Stationarity and the law from one start
Section of Discrete-time Markov chains, part of Stochastic processes
Stationarity and the law from one start
| Type | Section |
| Taught in | Discrete-time Markov chains |
| Part | Stochastic processes |
If the chain starts from \pi, every later marginal equals \pi. The path still moves. Stationarity is a statement about marginal distributions, not a claim that the chain freezes.
If the chain starts in a single state i, the law \mu_0 P^n may take time to approach \pi. In a periodic chain it may fail to approach \pi at individual times, even though \pi exists. The class structure, the period, and the conditions that turn a stationary law into a limit from every start are on Classes, recurrence, and reversibility. The limit theorem for an irreducible aperiodic finite chain, with the stepping-stone calculation, is on Markov chains.
One row of P is the law of the next state given the current state. Hitting probabilities and mean hitting times solve first-step equations with boundary values on the target. A vector \pi=\pi P is unchanged by one step. Convergence of P^n to that vector needs more than the equation \pi P=\pi.
Used in
See also
Discrete-time Markov chain · Stationary distribution · Transition matrix · Hitting probabilities · Spectral view for two states · Markov assumption as a chosen state · Doubly stochastic matrix