The invariance principle for linearly positive quadrant dependent random fields

J. Korean Math. Soc. 33 (1996), No. 4, pp. 801-811

THE INVARIANCE PRINCIPLE FOR LINEARLY POSITIVE QUADRANT

DEPENDENT RANDOM FIELDS

TAE-SUNG KIM AND HYE-YOUNG SEO

1. Introduction

Let Zd denote the set of all d-tuples of integers(d ~ 1, a positive integer). The points in Zd will be denoted by m,ll, etc., or sometime, when necessary, more explicitly by

(mI,

m2, ... ,md), (nI,n2, ... ,nd) etc. Zd is partially ordered by stipulating m ::;IIiffmi ::; ni for each i, 1 ::; i ::; d. We write

Q,l

and

nl

respectively for points (0,0,··· ,0),

(1,1,··· ,1)

and (n, n,··· ,n) in Zd. For

11 =

(nI,··· ,nd), let 1111 stand for the product nl x n2 x ... x nd. Define

ItI

similarly for

t

^E [Q,l]·

Let {Xi: j =

(jI,h)

E Zd} be

a

random field, i.e., a collection of random v~iablesindexed by time set Zd, on some probability space (n,F,p) with

EXi

=

O,EXl <

^00. Forn EN put

(1.1)

assume (1.2)

Define

Snl=

LXi'

l$i$nl

[ntll [ntdl

(1.3) Wn(t) = Wn((Q,i])

= (a

²

n f

_)-1

L ... LXi'

il=l id=l

Received July 24, 1~95.

1991 AMS Subject Classification: 60F17, 60G60.

Key words: linearly positive quadrant dependence, random field, invariance principle(functional central limit theorem).

This work was supported by NON DIRECTED RESEARCH FUND, Korea Research Foundation.

802 Tae-Sung Rim and Rye-Young Seo

(1.6)

where Wn(t)

=

0 for someti

=

0 and for B

=

(§.,t] C [Q,1.]

(1.3)' Wn(B) =

(0-

²nt)-1 LXi..:

[n.!J<i$[nll

Then Wn is a measurable map from (0,.1') into (Dd,B(Dd» , where Dd is the set ofall functions on [0, 1]dwhich have left limits and are continuous from the right, and B(Dd) is the Borel0- -field induced by the Skorohod topology. {Xi:

1..

E Zd} fulfills the invariance principle (functional central limit theorem) if Wn converges weakly to the d- parameter Wiener process W on Dd.

A random field{Xi:

1..

^E Zd}is said to be pairwise positive quadrant dependent (pairwise PQD) if for any real Ti'Tiand

i

⁼¹⁼

1..

(1.4) P{Xi

>

_Ti,Xi

>

Ti} _~ P{Xi

>

Ti}P{Xi

>

Ti}'

A much stronger concept than PQD was considered by Esary, Proschan and Walkup[6]; A random field {Xj :

i

E Zd} is said to be associated if for any finite collection{Xj(l)"" ~X7<m)}and any real coordinatewise increasing functions

!,

g on Rm -

(1.5) Cov[f(Xi(l) , ... ,Xi(m»,g(Xi<l),'" ,Xt.(m»] ~ 0,

whenever the covariance is defined. Newman[ll] first introduced the concept of linearly positive quadrant dependent notion. We extend this notion to the random field, that is, we say that a random field {Xi:

1..

E Zd} is linearly positive quadrant dependent(LPQD) if for any disjoint subsets A, B of Zd and positive

Tls

LTiXi and LriXi are PQD.

lEA iEB

Linearly positive quadrant dependence implies, in particular, nonnega- tive correaltions ofthe random variables Xj. The following theorem is an extension of the central limit theorem for-associated random fields of Cox and Grimmett[4J to linearly positive quadrant dependent random fields by using conditions on the coefficient of maximal covariance

u(r) = sUPl.$!$nl.

L

^Cov(X

_i

^,X.0

i:lli-!II~r

where

111..11

= max(lill,···

,lid!)'

The invariance principle for LPQD random fieids 803 THEOREM A(KIM, 1995). Let {Xj :j E Zd} bea linearly positive quadrant dependent random field with EXj

=

0. Assume

(i) infl~j~nlVar(Xj )2:: 0, - (ii) sUPl~i~nlEIXjI3 < 00,

(iii) u(o)

<:

^{00,u(r)-+ 0, as r -+ 00.}

1

Then (Snl - E(Snl))/(Var(Snl))2 is asymptotically normally dis- tributed asn -+ 00.

Theorem A can be extended to the invariance principle if instead of LPQD the stronger concept of association is required:

THEOREM B(KIM 1995). Let {Xi: j E Zd} be an associated ran- dom field with EXi

=

0,EX]

<

~ ~d define WnC') as in (1.3).

