Foster's theorem, often discussed in the context of stochastic processes and in particular for Markov chains and Markov decision processes, provides insights into the long-term behavior of certain types of random processes. One common application of Foster's theorem is in the study of Markov chains with continuous state spaces. In its simplest form, Foster's theorem relates to the existence of a stationary distribution for a Markov chain.
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For a finite connected unweighted loopless graph with graph vertices,The edge-inclusion formula for a uniform spanning tree gives the left-hand side as , and every spanning tree has edges. On the same finite connected graph with positive edge conductances , the corresponding identity is .