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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 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 some finite set and strictly positive . [2]