Stationary Distributions and Long-Run Behaviour
A distribution left unchanged by one step is an eigenvector of the transition matrix for eigenvalue one. Irreducibility and aperiodicity are the conditions under which every starting distribution converges to it.
What a learner can do afterwards
- Solve for a stationary distribution and check that it sums to one
- Decide whether a chain is irreducible and aperiodic
- Say what periodicity does to the long-run behaviour
1 · Read
A distribution across the states is a list of nonnegative weights summing to one. A stationary distribution is one the transition matrix leaves unchanged, written pi equals pi P. It is an eigenvector for eigenvalue one that also happens to be a probability vector. To find one, solve pi equals pi P together with the sum-to-one condition, then check the entries sum to one.
Take the two-state chain with rows (0.8, 0.2) and (0.3, 0.7). Write pi as (a, 1 minus a) and impose pi equals pi P. The first equation is a equals 0.8a plus 0.3(1 minus a), which simplifies to 0.5a equals 0.3. Hence a is 0.6, so the stationary distribution is (0.6, 0.4), and 0.6 plus 0.4 is indeed 1.
Whether the chain actually converges to pi depends on two words. Irreducible means every state can reach every other state. Aperiodic means the return times to a state have greatest common divisor 1. A chain that flips between two states every step is irreducible but periodic with period 2: each state reaches the other, yet returns take an even number of steps. Periodicity makes the powers oscillate instead of settling.
A stationary distribution can exist without convergence. The flip-flop chain has stationary vector (0.5, 0.5) even though its powers oscillate forever, so the fixed-point equation is a weaker demand than convergence. An absorbing state that loops to itself with probability 1 has period 1, since every positive time is a return time. Always separate the two questions: does pi exist, and do the powers converge to it.
Solve pi equals pi P for the resting weights, then check irreducibility and aperiodicity for convergence.
2 · Watch
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Where this leads
Jobs that lean on this skill. Follow one to see everything it is built on.
8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.