Studia Scientiarium Mathematicarum Hungarica 33. (1997)
1-3. szám - Berkes I.-Horváth L.: Almost sure invariance principles for logarithmic averages
2 I. BERKES and L. HORVÁTH Theorem A and standard properties of weak convergence (see e.g. Dudley [7], Theorem 8.3) imply that under (1.2), (1.3) the relations are equivalent. In particular, a sufficient condition for the a.s. central limit theorem (1.5) is Sn/an^N( 0,1). Condition (1.3) is satisfied, e.g., if n~7an is nondecreasing or if an =npL(n) where p > 7 and L is a slowly varying function. The purpose of the present paper is to prove a.s. invariance principles corresponding to relation (1.4). Our first result is the following Theorem 1. Let X\,X^, • • • be independent random variables, f: R-rR a bounded measurable function and (an) a positive numerical sequence such that (1.5) lim -----— — l\ — < x 1 = <&(x) a.s. for all x iV->oo log N k (afc J and lim ----— — P<[ — < x 1 = $(3:) for all x N-* 00 log N k \ ak J (1.6)E Sna <K (n = 1,2,...) (1.7) E sup mh -f forCa-l'2^h<\ (1.8) ai/ak^C(l/ky (1 gfcgJ) (1.9) Xn ■— Var £ \ f ( C7(log N)6 kZN for some positive constants K, C, a, ß, 7, Á satisfying (1.10) a> 8, ß>8, A >5/6. Then there exists a Wiener process W such that (L11) Ej(/(S)-^(t))=^)+o(4D -for some positive constant 77.