Stationary distributions
Objective. Find and interpret long-run distributions.
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- 2. Worked example
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Learn: the key idea
A stationary distribution of a Markov chain is a probability vector π that is unchanged by the transition matrix, satisfying πP = π, and it describes the long-run proportion of time spent in each state. For an irreducible, aperiodic chain, the distribution after many steps converges to this unique stationary π regardless of the starting state, so solving πP = π together with the normalisation Σπᵢ = 1 characterises long-run behaviour. Finding π reduces to finding the left eigenvector of P with eigenvalue 1, normalised to sum to one. Example: for P = [[0.5,0.5],[0.2,0.8]], solving πP = π gives π = (2/7, 5/7), the long-run fraction of time in each state.
Worked example
Find the stationary distribution of the Markov chain with transition matrix P = [[0.5, 0.5], [0.2, 0.8]].
- 1Write π = (π₁, π₂) with π₁ + π₂ = 1 and solve πP = π.
- 2First equation: 0.5π₁ + 0.2π₂ = π₁, which simplifies to 0.2π₂ = 0.5π₁, so π₂ = 2.5π₁.
- 3Substitute into π₁ + π₂ = 1: π₁ + 2.5π₁ = 1, so 3.5π₁ = 1, giving π₁ = 2/7.
- 4Then π₂ = 1 − 2/7 = 5/7, so the stationary distribution is π = (2/7, 5/7).
Answer: π = (2/7, 5/7)
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Start practisingCommon mistake. A common error is finding a right eigenvector of P (solving Pv = v) instead of a left eigenvector (solving πP = π); for a non-symmetric transition matrix these generally differ.
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