Assumption: Let $P$ be an irreducible Markov chain on a finite state space $S$.

Theorem: For any target state $i \in S$ and any initial state $j \in S$, the tail probability of the hitting time $T_i$ decays exponentially. That is, there exist constants $m \in \mathbb{N}_+$ and $\epsilon > 0$ such that

$$ \mathbb{P}_j(T_i > km) \le (1 - \epsilon)^k, \quad \forall k \in \mathbb{N}_+. $$

Corollary: Since the tail probability is dominated by a geometric sequence, every positive integer moment of the hitting time exists and is finite:

$$ \mathbb{E}_j[T_i^n] < +\infty, \quad \forall n \in \mathbb{N}_+. $$