Acta Mathematica Academiae Scientiarum Hungaricae 39. (1982)

1982 / 4. szám - Peligrad, Magda: A criterion for tightness for a class of dependent random variables

Acta Math. Acad. Sei. Hungar. 39 (4) (1982), 311—314. A CRITERION FOR TIGHTNESS FOR A CLASS OF DEPENDENT RANDOM VARIABLES MAGDA PELIGRAD (Bucharest) 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 (Xj, Ш1) be a sequence of a square integrable random variables on the probability triple (fl, F, P) and put FH' = cr(Xi; n^iSm). For each m^O, define <Pm = sup sup \P(B\A)-P(B)\. и AtF%,BíF?,XZ,P<.A)* о We denote E(Xn\FJ by EmXn, by a„. We also assume that where [x] is the greatest integer contained in x. We shall give sufficient conditions for the weak convergence of Wn, in Skorohod’s space D=D[0, 1], cf. [1], to the standard Brownian motion process on D, denoted by W. For an event A let IA denote its indicator. Theorem. that for every Let (X;, I si) be a stochastic sequence satisfying (1) and assume £> 0, (2) and (3) Wn^W (i.e. the finite-dimensional distributions converge). Then W„ converges weakly to W. (1) EXi = 0 For each í£[0, 1] put for all i, ^(jpi) =0(n). Wn(t) = [«!] 2Ъ t=i lim---- У, EX2I(iy. ^ = 0 nan fri ' (\xi\>eaJn> Acta Mathematica Academiae Scientiarum Hungaricae 39, 1982

Next