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Limit behaviour for stochastic monotonicity and applications

Published online by Cambridge University Press:  01 July 2016

Harry Cohn
Harry Cohn*
Affiliation:
The University of Melbourne
*
∗Postal address: Department of Statistics, University of Melbourne, Parkville, VIC 3052, Australia.
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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.

Keywords

MARKOV CHAINMARTINGALEDIFFUSIONBRANCHING
Type
Research Article
Information
Advances in Applied Probability , Volume 20 , Issue 2 , June 1988 , pp. 331 - 347
Copyright
Copyright © Applied Probability Trust 1988 

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Footnotes

Research partly carried out while visiting the Center for Stochastic Processes, University of North Carolina and supported by AFOSR # F49620 82 C 0009.

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