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I'm stuck with this question saying that, we have a homogeneous MC which is indecomposable and aperiodic, if $\tau$ is the random time of return back to any fixed state. Show that there exist $C>0$ and $q<1$ such that $P(\tau>n) \leq Cq^{n}$. I'm not sure whether the question is missing some information about what's $q$ here. Does anyone have an idea?

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I guess you should assume that your MC is finite, otherwise $\tau$ can be infinite. – Ilya Nov 7 '11 at 10:23
yes, probably that's one assumption i need. but how do i start to prove that inequality anyway? – Math Nov 7 '11 at 10:37

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