= Infinite mean first passage of a simple symmetric random walk
{title2=$\mathbb ET_1=\infty,\quad\mathbb P(T_1<\infty)=1$}
For a <simple symmetric random walk> starting from zero, its first <hitting time> of $1$ is almost surely finite but has infinite <expectation>. Stopping at exit from $[-m,1]$, the <optional stopping theorem> for $S_n$ gives upper-exit <probability> $m/(m+1)$, while stopping $S_n^2-n$ gives mean exit time $m$. The first <probabilities> tend to one, and the exit times are bounded above by the first passage to $1$, proving both conclusions. Finite-time stopping and bounded stopped positions justify passage to the exit time.
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