A criterion for tightness for a class of dependent random variables

Abstract

1) EX, = 0For each tC [O, 1] put LOYNES [3] proved that if the finite-dimensional distributions of a sequence of martingales converge and if for each time t the variables are uniformly integrable, then weak convergence follows (in either C or D) provided the limiting process satisfies a certain condition; this condition is satisfied by the Wiener process. Using this result we prove a weak invariance principle for a class of dependent random variables, satisfying a Lindeberg-type condition. The weak invariance principle we obtain for (p-mixing sequences shows that the mixing rate used by MCLEISH in Theorem (3.8) of [4] and in Corollary (2.11) of [5], can be improved provided the finite-dimensional distributions converge.Let (X,, i~1) be a sequence of a square integrable random variables on the probability triple (f2, F, P) and put Fm=a(X~; n<=i<-m). For each m=>0, define ~o~,, = sup sup IP(BIA)-P(B)I. , ACr~,BC~',"+~,P(a)~0We denote E(X.IFm) by EmX,, ~q~ by a,. We also assume that i=1 f for all i, E Xi = O(n).where [x] is the greatest integer contained in x. We shall give sufficient conditions for the weak convergence of IV,, in Skorohod's space D=D[0, 1], el.[1], to the standard Brownian motion process on D, denoted by W. For an event A let I a denote its indicator.THEOREM. Let (Xi, i=>1) be a stochastic sequence satisfying (1) and assume that for every e>0, (2) lim 1 ~ EXi2i(rx, l>~a, gZ) = 0 n~ na n i=1 and (3) w.-~ w (i.e. the finite-dimensional distributions converge). Then IV, converges weakly to W.