Monotone Markov chains

  • Masaaki Kijima


In this chapter, we consider the monotonicity properties of discrete-time Markov chains where each monotonicity is characterized in terms of transition matrices. A Markov chain {X n }is said to be increasing (decreasing, respectively) if X n+1X n (X n X n+1) for all n = 0,1, ..., where ≻ denotes an ordering relation in some stochastic sense, and in either case we call {X n }internally monotone, or monotone for short. An external monotonicity is such that, for two Markov chains {X n }and {Y n , we have X n Y n for all n. Monotonicity properties are important both theoretically and practically because they lead to a variety of structural insights. In particular, they are a basic tool for deriving many useful inequalities in Markov chains for stochastic modeling.


Markov Chain Hazard Rate Probability Vector Discrete Random Variable Service Time Distribution 
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Copyright information

© M. Kijima 1997

Authors and Affiliations

  • Masaaki Kijima
    • 1
  1. 1.Graduate School of Systems ManagementUniversity of TsukubaTokyoJapan

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