Gompertz distribution: Difference between revisions
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{{Expert-subject|Mathematics|date=February 2009}} | |||
{{ about|the theorem in Markov probability theory|the theorem in electrical engineering|Foster's reactance theorem}} | |||
In [[probability theory]], '''Foster's theorem''', named after [[F. G. Foster]],<ref>[http://www.projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.aoms/1177728976 F. G. Foster, “On the stochastic matrices associated with certain queueing processes,” Ann. Math Statist., Vol. 24, pp. 355-360, 1953.]</ref> 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 aperiodic, irreducible discrete-time Markov chain on a countable state space <math>\mathbf{S}</math> having a transition probability matrix <math>P=[p_{ij}] \text{ } \forall \text{ } i, j \in \mathbf{S}</math>. Foster's theorem states that the Markov chain is positive recurrent if and only if there exists a [[Lyapunov function]] <math>V : \mathbf{S} \rightarrow \mathbf{R}</math>, such that | |||
# <math>V(i) \geq 0 \text{ } \forall \text{ } i \in \mathbf{S}</math> | |||
# <math>\sum_{j \in \mathbf{S}}p_{ij}V(j) \leq V(i) - 1</math> | |||
# <math>\sum_{j \in \mathbf{S}}p_{0j}V(j) < {\infty}.</math> | |||
==Related links== | |||
* [[Lyapunov optimization]] | |||
* [[Lyapunov function]] | |||
==Notes== | |||
{{reflist}} | |||
==References== | |||
# Pierre Brémaud - ''Markov Chains: Gibbs Fields, Monte Carlo Simulation, and Queues'', Springer, 1998 (Chapter 5) | |||
[[Category:Stochastic processes]] | |||
[[Category:Statistical theorems]] | |||
{{probability-stub}} | |||
Revision as of 18:01, 25 January 2014
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In probability theory, Foster's theorem, named after F. G. 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 aperiodic, irreducible discrete-time Markov chain on a countable state space having a transition probability matrix . Foster's theorem states that the Markov chain is positive recurrent if and only if there exists a Lyapunov function , such that
Related links
Notes
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References
- Pierre Brémaud - Markov Chains: Gibbs Fields, Monte Carlo Simulation, and Queues, Springer, 1998 (Chapter 5)