A Note on Feller's Strong Law of Large Numbers

Abstract

Let $X_n, n \geq 1$, be i..d. random variables with common distribution function $F(x)$ and $\gamma_n, n \geq 1$, be a sequence of constants such that $\gamma_n/n$ is nondecreasing in $n$. Set $S_n = X_1 + \cdots + X_n$. The main theorem of this paper gives an integral test which determines the infinite limit points of $\{S_n/\gamma_n\}$. This result extends and combines Feller's (1946) strong law of large numbers (SLLN) and the results Kesten (1970) and Erickson (1973).