Limit behaviour for stochastic monotonicity and applications

Abstract

A transition probability kernel P(·,·) is said to be stochastically monotone if P(x, (–∞, y]) is non-increasing in x for every fixed y. A Markov chain is said to be stochastically monotone (SMMC) if its transition probability kernels are stochastically monotone. A new method for tackling the asymptotics of SMMC is given in terms of some limit variables {Wq}. In the temporally homogeneous case a cyclic pattern for {Wq} will describe the limit behaviour of suitably normed and centred processes. As a consequence, geometrically growing constants turn out to pertain to almost sure convergence. Some convergence criteria are given and applications to branching processes and diffusions are outlined.