Stationary distribution
Quantitative Finance glossary · Probability, Statistics & Linear Algebra
Definition
A probability distribution over the states of a Markov chain that is left unchanged by the chain's dynamics.
Stationary distribution in plain English
Picture a system that hops randomly between states — like weather flipping between sunny, cloudy, and rainy, where tomorrow's odds depend only on today. A "stationary distribution" is the special mix of how often you'd find the system in each state that stays fixed forever: if today's chances match this mix, then after one more random hop the chances are exactly the same. It's the long-run equilibrium the system settles into no matter where it started — so if the stationary distribution says 40% sunny, then over many years about 40% of days end up sunny. Quants care because it tells you the steady-state behavior of anything modeled as random hops, from credit ratings drifting to markets switching between calm and volatile regimes.
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