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Stochastically monotone Markov Chains

  • D. J. Daley
Article

Summary

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.

Keywords

Markov Chain State Space Simple Model Stochastic Process General State 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag 1968

Authors and Affiliations

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

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