# Numbers (Extended Mix)

Consider a sequence $X_1, X_2, \ldots$, of i.i.d. random variables. For each integer $m \geq 1$ let $S_m$ denote the $m$th partial sum of these random variables and set $S_0 = 0$. Assuming that $EX_1 \geq 0$ and the moment generating function $\phi$ of $X_1$ exists in a right neighborhood of 0 the Erdos-Renyi strong law of large numbers states that whenever $k(n)$ is a sequence of positive integers such that $\log n/k(n) \sim c$ as $n \rightarrow \infty$, where $0

## Numbers (Extended Mix)

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