# Show that $\lim_{t\to \infty} 1/t \; \max_{n \leq t} S_n \to E[X]$ a.s

This is a twist on the strong law of large numbers.

$S_n = \sum_{k=1}^n X_k$ where $X_k$ are i.i.d.

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There is nothing stochastic here: to wit, assume that a real valued sequence $(s_n)$ is such that $s_n/n$ converges to a nonnegative limit $\ell$ and define a new sequence $(m_n)$ by $m_n=\max\{s_k;k\le n\}$, then $m_n/n$ converges to $\ell$.