A remark on the strong law of large numbers

Abstract:

Let ${X_1},{X_2}, \ldots$ be mutually independent random variables such that $E({X_n}) = 0$ and $E(X_n^2) = \sigma _n^2 = 1$ for all $n = 1,2, \ldots$. For each $n = 1,2, \ldots$ let ${S_n} = \sum \nolimits _{j = 1}^n {{X_j}}$; then, by the Kolmogorov criterion for mutually independent random variables, ${S_n}/{n^{1/2 + \alpha }} \to 0$ almost surely as $n \to \infty$ for any positive constant $\alpha$. A deeper understanding of this theorem will be facilitated if we know the order of magnitude of $E\{ {N_\infty }(\alpha ,\varepsilon )\}$ as $\varepsilon \to {0^ + }$, where ${N_\infty }(\alpha ,\varepsilon )$ is the integer-valued random variable defined by ${N_\infty }(\alpha ,\varepsilon ) = \sum \nolimits _{n = 1}^\infty {{\chi _{(\varepsilon {n^{1/2 + \alpha }},\infty )}}(|{S_n}|)}$. The present note does the work for a wide class of random variables by using Esseen’s theorem and Katz-Petrov’s theorem.