Assume -

(1.8) EIWn(B)I2+c5 _~

GIBI

for some finite G

where, B

= ^(.2.,t]

^C [Q,l] and Wn(B)

= ^((j

^2nt)-1 I:[n~<i~[nil^Xi' Then {Xi:

i

^E Zd} fulfills the invariance principle. -

In this note we show that Theorem B still holds for LPQD random fields and present that the LPQD random field with Lebowitz inequal- ity property satisfies the invariance principle.

All result are stated in Section 2. The proofs of our theorems as well as some lemmas are given in Section 3. In Section 4 we also apply it to the random measures.

2. Results

THEOREM 2.1. Let {Xi:

i

E Zd} be an LPQD random field with EXi = 0,EXl

<

00. Assume

(2.2) ((Wn(t)-_Wn(~.)?:

t

^E

[0,

l]d,n 2:: 1} is uniformly integrable,

804

and for every c

> °

Tae-Sung Kim and Bye-Young Seo

where w(Wn,15) = sup{IWnC~)-

Wn(OI : lit - :ill <

t5}.

Then {Xi.. :

l

E Zd} fulfills the invariance principle.

The following theorem shows that condition (2.1) is necessary for the invarianceprinciple:

THEOREM 2.2. Let {Xi.. :

l

^E Zd} be an LPQD random field with EXi.. = O,EX]

<

⁰⁰ and define WnO as in (1.3). H {Xi.. :

l

E Zd}

fulfi1ls the invi;riance principle, then condition (2.1) holds.

THEOREM 2.3. Let {Xi.. :

i

^E Zd} be an LPQD random field with EXj = 0,EX]

<

00. Assume

where B

=

(:i,t],!l~

:i,t

_~1 andWn(B)

=

(/72n !)-1L:[n~<i~[n:!l^Xi' Then {Xi.. :

l

E Zd} fu1fi11s the invariance principle.

We say that the random field {Xj :j E Zd} satisfies the Lebowitz

Inequality if - -

(2.6) E(XiXiX!£Xll _~ E(XiXi)E(X!£Xll

+

E(XiX~E(XiXll

+

E(XiXllE(XiX~

THEOREM 2.4. Let {Xi.. :

i

^{E Zd} be an} LPQD random field with EXi = 0,EXj

<

00 and satisfy the Lebowitz Inequality and define

Wn (·) as in _(1~3). Assume (1.2), (iii) of Theorem A, and (2.4). Then

{Xi:

i

E Zd} fulfi11s the invariance principle.

3. Proof

The invariance principle forLPQn random fields 805

LEMMA 3.1(LEHMANN, 1966). Let Xi,Xi ^be PQD random vari- ables with finite variance. Then

(i) Cov(Xi , Xi) 2: 0,

(ii) Cov(Xi,Xi)

= °

^{ifand onlyifXi, Xi are}independent.

LEMMA 3.2(BIRKEL, 1993). For each k

?:

1let Xl^k),X?) bePQD

random variables such that -

Then Xi,Xi ^are PQD.

It is easy to see that Theorem 2.1 ofKim[9] still holds for random variables which arenonnegatively correlated. Hence we obtain :

LEMMA 3.3. Let {Xj : j E Zd} be an LPQD random field with EXi= 0,EXl < 00.

As;um~

(3.2) E{W~(t.)} ~n

1t.1

for t.E [Q,1].

Then the following conditions areequivalent:

(i) E{Wn(~)Wn(t.)} ~n

I§.I

for^Q

:5 §.:5

t.

:51,

(ii) E{(Wn(t.) - Wn(§.»(Wn(Q) - Wn(y.»)} ~n 0,

A subset B of [Q,1] is called a block if it is of the form

(§.,

t]

IIj=I(Sj,tj] where

§.

⁼ (SI,'" ,Sd), t = (t},· .. ,td), and (Sj,tj)'s are half closed subintervals of

[Q,l].

For each i, 1

:5

ⁱ

:5

^d, let

°

^{= a~i)}

^<

b~i)

<

_a~i)

<

_b~i)

< ... <

_a~l

<

_b~l = 1 be real numbers. Call a

collection of blocks in

[Q,ll

'strongly separated' if it is of the form {IIt=1(a~:), b~:)];¹

:5

ki

:5

ni, 1

:5

ⁱ

:5

dJ, or if it is a subfamily of such a family of blocks.

The following lemma is obtained by Deo[5](see Lemma 3 of Deo[5]):

806 Tae-Sung Kim and Hye-Young Seo

LEMMA 3.4(DEO, 1976). Let {Wn } be a collection stochastic pro- cesses in 1Jd such that,

(i) EWn(f) --+n 0, EW~(t.) ^--+

ItI

^{for each}i E(Q,l], (ii) {W~(t)} is uniformly integrable for eachi E (Q,l],

(iii) ifBb· .. ,Bk are a collection of strongly separated blocks then the incrementsWn(B1 ),Wn(B2 ) , · · · ,Wn(Bk) areasymptotically inde- pendent,

(iv) for each

6>

0, rt

> °

^{we can find a}

^6> °

such that P{w(Wn ,6)

>

6}

<

rt for all sufficiently large n. Then {W_{n }} converges weakly in 1Jd, to the Wiener process.

