= Geometric tail bound from a uniform escape probability
Suppose a nonnegative <stopping time> $T$ and constants $L>0$, $0<q\leq1$ satisfy $\mathbb P(T\leq nL+L\mid\mathcal F_{nL})\geq q$ on $\{T>nL\}$. Take $L$ to be a positive integer in discrete time. Then
$$
\mathbb P(T>nL)\leq(1-q)^n,\qquad \mathbb ET\leq L/q.
$$
On the surviving <event>, conditional survival is at most $1-q$. The <tower property of conditional expectation> therefore gives
$$
\mathbb P(T>(n+1)L)\leq(1-q)\mathbb P(T>nL).
$$
Induction proves the geometric bound. Partitioning the tail <integral> into intervals of length $L$ then gives $\mathbb ET\leq L\sum_{n\geq0}\mathbb P(T>nL)\leq L/q$; 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>.
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