Assume that $X$ is a Markov chain on a finite state space $S$ with $A\subset S$ a set of absorbing states for $X$. Assume that for any state $x\in S$ the the transition probability for $X$ to go from $x$ to $A$ is bounded from below as $\inf_{x\in S} p_{x,A}\geq \alpha > 0$. Is there a criterion which allows me to say something about the expected time to absorption $T_A$? Intuitively it should be something like $\mathbb{E}[T_A]\leq C\alpha^{-1} $ where $C$ is some constant.
May you give me a hint or am I completely mistaken?
