= Accessible state of a Markov chain
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{title2=$i\to j\iff\exists n\geq0:(P^n)_{ij}>0$}
= Accessibility in a Markov chain
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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 $n\geq0$, allowing a zero-step path when $i=j$. Accessibility is reflexive and transitive; <communicating states> require accessibility in both directions. See https://www.columbia.edu/~ww2040/3106F13/lect0924.pdf[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.
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