Studia Scientiarium Mathematicarum Hungarica 33. (1997)
1-3. szám - Berkes I.-Horváth L.: Almost sure invariance principles for logarithmic averages
Studio. Scientiarum Mathematicarum Hungarica 33 (1997), 1-24 ALMOST SURE INVARIANCE PRINCIPLES FOR LOGARITHMIC AVERAGES I. BERKES and L. HORVÁTH Dedicated to Endre Csáki on his sixtieth birthday 1. Introduction Let X\, X2,... be independent, identically distributed random variables with EX\ = 0, EXf = 1 and let Sn = X\ + • • • + Xn. By the a.s. central limit theorem (Brosamler [4], Schatte [15], Lacey and Philipp [11], Fisher [9]) (1.1) lim -—i < x l = ^»(x) a.s. for all x, yv^oo log N 1 y/k J where / denotes indicator function and 4> stands for the standard normal distribution function. Several papers dealt with ‘logarithmic’ limit theorems of the type (1.1) and many generalizations of (1.1) have been obtained. In particular, the following theorem extends (1.1) for a large class of independent sequences: Theorem A (Berkes and Dehling [2]). Let Xi,X2,..- be independent random variables and (an) a positive numerical sequence such that setting Sn = X 1 H— ■ + Xn we have (1.3) at/akZC(l/ky (lZkgl) for some positive constants C, K, 6 and 7. Then for any bounded Lipschitz 1 function f on R we have 1991 Mathematics Subject Classification. Primary 60F15; Secondary 60F17. Key words and phrases. Pointwise central limit theorem, logarithmic averages, a.s. invariance principle. The first author’s research has been supported by Hungarian National Foundation for Scientific Research Grants No. T 16384 and T 19346. 0081-6906/97/$ 5.00 ©1997 Akadémiai Kiadó, Budapest (1.2) E^log logSn 1+Í (n = 1,2,...) (1.4) um rAvEr (f(-)-El(-))=° yv-ioo logN \ \akJ \ak J ) (VI AG YAH ntíOOiVIÁNYOS AKADÉMIA KÖNYVTÁRA