A certain class of stochastic summability methods of mantissa type is introduced and its connection to almost sure limit theorems
is discussed. The summability methods serve as suitable weights in almost sure limit theory, covering all relevant known examples
for, e.g., normalized sums or maxima of i.i.d. random variables. In the context of semistable domains of attraction the methods
lead to previously unknown versions of semistable almost sure limit theorems.
is also valid, where (dk) is a logarithmic weight sequence and DN = ∑k=1Ndk. In this paper we extend the last result for a very large class of weight sequences (dk), leading to considerably sharper results. We show that logarithmic weights, used traditionally in a.s. central limit theory,
are far from optimal and the theory remains valid with averaging procedures much closer to, in some cases even identical with,