State Communication

Topics: Stochastic Process

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(definition)

Given a Markov chain, we say that the state $j$ is accessible from the state $i$ if $p_{ij}^{(n)}>0$ for some $n\ge 0$.

Do note that we can have the case of not having the state $i$ being accessible from the state $j$, but having the state $j$ accessible from $i$.

When both cases are satisfied (i.e. $i$ accessible from $j$, and $j$ accessible from $i$), we say that both states communicate.

We can partition the set of states in a given chain by grouping them into classes, each one containing states that all communicate.

Properties §

• Every state communicates with itself (since $p_{ii}^{(0)}=1$; reflexive).
• If $i$ communicates with $j$, then $j$ communicates with $i$ (symmetric).
• If $i$ communicates with $j$, and $j$ communicates with $k$, then $i$ communicates with $k$ (transitive).