Acta Mathematica Academiae Scientiarum Hungaricae 39. (1982)

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

312 MAGDA PELIGRAD In order to prove this theorem we need the following Lemma. Let (Xh i=l) be a stochastic sequence satisfying (1) and (2). Then for every t£[0, 1], Wn(t)=Z„(t) + Vn(t), where for every n, Zn(t) is a martingale and V„(t) converges to 0 in probability. Proof. Condition (2) implies that there exists a sequence of positive numbers d„ converging to 0 as n — such that (4) lim П-*- OO—i-y- УЕХ21,у. . na„d„ ifi ' IW = 0. For every и si, let us denote Xi (") = XiI{\Xt\»andnfi)-EXiI(\Xl\SandJT,)' Y'i (») = xih\xl\^oninyir)-EXiI(\xi\^aninarr and note that Xt = X' (n)+Y{ (n). Put and define the following random functions Obviously for every í£[0, 1], IV„(t)=Z„(t) + Y„(t) — U„(t) and for every n,Z„(t) is a martingale. We shall prove that the random elements U„ and ¥„ converge to 0 in probability. By the properties of Skorohod’s metric it is sufficient to prove that for every e>0 P(sup |Y„(i)l > e) - 0 (n - oo) i£[o.l] P( sup |C7„(i)| > e) - 0 (n - oo). We have P( sup |F„(0I > 8) ^ pf-L 2\Y,'(n)\ > e) ^ teio.i] If n i=i ) 2ЕЩ1^апЛпГп) ----%—Z EXfI( . . ^ ea„d„n ,=i ' (lxi\=~a',d„V") whence by (4) we obtain (5). In order to prove (6), note that, by Lemma 1.1.8 [2], it follows for j^i that \E.,X'j{n)\ S 2and„ fty'i-j a.s. (5) and (6) Zfn) = ZEiX'jin), j=i Ut(n) =n Z EiX'j(n), Yfn) = 2.Yj(n) l=i+1 j = 1 z„(0 =Z[„,](n) ft ’ u.(0 = ft ’ 7.(0 = Vn Acta Mathematica Academiae Scientiarum Hunqaricae 39, 1982

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