Stochastically monotone Markov Chains

  • D. J. Daley


A real-valued discrete time Markov Chain {Xn} is defined to be stochastically monotone when its one-step transition probability function pr {Xn+1y¦ Xn=x} is non-increasing in x for every fixed y. This class of Markov Chains arises in a natural way when it is sought to “bound” (stochastically speaking) the process {Xn} by means of a “smaller” or “larger” process with the same transition probabilities; the class includes many simple models of applied probability theory. Further, a given stochastically monotone Markov Chain can readily be “bounded” by another chain {Yn}, with possibly different transition probabilities and not necessarily stochastically monotone, and this is of particular value when the latter process leads to simpler algebraic manipulations. A stationary stochastically monotone Markov Chain {Xn} has cov(f(X0), f(Xn)) ≧ cov(f(X0), f(Xn+1))≧0 (n =1, 2,...) for any monotonic function f(·). The paper also investigates the definition of stochastic monotonicity on a more general state space, and the properties of integer-valued stochastically monotone Markov Chains.


Markov Chain State Space Simple Model Stochastic Process General State 
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Copyright information

© Springer-Verlag 1968

Authors and Affiliations

  • D. J. Daley
    • 1
  1. 1.Statistical LaboratoryCambridgeEngland

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