"Almost sure limit theorems for sums and maxima from the domain of geometric partial attraction of semistable laws"

István Berkes, Endre Csáki, Sándor Csörgö and Zoltán Megyesi

Summary: The possible limiting distributions of sums of independent identically distributed random variables along geometric subsequences are the semistable laws and the domain of geometric partial attraction of a semistable law consists of distributions attracted to it along such a subsequence. The aim of this paper is to show that sums and maxima from the domain of geometric partial attraction of a semistable law satisfy almost sure limit theorems along the whole sequence of natural numbers, despite the fact that ordinary convergence in distribution typically takes place in both cases only along subsequences. We describe the class of all possible almost sure asymptotic distributions both for sums and maxima.

Keywords: Semistable laws; domains of geometric partial attraction; sums and maxima; asymptotic distributions; logarithmic averages; almost sure limit theorems.

AMS 1991 subject classification: 60F15; 60E07.