Communicating classes
Section of Classes, recurrence, and reversibility, part of Stochastic processes
Communicating classes
| Type | Section |
| Taught in | Classes, recurrence, and reversibility |
| Part | Stochastic processes |
| Questions using it | 2 |
State j is accessible from i, written i\to j, when p_{ij}^{(n)}>0 for some n\ge 0. The case n=0 counts: every state is accessible from itself. States i and j communicate when each is accessible from the other. Communication is an equivalence relation, so it partitions S into classes. A class is closed when no transition leaves it. A chain with a single communicating class is irreducible. An absorbing state is a closed class of one state, with p_{ii}=1.
On a finite set, the states that can still reach a given closed class, and are not in it, are on their way out. They are visited only finitely often.
Used in
- Taught in Classes, recurrence, and reversibility
- Also used in Markov chains
- Questions that use it (2) — The limit of the three-state chain (answer) · One closed class (answer)
- All 3 questions on Classes, recurrence, and reversibility
See also
Aperiodic Markov chain · Irreducible Markov chain · Transient and recurrent states · Recurrence and period · Finite irreducible chains · Detailed balance · Return and period · Global balance and detailed balance