Strong Limit Theorems for Certain Arrays of Random Variables

Abstract

A lemma concerning real sequences is proved and applied to sequences of random variables $(\mathrm{rv}) X_1, X_2\cdots$ to determine conditions under which $\lim\sup_{n\rightarrow\infty} b_n^{-1} \sum^n_{m=1} f(m/n)X_m < \infty$ a.s. for all $f$ in a particular collection of absolutely continuous functions and for nondecreasing positive real sequences $\{b_n\}$. Theorems in the case $b_n = (2n \log \log n)^\frac{1}{2}$ are proved for generalized Gaussian rv, for equinormed multiplicative systems and for certain martingale difference sequences.