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Stochastically monotone Markov Chains

1968; Springer Nature; Volume: 10; Issue: 4 Linguagem: Inglês

10.1007/bf00531852

0044-3719

D. J. Daley,

Bayesian Methods and Mixture Models

A real-valued discrete time Markov Chain {X n} is defined to be stochastically monotone when its one-step transition probability function pr {X n+1≦y¦ X n=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 {X n} 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 {Y n}, 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 {X n} has cov(f(X 0), f(X n)) ≧ cov(f(X 0), f(X n+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.