When a stationary law fails to be unique
Section of Classes, recurrence, and reversibility, part of Stochastic processes
When a stationary law fails to be unique
| Type | Section |
| Taught in | Classes, recurrence, and reversibility |
| Part | Stochastic processes |
Suppose states 0 and 1 are both absorbing, with no transition between them. Both point masses \delta_0 and \delta_1 are stationary, and so is every mixture a\delta_0+(1-a)\delta_1. From a given start, the long-run law is the class the path enters. A finite chain has a unique stationary distribution only when its closed-class structure supports one.
Each finite closed irreducible class has its own unique stationary law, by the theorem above. Any stationary law on the whole chain is a mixture of those class laws, with zero mass on the transient states. That decomposition solves the stationary equations before any linear algebra, and it explains why eigenvalue 1 of P can have more than one independent left eigenvector.
Used in
See also
Recurrence and period · Finite irreducible chains · Detailed balance · Communicating classes · Return and period · Global balance and detailed balance · Two families with an explicit stationary law · Period and time averages