If I have a transition matrix defined as $p_{ij}=1$ for $j=i+1$ and $p_{ij}=0$ otherwise, where the state-space is countably infinite, what would this be? It doesn't look like a communicating class because no 2 states communicate but then the "communication" relation partitions the state-space, hence the entire state-space must be one big communicating class here? Then the whole thing is closed? But that shouldn't be right...
Clarification on the definition of a closed communicating class
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$\begingroup$
stochastic-processes
markov-chains
1 Answers
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Correct me if you use the wrong definition of the communication, I base on the book 'Markov Chains and Stochastic Stability'. There $i,j$ communicate iff either $i=j$ or $p^m_{ij}>0$ and $p^n_{ji}>0$ for some $0
To be precise, in your question the statement
no 2 states communicate
is incorrect since any state communicate with itself by definition.