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
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
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