Stationary Distribution¶

Definition¶

A stationary distribution for a time-homogeneous Markov chain with transition matrix $\boldsymbol{P}$ is a probability row vector $\boldsymbol{\pi}$ such that

\boldsymbol{\pi}\boldsymbol{P}=\boldsymbol{\pi},\qquad \pi_i\ge0,\qquad \displaystyle\sum_i\pi_i=1.

Starting the chain with distribution $\boldsymbol{\pi}$ leaves its distribution unchanged at every subsequent step.

Interpretation¶

Stationarity describes a distribution that is preserved by a transition. It does not by itself say that the chain approaches this distribution from a different starting point. In a bonus-malus model, stationary class proportions produce a mean premium by weighting each class premium by its stationary probability.

Key Results¶

For a finite state space, at least one stationary distribution exists.
Finite irreducibility gives uniqueness and strictly positive entries. Additional aperiodicity gives convergence from every initial distribution.
To calculate a finite stationary distribution, solve the linear system $\boldsymbol{\pi}(\boldsymbol{P}-\boldsymbol{I})=\boldsymbol0$ together with the normalization equation. One balance equation can be replaced by the normalization equation when solving the system; always check nonnegativity.
A countably infinite irreducible chain need not have a stationary probability distribution; the finite-state conclusion cannot be transferred without further assumptions.

Examples¶

For $\boldsymbol{P}=\begin{pmatrix}0.8&0.2\\0.3&0.7\end{pmatrix}$ $(0.8 0.3 0.2 0.7)$ , the balance equation is $0.2\pi_0=0.3\pi_1$ $0.3 π_{1}$ . Together with $\pi_0+\pi_1=1$ $1$ , this gives $\boldsymbol{\pi}=(0.6,0.4)$ $(0.6, 0.4)$ .

For the alternating chain $\boldsymbol{P}=\begin{pmatrix}0&1\\1&0\end{pmatrix}$ $(0110)$ , $(1/2,1/2)$ is stationary, but a chain started in state 0 alternates forever. A stationary distribution and convergence are different properties.

Sources¶

MIT Introduction to Probability, Lectures 24–25.
Course placement: RUG Stochastic Models, 2026–2027. These foundation notes have no assigned lecture yet.

Prerequisites¶

Aliases: invariant distribution, stationary probability vector Symbols: pi, sum