A Proof of the Markov Chain Tree Theorem

Abstract

Let X be a finite set, P be a stochastic matrix on X, and P = lim ,,_,(l/n)X$LAPk. Let C=(X, E) be the weighted directed graph on X associated to P, with weights p,;. An arborescence is a subset a c E which has at most one edge out of every node, contains no cycles, and has maximum possible cardinahty. The weight of an arborescence is the product of its edge weights. Let _z? denote the set of all arborescences. Let dI, denote the set of all arborescences which have j as a root and in which there is a directed path from i to j. Let 1) & 11, resp. II_@‘,, 11, be the sum of the weights of the arborescences in &, resp. &,j. The Markov chain tree theorem states that p,, = Ij zz!,, II/ II _&II. We give a proof of this theorem which is probabilistic in nature.

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Cite this paper

@inproceedings{Anantharam1989APO, title={A Proof of the Markov Chain Tree Theorem}, author={V. Anantharam and Pantelis Tsoucas}, year={1989} }