Proof of Theorem 2.1. Condition (2.1) implies (3.2) and hence, we have (ii) of Lemma 3.3. We will apply Theorem 19.1 of Billingsley [2]. Since Wn(.Q) = 0, (2.3) here and Theorem 15.5 of Billingsley [2]

yield the tightness of the sequence {W_{n :} n 2 I}. Let X be a limit in distribution of a subsequence of{Wn :n ~ I}. ThenP{X E C[Q,l]} = 1 by Theorem 15.5 of [2]. It suffices to show that X is distributed like

w:

By (2.2) and (3.2) {Wn(i) : n

2

I} and {W~(i) :n

2

I} are uniformly integrable for everyi E [Q,l]. As

in distribution (for a subsequence), Theorem 5.4 of Billingsley [2] and (3.2) imply

EX(i) = 0, EX²(t) =

It\.

According to Theorem 19.1 of Bi1lingsley [2], X is distributed like W ifX has independent increments, that is

(3.3)

X(il) - X(!o) , ...

,X(h) -

X(h_l)

are independent for all k 2 1,

To show (3.3) put

Uni= Wn(ti) - Wn(ii-l)' 1 :::;i :::;k.

Since(Unb ··· ,Unk ) ~n(X(h) -

X(t

_o),-··

,X(h)

-X(ik-l» in dis- tribution(for a subsequence), and since theUni are pairwise PQD,

The invariance principle for LPQD random fields 807

are pairwise PQD, according to Lemma 3.2. A similar argument as above(using Theorem 5.4 of Billingsley[2]) yields, for i =1=j,

according to (ii) of Lemma 3.3. Hence the X(ti) - X(ti-I) are pair- wise PQD and uncorrelated random varfables and thus independent by Lemma 3.1. This proves (3.3). This completes the proof.

Proof of Theorem 2.2. Since the invariance principle is fulfilled {W~(t) :n _~

I}

is uniformly integrable and hence

(3.4) E{W~(:t)} ---+n E{W²(t)}

= It/

for

t

E

[Q,ll,

according to Theorem 5.4 of Billingsley[2]. By Lemma 3.3 it remains to show

for

12 ::;

2. ::;

t ::;

^{.Y. ::;12. ::;}

1.

To prove (3.5), let Q ::; 2. ::; t ::; .Y. ::; 12. ::;

1

be given. Since the invariance principle is fulfilled, {W;(t) :n ~ I} is uniformly integrable.

Hence

is uniformly integrable by (3.4) . As

in distribution, according to Theorem 5.4 of Billingsley[2] and (3.6) E{(Wn(t) - W(~»(Wn(12.)- Wn(.Y.»}

---+n E{(W(D - W(2.»(W(12.) - W(.Y.»}.

But

E{(W(t) - W(~»(W(12.) - W(.Y.»}

= E{W(t) - W(2.)}E{W(12.) - W(1£)}

=

0,

808 Tae-Sung Kim and Hye-Young Seo

which proves (3.5). Thus by Lemma 3.3 we obtain

Proof of Theorem 2.9. The proofofTheorem 2.3 can now be com- pleted by applying Lemma 3.4(Lemma 3 of Deo[5]) to the sequence {W_{n } :} That the conditions (i) and (ii) of this lemma are satisfied by our sequence {W_{n }} here is a straightforward verification from assump- tions EXj = 0 (2.4) and (2.5). The condition(iii) of Lemma 3.4 is also satisfied by {W_{n }} from Lemmas 3.1 and 3.3 and condition (2.4) here.

Finally condition(iv) ofthis lemma is satisfied by{Wn } because of as- sumption (2.5) of Theorem 2.3 here and the equation (1) and Theorem 1 of Bickel and Wichura[lJ. This completes the proof.

Proof of Theorem 2.4. Let B = (~,tl,Q5 ~5 t 51- First note that

where Wn(B) = (0-2n~)-1 L[n.cl<j~[n:!l^{Xi for B =} (~,tJ ^C [Q,tJ. The Lebowitz Inequality and the linear positive quadrant dependence give

E(W_n(B))4

= E((0-2n~)-1

L

^Xi)4

[n.cl<i~[n:!l

1 1

=

(o-S)(n2d)

L LL L

E(XiXiX!£XJJ [n.cl <i, i, 1£, l~[n:!l

1 3 .

