Acta Mathematica Academiae Scientiarum Hungaricae 56. (1990)

1990 / 1-2. szám - Szyszkowski, I.: An invariance principle for dependent random variables

Acta Math. Hung. 56 (1—2) (1990), 45—51. AN INVARIANCE PRINCIPLE FOR DEPENDENT RANDOM VARIABLES I. SZYSZKOWSKI (Lublin) 1. Introduction and notations Let {X„,nS 1} be a sequence of random variables on a probability space (Q, F, P). Let FTM = g(Xí : Define two measures of depend­ence by <p(m) = sup {\P(B\A)-P(B)\: A£F", P(A) * 0, BdF~+m, n is 1}, i//(m) — supI P(AOB) 1 P(A)P(B) AdF{, BeF„~m, P(A)P(B) ^0,nSlj. The sequence {Xn, йё 1} is said to be /р-mixing or (//-mixing according as cp(n)-»0 ori//(«) — 0, respectively, as Clearly a (//-mixing sequence is ^-mixing. Assume EXn=0, for every nisi and let Sn= ^ X-t and sl=ES%. i=1 We will always assume that л2—°° as n—°°. Consider a sequence {/c„, /г=0} of real numbers satisfying 1 (1) 0 = k0 < < k2 < lim max (A:f—Ä:,-_X)/A:n = 0, П oo and for each йё1 define — Sm„(t)/sn’ *£[0,1], where m„(7)=max {7^0: fc;s?k„}. The function со->-)Т„((, co) is a measurable map from (Í2, F) into (Z>[0, 1], /?), where D[0, 1] is the set of all functions, defined on the interval [0, 1], which have left hand limits and are continuous from the right at every point, and В is the Borel /т-field on D[0, 1] induced by the Skorohod topology (cf. [I]). In this paper we present some sufficient and necessary conditions for the weak convergence, in D[0, 1], of the random elements {Wn, йё 1} to the standard Brow­nian process on D[0, 1], denoted by W in the sequel. We give the invariance principle for nonstationary, mixing-type sequences under Lindeberg’s condition. This result improves the moment conditions used by McLeish and Peligrad in [6]—[11]. For example, we do not assume that lim sHn=a2>() and {T|, n^l} is uniformly integrable. Moreover, we consider the processes {SmnW/s„, Ostsl), while the authors of the above mentioned papers have investigated the processes {S^/an112, O^f^l}. Thus we obtain a version for the dependent case of an invariance principle of Prohorov which treats the independent case under Lindeberg’s condition (cf. [1, Problem 1, p. 77]).

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