hitting time

Let ${(X_n)}_{n\ge 0}$ be a Markov Chain. Then the hitting time for a subset $A$ of $I$ (the indexing set) is the random variable:

$$H^A=inf\{n\ge 0:X_n\in A\}$$

(set $inf\mathrm{\varnothing }=\mathrm{\infty }$).

This can be thought of as the time before the chain is first in a state that is a member of $A$.

Wite $h_i^A$ for the probability that, starting from $i\in I$ the chain ever hits the set A:

When A is a closed class, $h_i^A$ is the absorption probability.