Is the following statement true? An irreducible chain with a stationary distribution converges to it

No. The two-state chain that swaps deterministically (period 2) π = (½, ½) is stationary and unique, but pₙᵢⱼ oscillates between 0 and 1. Aperiodicity is required.

Counterexample

Is this true?

An irreducible chain with a stationary distribution converges to it

No — it is false.

The counterexample

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

Why it fails

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

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Keep practicing

Hypothesis dropped

A theorem applied without checking one of its preconditions.

Convergence in probability ⇒ almost sure convergence
Uncorrelated ⇒ independent
Pairwise independent events are mutually independent

See real exam options built on this mistake Drill statements like this

probability markov

Go deeper· Explained in full in Markov chains: classification and stationary behaviour· Asked in CSIR NET December 2025· From Limit Theorems and Markov Chains › Markov chains: classification of states, stationary distributions

Published 23 Aug 2026. Found a mistake? Tell us — corrections are the fastest thing we act on.

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