= Finite-interval exit times have geometric tails
For a nearest-neighbour <Markov chain> on $\{L,\ldots,U\}$ with positive right-step probabilities in its interior, let $m=U-L$ and $p_* >0$ be their minimum. From every interior state, at most $m$ consecutive right steps reach $U$, with probability at least $p_*^m$. The <Markov property> then gives $\mathbb P(T>km)\leq(1-p_*^m)^k$ for the first boundary <hitting time>. Thus $T$ is almost surely finite and has finite expectation, regardless of the <holding probability of a Markov chain> at each interior state.
// Target: probability-theory.bigb
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