J. Korean Math. Soc. 32 (1995), No. 2, pp. 321-328
THE INVARIANCE PRINCIPLE FOR p-MIXING RANDOM FIELDS
TAE-SUNG KIM AND EUN- YANG SEOK
1. Introduction
Ibragimov(1975) showed the central limit theorem and the invari- ance principle for p-mixing random variables satisfyinga2(n) = nh( n)
-+ 00 and EI~oI2+6 < 00 for some 8 > 0 where a2(n) denotes the variance of the partial sum Sn and h(n) is a slowly varying function.
In this paper we extend the concept of p-mixing to random fields and obtain an invariance principle for such random fields. This theorem may be regarded as a generalization of Theorem 3.1 of Ibragimov(1975) to multivariate time.
Let Zd denote the set of all d-tuples of integers(d ::: 1), and let {~nl,n2"" ,nd : (nI,nz,'" , nd) E Zd} be a random field, i.e., a collection of random variables indexed by time set Zd. For each j (1 ~ j ~ d) and r ~ 0, let A+ (j;r) be the a-field generated by {~nl,n2 , ... ,nd : nj :::
r, other n/8 unrestricted} and let A -(j; r) be the a-field generated
by {~nl,n2,'" ,nd : nj ~ r, other nis unrestricted} and we shall write
L(A+(j; r))[L(A-(j; r))] for the collection of all A+(j; r)[A-(j; r)] me- asurable random variables with finite variance. For r ::: I, we write
(1.1 ) . E[(x - Ex)(y - Ey)]
p();r) = supE~(x_ Ex)E!(y _ Ey)'
where the supremum is taken over all random variablesx EL(A+(j:r )), yE L(A-(j;r)),and
(1.2) p(r) = max p(j; r).
i « d_J_
Received March 10, 1994. Revised December 23, 1994.
1991 AMS Subject Classifications: 60F05, 60G10.
Keywords and phrases: p-mixing, random field, invariance principle
The first auther was supported by NON DIRECTED RESEARCH FUND, Korea Research Foundation.
Clearly{p(r )} is a decreasing sequence of real numbers. Hp(r) -+ 0 we say that the random field is p-mixing. This is a natural extension to multivariate time parameter of the well known concept of p-mixing for sequences of random variables.
Let Td be the d-fold product of the closed unit interval [0,1], let Ddbe the Skorokhodfun~tionspace onT d;and let {W(tl, t2," . ,td) : (t}, t2,··· ,td) E T d} be the d-parameter Wiener process. Assume E(eo,o, ... ,0) = 0,E(e~,o,... ,0) < 00. For nt ~ 1,n2 ~ 1,'" ,nd ~ 1, define the partial sum,
nl n2 nd
(1.3) Snl,n2,···,nd =L L ... L eit,h,"',id' it=l h=l id=l
H some ni = 0 and other ni are greater than or equal to 0, it is convenient to set Snt,n2,'" ,nd = O. Let q2(nt,n2,··· ,nd) denote the variance ofSnl,n2,'" ,nd. Construct a sequence {Wn(tI, t2,··· ,td) : n ~ 1,(ft,t2, ... ,td) E Td}of stochastic processes inDd by
(1.4) Wn(tI,t2,··· ,td)= u(n,n,'" ,n)-IS[ntt],[nt2],... ,[ntd]' (tl' t2,··· ,td)E Td, where [.] is the usual greatest integer function.
Inthe case of univariate time parqmeter, i.e., d = 1,the invariance principle has been proved under two different sets of hypotheses. In one case no assumption is made about existence of a moment of eo
of order greater than two; however a condition is imposed on the rate at which the mixing coefficient per) goes to zero. Ibragimov(1975), however, had, a theorem([8],The01~em 2.1), in which convergence of normalized partial sums of p-mixing sequences to normal distribu- tion is proved without assuming the condition imposed on the rate of p-mixing coefficient. What is assumed instead is
This theorem was casted in the form of the invarance principle by Ib~a,giIJ;1~y(1~75).
THEOREM O. (Ibragimov, 1975) Let {en: n E N} be a p-mixing processes with Een = 0. If (1.5) is fulfilled then {en} satisfies thr invarianceprinciple.
The invariance principle for p-mixing random fields 323
The object of this note is to extend Theorem 0 to random fields, that is, toprove an invariance principle for general p-rnixing random fields.
In Section 2 we state some assumptions and main theorem. The proofs of main theorem as well as lemmas are included in Section 3.
2. Preliminaries and main theorem
The main difficulty seems to be in the proper generalization of the one-dimensional condition that 0'2(n) = Var(Sn) -+ 00 as n -+ 00.
A natural generalization would be to require that 0'2 (nI, n2, ... ,nd) = Vare Snl ,n2,'" ,nd) -+ 00as (nI,n2, ... , nd) -+ 00,where (nI,n2, ... ,nd)
-+ 00 means, as usual, minl<i<d ni -+ 00. That this is not enough is shown, by the following ex~Pie. Let {1]m,n : -00 < m < 00, -00 <
n < oo} be a collection of independent standard normal variables.
