Period of Markov Chain

Definition

Let $\{X_t\}$ be a HMC defined on the state space $E$, and the set $I_i$ be the set of the time steps returning from the $i\in E$ to oneself $I_i=\{t:p_{ii}^{(t)}>0,t\ge 1\}$.

The period of $i$ is the G.C.D of every element of $I$