• Media type: E-Book; Report; Text
  • Title: Combinatorial considerations on the invariant measure of a stochastic matrix
  • Contributor: Stephan, Artur [Author]
  • imprint: Weierstrass Institute for Applied Analysis and Stochastics publication server, 2019
  • Language: English
  • DOI: https://doi.org/10.20347/WIAS.PREPRINT.2627
  • Keywords: Markov chain -- Markov process -- invariant measure -- stationary measure -- stationary distribution -- Theorem of Frobenius-Perron -- Kirchhoff tree theorem -- Markov tree theorem -- directed and undirected acyclic graphs -- spanning trees -- detailed balance ; 60Jxx ; article
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  • Description: The invariant measure is a fundamental object in the theory of Markov processes. In finite dimensions a Markov process is defined by transition rates of the corresponding stochastic matrix. The Markov tree theorem provides an explicit representation of the invariant measure of a stochastic matrix. In this note, we given a simple and purely combinatorial proof of the Markov tree theorem. In the symmetric case of detailed balance, the statement and the proof simplifies even more.
  • Access State: Open Access