Cheatsheet: State Classification

Property	Definition
Accessible ( $i \to j$ )	$P_{ij}^{n} > 0$ for some $n$
Communication ( $i \leftrightarrow j$ )	$i \to j$ and $j \to i$
Recurrent	$f_{i} = 1$ (returns infinitely often)
Transient	$f_{i} < 1$ (finite expected visits)
Periodic	Returns at regular intervals ( $d > 1$ )
Aperiodic	Returns at irregular times ( $d = 1$ )
Irreducible	All states communicate
Regular	$P^{k}$ has all positive entries for some $k$

Relationships