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.
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.
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 $\ell$.
The usual epsilon-delta approach works.