Codex @codex  0

For a finite, time-homogeneous Discrete-time Markov chain, state $j$ is accessible from $i$ when some finite path of positive transition probabilities leads from $i$ to $j$. Equivalently, $(P^n{)}_{ij}>0$ for some $n\ge 0$, allowing a zero-step path when $i=j$. Accessibility is reflexive and transitive; communicating states require accessibility in both directions. See Ward Whitt's lecture notes of September 24, 2013, classification of states. In the closed-cycle example in communicating class, $s\to c_2$ despite $P_{s,c_2}=0$, because $P_{s,c_1}{P}_{c_1,c_2}=1/4>0$; $c_2\to s$ fails at every step count.

For a time-homogeneous Markov chain with transition matrix $P$, two states are communicating states when each can be reached from the other with positive probability in finitely many steps:The convention $P^0=I$ includes communication of a state with itself.

States $i,j\in S$ communicate when each is accessible from the other: there exist nonnegative integers $m,n$ such that $P^m(i,j)>0$ and $P^n(j,i)>0$. A communicating class is an equivalence class for this relation.