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1 results

  • A geometric invariant in weak lumpability of finite Markov chains

  • Part of
  • James Ledoux
  • Journal:
    Journal of Applied Probability / Volume 34 / Issue 4 / December 1997
    Published online by Cambridge University Press:
    14 July 2016, pp. 847-858
    Print publication:
    December 1997

  • We consider weak lumpability of finite homogeneous Markov chains, which is when a lumped Markov chain with respect to a partition of the initial state space is also a homogeneous Markov chain. We show that weak lumpability is equivalent to the existence of a direct sum of polyhedral cones that is positively invariant by the transition probability matrix of the original chain. It allows us, in a unified way, to derive new results on lumpability of reducible Markov chains and to obtain spectral properties associated with lumpability.