Acta Mathematica Academiae Scientiarum Hungaricae 56. (1990)
1990 / 1-2. szám - Szyszkowski, I.: An invariance principle for dependent random variables
46 I. SZYSZKOWSKI 2. Results The main results of this paper are given in the following three theorems: Theorem 1. Let {X-t, /£l} be a centered ф-mixing sequence of random variables having finite second moments, satisfying (2) lim .VjT2 2 EXfI(\Xi\ > £.9„) = 0 for every s =- 0 i = l and (3) lim y^~1( max E\Xj\) 2 <А(0 = 0. П-+оо 1^ J=n i_i Then (4) W„ -► W weakly in £>[0, 1], n — <=>, provided the sequence {kn, n gt 0} satisfies (1) and (5) s\ = k„h(k„), where h: R+ -* R+ is a slowly varying function. Theorem 2. Let {Xh /=1} be a centered ф-mixing sequence of random variables having finite (2 + S)-th moments for some <5 >0, (2') lim sk2-5 2 ВД|2+г = 0 i=i and (3') 2\^(i)](2+mi+6) 1=1 Then, for every sequence {k„, nS0} satisfying (1) and (5), W„W weakly hi D[0, 1], as и-*-«>. Theorem 3. Let {Xh /Si} be a centered cp-mixing sequence of random variables having finite second moments satisfying (2") lim sy2(L 2 \Xi\I(\Xi\ > £-0)2 = 0 for every e > 0, i=i and (3'0 2 <P(0 < °°;=i Then, for every sequence {kn, nSO} satisfying (1) and (5), the invariance principle (4) holds. Remark 1. (i) If <p(l)<l, then condition (2) is necessary for the invariance principle. The proof of this fact is essentially the same as that given by Peligrad [11, Proposition 2.2]. (ii) Condition (5) (with {kn, nsO} satisfying (1)) is necessary for (4), too. In fact it is enough to apply the method presented by Herrndorf in [2, Remark 2.3] Acta Mathematica Hungarica 56, 1990