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TL;DR

A probability distribution \( \pi \) satisfying \( \pi P = \pi \) for a Markov chain with transition matrix P. If the chain is irreducible and aperiodic, the stationary distribution is unique and the chain converges to it.

By Valenke Exam Prep Team·Last updated 2026-06-03

Stationary Distribution

A probability distribution \( \pi \) satisfying \( \pi P = \pi \) for a Markov chain with transition matrix P. If the chain is irreducible and aperiodic, the stationary distribution is unique and the chain converges to it.

Why it matters for interviews

Stationary distributions describe the long-run behavior of Markov chains, used in MCMC sampling, equilibrium analysis of market models, and credit rating long-run forecasts.

Definition and Mathematical Foundation

A probability distribution \( \pi \) satisfying \( \pi P = \pi \) for a Markov chain with transition matrix P. If the chain is irreducible and aperiodic, the stationary distribution is unique and the chain converges to it.

Application in Quantitative Finance

Stationary distributions describe the long-run behavior of Markov chains, used in MCMC sampling, equilibrium analysis of market models, and credit rating long-run forecasts.

Related Terms

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Frequently Asked Questions

When does a unique stationary distribution exist?▼
When the Markov chain is irreducible (every state reachable from every other) and positive recurrent. For finite irreducible chains, a unique stationary distribution always exists. Aperiodicity additionally guarantees convergence.
What is detailed balance?▼
A condition \( \pi_i P_{ij} = \pi_j P_{ji} \) for all i,j. If detailed balance holds, \( \pi \) is stationary. This is stronger than stationarity and is the basis for MCMC algorithms like Metropolis-Hastings.
How is the stationary distribution used in MCMC?▼
MCMC constructs a Markov chain whose stationary distribution is the target distribution (e.g., a Bayesian posterior). By running the chain long enough, samples approximate draws from the target distribution.