Here is my question:
Consider a homogeneous ergodic Markov chain on a finite state space $X=\{1,\ldots,r\} $. Define the random variables $\tau_n \,,n\ge1$ as the consecutive times when the Markov chain is in state 1, that is $\tau_1 = \inf(i\ge0 : w_i=1) $, $\tau_n = \inf(i > \tau_{n-1} : w_i =1)\,,n>1$. Prove that $\tau_1\,,\tau_2-\tau_1\,,\tau_3-\tau_2\,,\ldots $ are independent random variables.
Notice that "homogeneous Markov chain" means that the transition matrix are time independent. $w_i$ stands for the state of the Markov chain at step $i$.