Transient and recurrent states
Topic of Classes, recurrence, and reversibility, part of Stochastic processes
Transient and recurrent states
| Type | Topic |
| Taught in | Classes, recurrence, and reversibility |
| Part | Stochastic processes |
| Questions using it | 2 |
State i is recurrent when a chain started there returns with probability one, and transient otherwise. A recurrent state is positive recurrent when the expected return time is finite and null recurrent when it is infinite. In a finite chain every state of a closed class is positive recurrent.
In Classes, recurrence, and reversibility
State i is recurrent when \mathbb{P}_i(T_i^+<\infty)=1, and transient otherwise. A recurrent state is positive recurrent when \mathbb{E}_i T_i^+<\infty, and null recurrent when the return time is almost surely finite and has infinite mean. The period of i is
d(i)=\gcd\{n\ge 1: p_{ii}^{(n)}>0\}.
Used in
- Taught in Classes, recurrence, and reversibility
- Also used in Galton–Watson branching · Random walks and first passage · Continuous-time Markov chains
- Questions that use it (2) — The variance recurrence (answer) · Checking the duration formula
- All 3 questions on Classes, recurrence, and reversibility
See also
Recurrence and period · Communicating classes · Finite irreducible chains · Detailed balance · Return and period · Global balance and detailed balance · Two families with an explicit stationary law · When a stationary law fails to be unique