The bookUnit 4 · Limit Theorems and Markov Chains75 / 83

Markov chains: classification of states, stationary distributions

Why this is asked: Classify states (recurrent/transient, periodicity), then use irreducible + aperiodic + positive recurrent ⇒ unique stationary distribution with πⱼ = 1/mⱼⱼ.

Markov chains: classification and stationary behaviour

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Why periodic chains never settleinteractive

A cycle with stationary distribution (⅓,⅓,⅓) that never converges to it — until one slider makes it aperiodic.

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The trap here

“An irreducible chain with a stationary distribution converges to it” — false

The two-state chain that swaps deterministically (period 2)

$\pi = ($ ½, ½) is stationary and unique, but $p_{nij}$ oscillates between 0 and 1. Aperiodicity is required.