Abstract
Let X be a finite set, P be a stochastic matrix on X, and P̄ = lim n → ∞ (1/ n)∑ n−1 k=0 P k . Let G = ( X, E) be the weighted directed graph on X associated to P, with weights p ij . An arborescence is a subset a ⊆ E which has at most one edge out of every node, contains no cycles, and has maximum possible cardinality. The weight of an arborescence is the product of its edge weights. Let A denote the set of all arborescences. Let A ij 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 ∥ A ∥, resp. ∥ A ij ∥, be the sum of the weights of the arborescences in A , resp. A ij . The Markov chain tree theorem states that p̄ ij = ∥ A ij ∥/∥ A ∥. We give a proof of this theorem which is probabilistic in nature.
Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
