Discrete-time convergence to equilibrium needed a genuine hypothesis — aperiodicity — that periodic chains fail. The continuous-time story is, surprisingly, simpler: the randomness of exponential holding times automatically rules out any periodicity-style obstruction, so irreducibility alone (plus a stationary distribution) is enough to guarantee convergence. This note carries the discrete-chains topic's stationary-distribution and detailed-balance machinery over to continuous time, replacing with the generator-matrix condition .
Learning objectives. By the end of this note you should be able to:
- state irreducibility for a CTMC and explain why aperiodicity is automatic;
- state and use the criterion for stationarity;
- state the convergence theorem and the detailed balance condition for CTMCs;
- derive the birth-death stationary-distribution formula.
In this lesson
- Irreducibility and the disappearance of periodicity
- Stationary distributions
- Detailed balance for CTMCs
The full lesson, with worked examples and exercise solutions, is for members.
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