5

(o-S)(n 2d )

L LL L

[sup{E(XiXi)E(X!£XJJ}]

[n.cl<i, i, ls., l~[n:!l

3 1

5

(o-s)(n2d)

L L

[n.cl<i,ls.~[n:!l

[supi

L

Cov(Xi,Xi))sUP!£(

L

Cov(X!£,XJJ)J

i;li-il~O l;Ils.-ll~O

5 (

_{3s )(} ;d )n2d

lt - _~12u2(O)

0- n

=CIBI

^2•

The invariance principle for LPQD random fields 809

Thus we have

EIWn(BW

_~

CIBli.

This completes the proof according to Theorem 2.3.

4. Applications

In this section we will apply the notions of LPQD random fields to the random measures, that is, a simple argument using Chebyshev's in- equality allows us to extend the invariance principle for LPQD random fields to random measure. Bd denotes the collection of Borel subsets of d-dimensional Euclidean space Rd. The space M of all nonnega- tive measure p defined on (Rd,B^d) and finite on bounded sets will be equipped with the smallest u-field containing basic sets of the form {p EM: peA) _~ r} for A E Bd,

°

^{~ r}

^<

^00. A random measure X is a measurable map from a probability space (O,:F,P) into (M,M), the induced measure Px = PoX-Ion (M, M) is the distribution of X and ifX is a random measure and Bd is a Borel subset of R^d then X(B) represents the random mass ofthe regionB. (see Kallenberg[7J).

For the random measure X define the K-renormalization of X to be the signed random measure XK where

(4.1) XK(B) = X(KB) - ~X(KB)

uK"2

(4.2)

for t E [0,

oo)d.

Let {XK} be a sequence of random measures onRd. A set function XK satisfies the central limit theorem if for any bounded B E Bd,XK(B) converges in distribution to N(O,IBI) as K - 00

whereXK(B) is difined in (4.1) and IBI denotes the Lebesgue measure of B and the random measureX satisfies the invariance principle if XK

converges weakly to the d-dimensional Wiener measure W.

DEFINITION 4.1. A random measureX is linearly positive quadrant dependent if and only if the family of random variables :F= {X (B) :

B a Borel set } is LPQD.

810 Tae-Sung Rim and Bye-Young Seo

THEOREM 4.2. LetX be anLPQD random measure withEX(B)

=

0,

EX 2(B)

<

⁰⁰ and defineXK(t) asin (4.2). Assume (4.3) E{XK(.§.)XKCf)} -tK

).§.I

for Q~.§. ~

f

~

1.

For A E Bd,A bounded,IAI

>

1,there exists constants C

<

^00, and 6

> °

^{such that}

Tben X satisfies tbe invariance principle.

Proof. Note that for a block B

C [O,l]d

(4.5) XK(B) = X(KB) - ~X(KB)

uK"2

where, ifB = nf=l(Si,ti], then KB = nf=l(Ksi,Kti].

As the similar arguments to the proof of Theorem 2.3 the proof of Theorem 4.2 can be now completed by applying Lemma 3.4 to the sequence {XK }.

From (4.5) and condition (4.3) it is easily seen (4.6) E(XK(f» = 0, EX1(t)^{- t}K

ltl

which satisfies condition (i) of this lemma. By condition (4.4), for K large enough,

(4.7) E(IXK(fW+O)

~ (UK~)2+6C(U2KdltI)1+! ⁼ CltlI+~

and so {XK(!.)} and

{X1CO}

are uniformly integrable for every t E

[O,l]d,

that- is, condition"

(ii)

of this lemma is satisfied. To prove that {XK} satisfies condition (iii) of this lemma let Bl ,'" ,Bm C [O,l]d be strongly separated blocks, and let Bi

= ^(.§.,f],

^Bj

=

^{(:!!dZ.] , where}

Q _~ s ~ f ~ :!! _~ Q. ~

1.

Since the random variables X (Ij )'^S are nonnegative correlated it follows from (4.3) that -

(4.8)

Cov(XK(Bi),XK(Bj» _~ COV(XK(t) -XK(.§.),XK(Q.) -XK(:!!» -tK

°

The invariance principle for LPQD random fields 811

according to Lemma 3.3, where Ij = (j -l,j] for 1;:;j E Zd.

SinceXK(Bj)'s are LPQD, byTh~ore;'6 ofNewman[lD and (4.8) the XK(Bj)'s ~ independent as K _~ 00. Finally condition (iv) of this lemma-is satisfied by {XK} because of (4.7) here and the equation (1) and Theorem 1 of Bickel and Wichura[l]. This completes the proof.

ACKNOWLEDGMENTS. The authors wish to thank the referee for very thorough review of this paper.

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6.Esary J., Proschan F., and Walkup D., Association of random variables with application, Ann. Math. Statist.38 (1967), 1466-1474.

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Department of Statistics Won-Kwang University Iksan 570-749, Korea