"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.