I am trying to understand the concept of Markov chains, classes of Markov chains and their properties. In my lecture we have been told, that for a closed and finite class of a discrete Markov chain it holds that $$P_j(\text{infinitely often visit }k)= 1 $$ for any $j,k$ in this closed and finite class. Is there an easy way to see that this statement ist true? Is this the case because finite and closed classes are recurrent?
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What do you mean by "closed" here? Typically it refers to an absorbing subset of the state space, but I'm not sure what it means applied to the Markov chain itself. – Frank Seidl May 19 '21 at 13:16
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@FrankSeidl, that's what it means here. A communication class of states is called closed if the Markov chain cannot leave the class once inside. – Joe May 19 '21 at 13:23
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Yes, thats exactly how we defined it – putti.123 May 19 '21 at 13:26
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When you say "easy way to 'see' that this statement is true", does that mean an easy proof, or an intuitive explanation? For the latter, yes, because the class is recurrent, it will return to $j$ infinitely often with probability 1; and each time it returns to $j$ it has the same positive probability to visit $k$ from $j$, so it will visit $k$ infinitely often. – Joe May 19 '21 at 13:45
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so this does mean that every finite and closed communication class is recurrent? That was a statement i didn't know was true, i just assumed it. But okay then i can see that it makes sense – putti.123 May 19 '21 at 13:49
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Yes, if a closed communication class is finite, then it must visit at least one state infinitely often. But since that state communicates with each of the other states in that class, it will visit those states infinitely often as well. – Joe May 19 '21 at 13:52
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okay thank you very much! it really helped to get an intuition for the definitions. – putti.123 May 19 '21 at 13:56
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You're welcome. – Joe May 19 '21 at 14:21