An interesting property of consecutive hitting times from Koralov&Sinai.
Consider a homogeneous ergodic Markov chain on the finite state space $X = \left\{1,\ldots,\ r\right\}$. Define the random variables $\tau_{n}$, $n \geq 1$, as the consecutive times when the Markov chain is in state $1$, that is $\tau_{1} = \inf (i \geq 0: \omega_{i} = 1 )$, $\tau_{n} = \inf(i > \tau_{n-1}: \omega_{i}=1)$, $n > 1$. Prove that $\tau_{1}$, $\tau_{2} - \tau_{1}$, $\tau_{3}-\tau_{2}$... is a sequence of independent random variables.
This should be pretty easy but I don't see it. Thanks