Is it possible to derive strong law of large numbers for a Harris recurrent stationary Markov chain form Birkhoff Ergodic Theorem? As I know that there is a link between SLLN for iid sample and Birkhoff Ergodic Theorem. In particular, does there any link exist in the literature in between Harris recurrence and ergodicity in Birkhoff sense?
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$\begingroup$ When you stationary, do you mean that there's an invariant distribution (not necessarily a finite measure)? If so, you can build an invariant measure for the Markov chain process and apply the ergodic theorem. This is mainly useful if there is a finite invariant distribution. You can see this done for the finite state case in Walters' book for example. $\endgroup$– Anthony QuasCommented May 23, 2015 at 5:18
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$\begingroup$ Yes, I mean under invariant distribution. Ergodic theorem for Markov chain has different version than Birkhoff Ergodic theorem. Ergodicity of a Markov chain is defined as time average in path space. So I'm looking for specific connection between Birkhoff Ergodic theorem and SLLN for Harris recurrent Markov chain. $\endgroup$– user74016Commented May 24, 2015 at 1:12
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$\begingroup$ So sure. There's a direct connection. If you have an invariant probability distribution for the Markov chain, then you can build from it an invariant measure (in the ergodic theory sense) on the space of paths in the space $S^\mathbb Z$. The Birkhoff ergodic theorem can then be applied to this measure (with the transformation being the shift transformation). $\endgroup$– Anthony QuasCommented May 24, 2015 at 2:02
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