Geometric tail bound from a uniform escape probability

Suppose a nonnegative stopping time $T$ and constants $L>0$, $0<q\le 1$ satisfy $\mathbb{P}(T\le nL+L\mid {\mathcal{F}}_{nL})\ge q$ on $\{T>nL\}$. Take $L$ to be a positive integer in discrete time. ThenOn the surviving event, conditional survival is at most $1-q$. The tower property of conditional expectation therefore givesInduction proves the geometric bound. Partitioning the tail integral into intervals of length $L$ then gives $\mathbb{E}T\le L{\sum }_{n\ge 0}\mathbb{P}(T>nL)\le L/q$; in discrete time, group the tail sum into $L$ successive integer indices instead. This proves finite expected hitting times when every surviving state has a uniform positive chance of escaping within a fixed time. It works in discrete or continuous time without assuming independence of successive survival events.

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 $m\ge 1$. A block of $m$ such positive increments forces an exit. Conditional on survival, every successive block has success probability at least $p^m$, so $\mathbb{P}(T>km)\le (1-p^m{)}^k$ and $\mathbb{E}T\le m/p^m$. This is an instance of the geometric tail bound from a uniform escape probability; centering of the increments is not required.