Accessible state of a Markov chain ($i\to j⟺\exists n\ge 0:(P^n{)}_{ij}>0$)

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.