has been devoted to the study of linear processes under various circumstances; Fakhre-Zakeri and Lee (1992) and Fakhre-Zakeri and Farshidi (1993) established central limit theorem under iid assumption on{Et} and Lee (1997) studied central limit theorem under strong mixing condition on{Et}.

In this paper we derive a central limit theoremJor a stationary linear process of the form' (3), which is generated by a strictly stationary associated sequence {Et} with EEt = 0, E(En < 00. And we also consider the functional central limit theorem.

2.' Main results

We start this section by introducing the maximal inequality for a sequence{Et} of associated random variables.

LEMMA 2.1. Suppose that {Et, t E Z+} is a sequence of strictly stationaryassociated random variables with EEt =0 and EE~< 00 and satisfying (1). Then for anyE> 0

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p(

^max ÎEI^{+ ... +}Êkl ^>Ê)

_{~ n~}

l~k~n E

where(72 is defined as in (1).

Proof. This lemma is a direct result of Newman and Wright (see, Theorem 2 of [8]);

p(

^max ÎEI ^{+ ... +}Êkl ^> Ê)

l~k~n

~ E(max IEI+ ... +Ekl)2/E2

l~k~n

by Chebyshev's inequality

~ E(EI + ... +En)2/E2

by Theorem 2of Newman and Wright (1981)

~ n(72/ E2 by (1).

THEOREM 2.2. Let {Et, t E Z} be a sequence of strictly stationary associated random variables defined on a probability space (0, F, P)

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with E€t= 0, E€; < 00 and satisfying (1).

Let us consider a stationary linear process of the form

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00

Xt⁼ 2:aj€t-j, t E Z+,

j=O

(8)

whereE':o lajl < 00.

Then {X_{t }} fulfills the central limit theorem, thatis,

(7) n-!t X_t

~ N(O, q2(t

^a

j) 2)

t=l J=O

asn ---+ 00.

Proof. Letting

00

o'j = 2:ai i=j+1 and

00

Et=

2:

^o'j€t-j,

j=O

whichiswell-defined since :E~olajl < 00, we have

00

Xt ⁼

L

^{aj€t-j ,}

j=O

00

- ao€t+

2:

^aj€t-j

j=l

(I=dj )€t - O-o€t+I=(aj-l - aj)€t-j

j=O j=l

- (i=aj)€t+Et-l-Et,

J=O

which implies that

t

^X

^t

⁼

^(~a; ⁾ (t ^'t)

^{+10 -} ^En

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Linear process generated by an associated process 467 Since CE~oaj)€t 'S are associated by Newman's central limit theorem we have

Note that (L~o^aj? is finite by the assumption L~o^lajI< 00.

Thus if

EO/v'ri ~ ^{0 as}ⁿ ^--t ⁰⁰ and

(10)

then (11)

will follow.

To prove (10), it suffices to prove that (12)

and (13)

EO/v'ri--t0 a.s. as n --t 00

En/v'ri --0 a.s. asn ^--t00.

But (12) and (13) follow from E(ln)2 < 00 (see Remark) and the fact that for any 6> 0

00 00

LP(llnl/v'ri> 15)= LP(llol > ,;:;it5) <00,

n=l n=l

because for an arbitrary random variable Z

00

EZ²< 00 {::}

L

^P(IZI ^> ^,;:;it5) ^< ^00.

n=l

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REMARK.

00 00 00

- L:aJEti-j +

I:I:

akalE(ft-kft-l)

j=o k=O 1=0k,pl

<

(fa~)Ef;

⁺

ff

lakllal\E(ft-kft-l)

j=O k=O 1=0(k,pl)

<

(~a;)

^El,⁺²

L L'<lla.lI"I(~

E(",,))

<

(~a;) (&1

⁺²

t'; &1")

^< ⁰⁰^by ^(1).

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[nu]

~n(U) ⁼ n-ir-¹

I:X

^{t ,} ^0::;^{U ::;}^1,

t=l

where [x] is the integer part ofx.

