= Random-walk exit bound from a positive-increment block
{title2=$\mathbb ET\leq m/p^m$}
For a <random walk> with <independent and identically distributed> increments and an exit time from $(0,r)$, suppose $p=\mathbb P(X_1>r/m)>0$ for some integer $m\geq1$. A block of $m$ such positive increments forces an exit. Conditional on survival, every successive block has success probability at least $p^m$, so $\mathbb P(T>km)\leq(1-p^m)^k$ and $\mathbb ET\leq m/p^m$. This is an instance of the <geometric tail bound from a uniform escape probability>; centering of the increments is not required.
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