Almost Certain Summability of Independent, Identically Distributed Random Variables

Abstract

The Strong Law of Large Numbers, valid for independent, identically distributed (i.i.d.) random variables $\{X_n, n \geqq 1\}$ with finite first moment, may be regarded as merely one of a host of summability methods applicable to the divergent sequence $\{X_n\}$. Here, a subclass of regular (Toeplitz) summability methods will be considered and concern will focus on the almost certain (a.c.) convergence to zero of the transformed sequence \begin{equation*}\tag{1}T_n = A_n^{-1} \sum^n_{j=1} a_jX_j\end{equation*} when centered where \begin{equation*}\tag{i}a_n \geqq 0,\quad A_n = \sum^n_{j=1} a_j \rightarrow \infty,\end{equation*} thereby ensuring regularity. If $T_n - C_n \rightarrow_{\operatorname{a.c.}} 0$ for some choice of centering constants $C_n$, the i.i.d. random variables $\{X_n\}$ will be called $a_n$-summable with probability one or simply $a_n$-summable. The Strong Law is the special case $(a_n \equiv 1)$ of Cesaro-one summability with $C_n \equiv EX$. Of course, if $X^\ast_n = X_n - X'_n, n \geqq 1$ are the symmetrized $X_n$ (i.e., $\{X'_n\}$ is i.i.d., independent of $\{X_n\}$ with the same distribution), then $a_n$-summability of $\{X_n\}$ implies $a_n$-summability of $\{X^\ast_n\}$ with vanishing centering constants, i.e. \begin{equation*}\tag{2}T^\ast_n = A_n^{-1} \sum^n_{j=1} a_jX^\ast_j \rightarrow_{\operatorname{a.c.}} 0.\end{equation*} It will be shown, on the one hand, that no such choice of $\{a_n\}$ and $\{C_n\}$ will render i.i.d. $\{X_n\}$ with the St. Petersburg (mass $2^{-n}$ at the point $2^n, n \geqq 1$) or Cauchy distribution $a_n$-summable. On the other hand, necessary and sufficient conditions for certain types of $a_n$-summability more refined than (implied by) Cesaro-one will be proffered. The prototype of these appears in Corollary 1 and Corollary 2.