Define ~m,n= 1]m+l,n -1]m,n· Then {~m,n}is clearly a p-mixing, two- parameter, random field with ~m,n having moments of all orders. Also 0'2(m,n) = Var(Sm,n) = 2n which goes to infinity as (m,n) -+ 00.
However in this case the limiting process of the sequence {Wn} is not the two-parameter Wiener-process, but the degenerate process {((tl,t2) : 0 $ tl,t2 $ I} where ((tl,t2) ::: ~V(t2), W being the standard one-parameter Brownian motion on [0,1].
Thus what seems to be required is a condition which ensures that the variance structure of Snl,n2"" ,nd is sufficiently homogeneous and the variances tend to infinity in all directions. The following condition accomplishes this. We assume that there exists a positive integer N such that for each i, 1$ i $ d, the following is true:
(2.1)
and that this limit is uniform in all values of nI,n2,' " , ni-I, ni+l,' .. , nd greater than N. Thus, e.g., ifd= 2, we require that
(2.2) lim 0'2(nl,n2) =00 for each n2 2: 1, nl-ex> 0'2(1,n2)
(2.3)
and that the limit in (2.2) is uniform in n2 ~ N and the limit in (2.3) is uniform innl ~N. To pnt condition (2.1) in perspective, note that it is implied by:
(2.4)
To formulate an analogue of the Ibragimov theorem([7},Theorem3.1), we assume that {enl,n2,'" ,nd : (nI, n2,'" ,nd) E Zd} is a p-mixing random field with E(eo,o,...,0) =0which satisfies (2.1) and the follow- ing condition,
(2.5) Eleo,o,...,012+6 < 00 for some ~ > O.
THEOREM 2.1. Let {enl,n2,'" ,nd :(nI, n2,'" ,nd) E Zd} be a p-mi xing random field. If(2.1) lind (2.5) are fulfilled then the sequence of stochastic processes{Wn } satisfies the invariance principle.
COROLLARY 2.2. Let {enl,n2,... ,nd : (nl,n2, .. · ,nd) E Zd} be a p-mixing random field with Eeo,o,...,0 = O. If(2.4) and (2.5) are ful- filled then {Wn } satisfies the invariance principle.
As is shown in [4J, since (2.4) itself is implied by
(2.6)
00 00 00
L L ... L IE(eo,o, ....oeit.h,···,id)1 < 00 and
;l=-ooh=-oo ;d=-OO
00 00 00
L L'" L E(eo,o, ... ,oeit.i2.···,jd)>0,
it=-00 h=-oo ;d=-OO wehave the following corollazy :
COROLLARY 2.3. Let {eil,h, ... ,jd :(h,h,'" ,id)E Zd} bea p-mi xing random field. If(2.5) and (2.6) are fulfilled then {e;l,h"".id}
satisfies the invariance principle.
3. Proof of Theorem 2.1
To alleviate the· notational burden we write·out details·for d·== 2:
For bigger d the proofissimilar but more tedious. We assume condi- tions of Theorem 2.1 inallthe following lemmas and write h(nl' n2)= (nln2)-lu2(nl,n2).
The invariance principle for p-mixing random fields 325 LEMMA 3.1. Hq2(nl,n2) -+ 00 then (a), (b) and (c) below hold.
( ) • [0" CtRt.R, ] . . -
a limnl ...00 0' nl,n,) = Cl, uniformlyIn n2 ~ N.
, -
(b) limn, ...oo[O'O' R~t~~:) ] = C2, uniformly in nl ~ N.
( ) • [0" Ctnl c,n, ]
C lim(Rl,R') ...OO 0' nt.'" = CIC2,
whereCl,C2 arepositive integers.
Proof. The proof is similar to that of Lemma 1of Deo[5].
LEMMA 3.2. Given 5>0, we can find N = N(5) such that
where M isa finite number.
Proof. It can be easily proved by the similar way to the proof of Lemma 6 in [6].
LEMMA 3.3. Under conditions of Theorem 2.1, if 6 < 1, then there exists A >0 such that
Proof. Apply the arguments in the proof of Lemma 2.1 in [8] to the sequence {Snl,n, - Snl-1,n, : 1 ~ nl < oo} and note that the resulting constant A is independent of n2 for all n2. The details are straightforward and therefore omitted. The remaining finite number of n~s can be handled by increasing A ifnecessary and applying the univariate-time Lenuna 2.1 in [8].
LEMMA 3.4. Under conditions of Theorem 2.1 there exist B >
0,"y > 1 and a positive integer L such that (3.2)
EIWn(tl,t 2)12+6 ~ B(t1t2)"Y for all 0 ~ tl, t2 ~ 1 and all n ~ L.
