= Uniqueness of stationary distributions for an irreducible countable chain
An <irreducible Markov chain> on a finite or countably infinite <state space> has at most one <stationary distribution>. On a countably infinite state space it exists exactly when the chain is <positive recurrent>. In that case, for every state $i$,
$$
\pi_i=\frac1{\mathbb E_iT_i^+},
$$
where $T_i^+$ is the <first return time>. The expression determines every component uniquely. The formula follows by dividing the one visit to $i$ in each return cycle by the <mean recurrence time>; the <strong law of large numbers> for successive cycles makes this occupation fraction precise.
Every irreducible <finite Markov chain> is positive recurrent and has exactly one stationary distribution. On an infinite state space, an irreducible <null recurrent Markov chain> or <transient Markov chain> has none. For example, the <simple symmetric random walk> on the integers is null recurrent. <Aperiodicity> affects convergence of the one-time laws, but is unnecessary for existence or uniqueness of a stationary distribution.
Back to article page
