Let $\{X_t\}_{t \geq 0}$ be a Markov Chain on some state space $S$, where there is exactly one absorbing state $a \in S$ and at least one state $b \in S \setminus \{a\}$ such that $b$ communicates with $a$. Let $X_t$ be irreducible on $S \setminus \{a\}$.
If the initial probability mass of $X_0$ is such that $\mathbb{P}(X_0 = a) = 0$, then clearly $\lim_{t \to \infty} \mathbb{P}(X_t = a) = 1$ by properties of the stationary distribution.
I want to know if it's also true that $\mathbb{P}(\lim_{t \to \infty} X_t = a) = 1$, i.e. that the chain $X_t \to a$ almost surely?
Thanks!