Convergence to an invariant distribution

A finite irreducible aperiodic Markov chain converges to its unique invariant distribution.

Table of contents
- Countable-state Markov chain convergence theorem Convergence to an invariant distribution

Countable-state Markov chain convergence theorem ( $P^{n} (i, j) \to π_{j}$ )

For an irreducible Markov chain on a finite or countable state space, a positive recurrent state implies a unique stationary distribution

π

. If the chain is also an aperiodic Markov chain, its transition probabilities satisfy the displayed limit for every pair of states. The same limit holds from any initial probability distribution by bounded convergence. Positive recurrence cannot be omitted in the countably infinite case, and aperiodicity is required for ordinary, rather than averaged, convergence. This is the standard discrete-time convergence theorem; the finite-state statement alone does not cover unbounded renewal countdowns.

Convergence to an invariant distribution

Countable-state Markov chain convergence theorem ( $P^{n} (i, j) \to π_{j}$ )

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Convergence to an invariant distribution

Countable-state Markov chain convergence theorem (Pn(i,j)→πj​)

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Countable-state Markov chain convergence theorem ( $P^{n} (i, j) \to π_{j}$ )

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