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.
Citation
R. J. Tomkins. "Strong Limit Theorems for Certain Arrays of Random Variables." Ann. Probab. 4 (3) 444 - 452, June, 1976. https://doi.org/10.1214/aop/1176996092
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