22 - Existence and Uniqueness of Stationary Distributions

Existence and Uniqueness of Stationary Distributions

Let $P$ be an irreducible transition matrix then

1. $P$ has stationary distribution if and only if $P$ is positive recurrent

2. If $P$ has a stationary distribution but it is unique and given by

$${\pi }_i=\frac{1}{m_i}\text{ for all }i$$

where $m_i$ is the mean return time to state $i$