If a chain starts in state i , is it bound to return to i at some later time?

Definition 12.30 A state i is called recurrent if Pi (Ti < ∞) = 1. A state is called transient if it is not recurrent.2 Here is a criterion for recurrence in terms of the transition matrix P and its powers.

Theorem 12.31 The state i is recurrent if and only if

∞X n=0 pi,i (n) = ∞.