Ifthe process {~n, n E Z+} converges weakly to Wiener measure on D[O,l], the space of all functions on [0,1] which have left limits and are continuous from the right, then we say that the sequence _{~n, n E Z+}

fulfillsthe functional central limit theorem.

Finally, we close this section by deriving that the linear process {Xt }

generatedby the associated sequence {ft} fulfills the functional central limit theorem.

THEOREM 2.2. Let {ft, t E Z} be a sequence of strictly stationary associated random variables defined on a probability space (n,:F,P) . withEct = 0 andEc; < ⁰⁰ andsatisfying (1). Let Xt ^be ^{defined as}ⁱⁿ (6). Then the stochastic process _{~n(u), nE Z+} fulfills the functional central limit theorem, that is, _{~n(u), 0 ::; U ::; 1} converges weakly to the Wiener measure W.

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Linear process generatedbyan associated process Proof As in Theorem 2.1 let

00

o'j = L ai

i=j+l

and

00

€t= Lo'j€t-j

j=O

with

00

Llaj! < 00.

j=O

Then we have

469

(15) and

Thus

[nu]

en(U) - n-!r-1L Xt

t=l

00 [nu]

= n-!r-1

(~aj)~

^€t

{16) +n-!r-1€o - n-!r-1€[nuJ'

First note that (I:~oaj )€t's are associated since €t 's are associated.

According to (12) and (13) the second term and the third term in the right-hand side of (16) converge in probability to zero, i.e.,

n-!r-1€[nuj ~ 0 and n-!r-1€o ~ O.

Hence by Theorem 4.2 of Billingsley (1968) it remains to prove

00 [nu]

(17) n-!r-1(Laj)L€t

~

^W,

j=O t=l

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where W is a Wiener measure on D[O,1] the space of all functions on [0,1] which have left limits and are continuous from the right.

Since ^Et's are associated it is well known that (18) holds according to Newman and Wrights' functional central limit theorem (see Theorem 3

of [8]). Hence the proof is complete. 0

References

[1] Billingsley, P., Converyence oJ Probability Measure, Wiley New York, 1968.

[2] Esary, J., Proschan, F. and Walkup, D:, Association oJ random variables with applications, Ann. Math. Statist. 38(1967), 1466-1474.

[3] Fakhre-Zakeri,I. and Farshidi, J., A centmllimit theorem with random indices for stationary linear processes, Statist. Probab. Lett. 17(1993), 91-95.

[4] Fakhre-Zakeri,I. and Lee, S.,Sequential estimation of the mean of a linear pro- cess, SequentialAnal.11(1992), 181-197.

[5].Lee,. S., Random central limit ~heoremfor linear process generated by a strong mixing process, Statist. Probab. Lett. 35(1997), 189-196.

[61 Lehmann, E. L., Some concepts oJ dependence, Ann. Math. Statist. 37(1966), 1137-1153.

[7] Newman, C. M., Normal fluctuations and the FKG inequalities, Comm. Math.

Phys. 74(1980), 119-128.

[8] Newman, C. M. and Wright, A. L., An invariance principle for certain dependent sequences,Ann. Probab. 9(1981), 671-675.

[9} Newman, C. M.,Asymptotic independence and limit theorems for positively and negatively dependent random variables, InequalitiesinStatistics and Probability (Y.L.Tong 00.),Inst. Math. Statist. Hayward, C. A., 1984.

Tae-Sung Kim

Division of Mathematical Science Wonkwang University

Ik-San, Chonbuk 570-749, Korea

E-mail: [email protected] Jong-Il Baek

Division of Mathematical Science Wonkwang University

Ik-San, Chonbuk 570-749, Korea

E-mail: [email protected]

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Linear process generated by an associated process

Jae-Hak Lim

Department of Accounting

Taejon National University of Technology Tae-jon 300-717, Korea