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Invariant Measure

Definition (Invariant measure)

Let $P$ be a transition matrix. A non-negative function $\lambda:S\to[0,\infty)$ is said to be an invariant measure if $\lambda=\lambda P$

Lemma (Positivity of Invariant Measure)

Let $P$ be an irreducible transition matrix, and $\lambda$ be an invariant measure. If $\lambda\not=0$ then $\lambda_{i}>0, \ \mbox{for }i\in S$

Theorem (Uniqueness of the Invariant Measure)

Let $P$ be an irreducible Markov chain, and $\lambda$ be an invariant measure with $\lambda_{k}=1$ then

$\lambda\ge\gamma^{k}$
If in addition $P$ is recurrent, then $\lambda=\gamma^{k}$

Linked from

Ergodic Theorem

Invariant Distribution for CTMC

Invariant Distribution

Invariant Measure