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I would like to ask if any of you know of a good resource containing rigorous proof (using sigma-algebras) of Markov Property and Strong Markov Property respectively in terms of Discrete Time Markov Chains.

(All resources I have found dealt with Continuous Time Markov Chains or with discrete MP/SMP in terms of Brownian Motion.)

Thanks, C.

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up vote 0 down vote accepted

Varadhan's "Probability Theory" has a proof of the SMP for discrete-time chains.

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You might also want to look at Kai Lai Chung's text as well, which spends some time looking at discrete MC's. – A Blumenthal Feb 26 '13 at 19:32
Thank you, that's just what I've been looking for. – Carlos Feb 26 '13 at 20:03

See Theorem 6.3.4. of Probability: Theory and Applications (4th edition) by Richard Durrett.

Also, a very accessible proof is found in Theorem 1.4.2 of Markov Chains by J.R. Norris.

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