Proof. Combine the preceding lemmas with a.rguments on page 693 of Davydov [2]. Davydov uses Karamata's representation of slowly
varying function. Inour case this is not available, but use of Lemma 3.2 makes it unnecessary. According to the preceding lemmas we obtain the following inequality,
Proof of Theorem.£.1. The proof of this theorem can now be com- pleted by applying Lemma3ofDeo[5] to the sequence{W~}. Condition (iv) of this lemma is satisfied by{Wn } because of Lemma3.4here and the equation (1) and Theorem 10fBickeland Wichura[l]. That the conditions (ii) and(iii) of Lemma3 ofDeo[5] axe satisfied by our se- quence {Wn } here is a str;:rightforward verification and the condition (ii) of Lemma 3 of Deo[5] is also satisfied by (3.2). This completes the proof of Theorem 2.1.
4. Applications
If X is a random measure and B is a Borel subset of Rd then X(B) denotes the mass that the random measure gives to B. All random measures will be assumed to be stationary. Let Ij be the unit interval (j -l,i), where i - 1 = (h - 1,j2 - 1,···:id - 1) andi == (it,}2~··· ,i~),i E.. pd-:If {XCI;) :i E Zd} satisfies (1.1) , (1.2) and per) -+0 we say that random~easureX is p-mixing. Let XK(t) == XK(tI,··· , td) be defined by
(4.1)
X (t) = X«O, Ktl] x ... x (0,KtdJ) -EX«O,Ktl] x ... x (0,KtdJ)
K - sZK~
for t ETd, where (4.2) S2 == lim
(nltn2,'" ,n,,)-oo
The invariance principle for p-mixing random fields 327 THEOREM 4.1. Let {XK} be a sequence of random measures. If {XK} fuliills (4.2) and the following conditions (4.3) and (4.4) then therandom measure X satisfies the invariance principle.
(4.3) EX(B) = O,EX2(B) < 00 for all Borel subsets BE Rd,
(4.4) EIX(Il) - EX(Ii)12+6 < 00 for some 8> O.
Proof. It is sufficient to show that XK(i) converges, weakly to the d-dimensional Wiener measure. By the definition (4.1)
",,[.K':l] ...",,[.K~cl][X(J.) _ EX(J.)]
XK(1.) = L,.,}l-l L..u-1 c l ) } J(2S 2
+X(([Kt.], Ki]) -:; EX(([Kt.] , Kt] .
J("'iS2
According to Theorem 2.1 the first term in the right hand side con- verges to the d-dimensional Wiener measure and the second term con- verges in probability to zero as K. Thus the proof is complete by The- orem 4.1 of Billingsley [2J.
Finally, we apply Theorem 4.1 to Poisson center random measures.
These are constructed as follow : let U be a stationary Poisson point random field with parameter p. Let V = {V£I.>t E Rd} be a collection of LLd. random measures with E[V£(Rd)J = ~ < 00.
Then we say that X is a cluster process with centers U and members Vif
X(B) = L V.~JB -.l.)
~:U~»O
for each bounded Borel set B. We denote X by tu,VJ. (see [10]) It is natural to hope that moment conditions on V will imply moment conditions on X regardless of "shape" ofV inRd. This is made precise in the following theorem :
THEOREM 4.2. Let X = tu,V] as above. IfX is p-mixing and E[V£(Rd)j2+6J
< 00 then X satisfies the invariance principle.
Proof. According to Theorem 3.1 of Burton and Kim [3]. For a rectangular box B in Rd and 0 S 8 S 2 there exists a constant K
depending only on 6 and IBI so that
EIX(B)12+6 ~ KE[(V£(Rd»2+6] and EIX(B) - EX(B)I2+6 ~ KIBI~.
Thus by Theorem4.1 the proof is complete.
References
1.Bickel, P. J. and Wichura, M. J., Convergence creteria for mmtiparameter stochastic processes and some applications, Ann. Math. statist 42 (1971), 1656- 1670.
2. Billingsley, P, Convergence of Probability Measures, Wiley, New York, 1968.
3.Burton, R. M. and Kim, T. S., An invariance principle for associated random fields., PacificJ. Math. 132 (1988), 11-19.
4. DaVydov, Yu. A., Convergence of distributions generated by stationary stochas- tic processes, Theory Prob. and Appl. 13 (1968), 691-696.
5.Deo, C. M., A functional central limit theorem for stationary random fields, Ann. Probab. 3 (1975), 708-715.
6. Deo, C. M., A note on p-mixing random fields., Theory Prob. and Appl. 21 (1976), 866-870.
7. Ibragimov,I. A. and Linnik, Yu. V., Independent and stationary sequence of random variables, Wolters-Noordhoff, Groningen the Notherlands, 1971.
8. Ibragimov, I. A., A note on the central limit theorem for dependent random variables, Theory Prob. Appl. 20 (1975), 135-141.
9. Herrndorf, N., A functional central limit theorem for p-mixing sequences, J.
Mul. Anal. 15 (1984), 141-146.
10. Kim, T. S and Seok, E. Y., The invariance principle for associated random fields, J. Korean Math. Soc. 31 (1994), 679-689.
Tae-Sung Kim
Department of Statistics Won Kwang University
!ri,570-749, Korea Eun-Yang Seok
Department of. Ma.thematics Won Kwang University
!ri,570-749, Korea
