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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

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This is simply because the Markov chain has no memory. That is, once you've reached state $1$, there's no record of how you reached it, and in particular how long it took you to reach it; therefore how long it will take you to hit it in the future ($\tau_2-\tau_1$) can't depend on how long it took you to hit it in the past ($\tau_1$).

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    @Dquik: I think the ergodicity is required not so much for the independence but for the existence of these times. It would be enough though if all states were recurrent; I don't see how we need aperiodicity or positive recurrence here.2012-10-23