Limit Behaviour for Stochastic Monotonicity and Applications

摘要

A transition probability function P is said to be stochastically monotone if P(x,-ALPHA,Y) is non-increasing in x for every fixed y. A (non-homogeneous) Markov chain or process is said to be stochastically monotone if its transition probability functions are stochastically monotone. Diffusions, random walks, birth-and-death and branching processes are examples of such models. It is shown that stochastically monotone processes exhibit two basic types of asymptotic behaviour. Chains with stationary transition probabilities display a cyclic pattern, and a suitably normed and centered chain turns out to converge almost surely if its is geometrically growing. Applications to diffusions and branching processes are added.