Foster's theorem

This article is about the theorem in Markov probability theory. For the theorem in electrical engineering, see Foster's reactance theorem.

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.

Consider an irreducible discrete-time Markov chain on a countable state space S having a transition probability matrix P with elements p_ij 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 $V:S\to \mathbb {R}$ , such that $V(i)\geq 0{\text{ }}\forall {\text{ }}i\in S$ and

$\sum_{j \in S}p_{ij}V(j) < {\infty}$ for $i \in F$
$\sum _{j\in S}p_{ij}V(j)\leq V(i)-\varepsilon$ for all $i \notin F$

for some finite set F and strictly positive ε.^[2]

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. p. 167. doi:10.1007/978-1-4757-3124-8_5. ISBN 978-1-4419-3131-3.

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Foster's theorem

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