Studia Scientiarium Mathematicarum Hungarica 33. (1997)
1-3. szám - Hurelbaatar G.: Almost sure limit theorems for dependent random variables
Studia Scientiarum A um hungarica 33 (1997), 167-175 ALMOST SURE LIMIT THEOREMS FOR DEPENDENT RANDOM VARIABLES G. HURELBAATAR Dedicated to Endre Csáki on the occasion of his sixtieth birthday Abstract For partial sums Sp of strongly mixing and associated random variables we prove that (1/logn) £(l/*)I{Sfc/a*€-}-+G(-) k^n with probability 1 if and only if (1/logn) £(l/fc)P(S*/a*S )-+G() k^n under the same moment condition as assumed for independent random variables. 1. Introduction One of the extensions of classical probability limit theorems is the socalled almost sure limit theorem. The basic result and starting point of these investigations is the almost sure central limit theorem, discovered by Brosamler [2] and Schatte [8] for i.i.d. random variables having finite (2 + <5)th moment and later proved by Fisher [3] and Lacey and Philipp [4] to hold under assuming only finite variance: Theorem. Let Xi,X2, EX2 = 1 and set Sn = X\ + • .. he i.i.d. random variables with EATi = 0, ■ • + Xn. Then A for any Borel-set A C E with A(cL4) = 0; moreover, the exceptional set of probability zero can be chosen to be independent of A. Here I denotes indicator function and A denotes the Lebesgue measure. Later Berkes and Dehling [1] proved a more general version of the almost sure central limit theorem and its functional version for independent, not necessarily identically distributed random variables. = (2n)~1/2 I e~t2/2dt a.s. lim -----n—>oo logn k<n 1991 Mathematics Subject Classification. Primary 60F15, 60F05. Key words and phrases. Strongly mixing, associated random variables, almost sure central limit theorem. 0081-6906/97/$ 5.00 ©1997 Akadémiai Kiadó, Budapest