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 S having a transition probability matrix P with elements pij for pairs i, j in S. Foster's theorem states that the Markov chain is positive recurrent if and only if there exists a Lyapunov function \( {\displaystyle V:S\to \mathbb {R} } \), such that \( {\displaystyle V(i)\geq 0{\text{ }}\forall {\text{ }}i\in S} \) and
\( \sum_{j \in S}p_{ij}V(j) < {\infty} \) for \( i \in F \)
\( {\displaystyle \sum _{j\in S}p_{ij}V(j)\leq V(i)-\varepsilon } \) for all \(} i \notin F \)
for some finite set F and strictly positive ε.[2]
Related links
Lyapunov optimization
References
Foster, F. G. (1953). "On the Stochastic Matrices Associated with Certain Queuing Processes". The Annals of Mathematical Statistics. 24 (3): 355. 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.
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