In probability theory, Foster's theorem, named after Gordon Foster,[1] is used to draw conclusions about the positive recurrence of Markov chains with countable state spaces. It uses the fact that positive recurrent Markov chains exhibit a notion of "Lyapunov stability" in terms of returning to any state while starting from it within a finite time interval.
Theorem
Consider an irreducible discrete-time Markov chain on a countable state space having a transition probability matrix with elements for pairs , in . Foster's theorem states that the Markov chain is positive recurrent if and only if there exists a Lyapunov function , such that and
- for
- for all
for some finite set and strictly positive .[2]
Related links
References
- ^ Foster, F. G. (1953). "On the Stochastic Matrices Associated with Certain Queuing Processes". The Annals of Mathematical Statistics. 24 (3): 355–360. doi:10.1214/aoms/1177728976. JSTOR 2236286.
- ^ Brémaud, P. (1999). "Lyapunov Functions and Martingales". Markov Chains. pp. 167. doi:10.1007/978-1-4757-3124-8_5. ISBN 978-1-4419-3131-3.