Almost sure limit theorems of mantissa type for semistable domains of attraction

Abstract

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.

A General Multiparameter Version of Gnedenko's Transfer Theorem

“…Although B n n k=1 (X k − b n ) does not converge in distribution, it is possible to achieve a limit distribution for random sums by a transfer type theorem as shown in [5]. For n ∈ N and the increasing sampling sequence (k n ) n∈N define k 0 = 1 and (3.3) ψ(n) = t n if n = k pn t n with p n ∈ N 0 given by k pn ≤ n < k pn+1 .…”

“…Further examples of random variables T n (corresponding to summability methods) with ψ(T n ) converging to the logarithmic distribution are given in [5]. , 0 ≤ r ≤ n denote the standardized random variables.…”

General multiparameter version of Gnedenko's transfer theorem

“…Although in general Bn n k=1 (X k − bn) does not converge in distribution, it is possible to achieve a limit distribution for random sums by a transfer type theorem as shown in [5]. For n ∈ N and the increasing sampling sequence (kn) n∈N define k 0 = 1 and ψ(n) = t n if n = kp n tn with pn ∈ N0 given by kp n n < kp n +1.…”

“…Further examples of random variables Tn (corresponding to summability methods) with ψ(Tn) converging to the logarithmic distribution are given in [5]. 3.3.…”