On the almost sure behaviour of sums of random variables (Q1903165)

For \(S_n = \sum^n_{i = 1} X_i\), where \((X_i)\) is a sequence of arbitrary random variables, it is examined that \(\lim_{i \to \infty} n^{1/2} \psi (u) |S_n |= \infty\) a.s. for some positive non-decreasing functions \(\psi (x)\) under conditions on the characteristic functions of \(S_n\) without any assumption on their moments. These results are generalizations and strengthenings of some previous results, and the Lévy concentration functions are used in proving the theorems. Special attention is paid to sequences of independent nonidentically distributed random variables.