Summary: Statistics & Probability Letters 8 (1989) 189-192
A PROOF OF THE MARKOV CHAIN TREE THEOREM
V. ANANTHARAM *
Field of Statistics, Center for Applied Mathematics, and School of Electrical Engineering, Cornell University, Ithaca, NY 14853,
SystemsResearch Center and Department of Electrical Engineering, University of Maryland, College Park, MD 20742, USA
Received July 1988
Revised September 1988
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,, = Ijzz!,, II/ II_&II. We give a proof of this
theorem which is probabilistic in nature.
Keywords: arborescence, Markov chain, stationary distribution, time reversal, tree.