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On the convergence of stochastically monotone sequences of random variables and some applications

Published online by Cambridge University Press:  14 July 2016

Harry Cohn
Harry Cohn*
Affiliation:
University of Melbourne
*
∗Postal address: Department of Statistics, Richard Berry Building, University of Melbourne, Parkville, Victoria 3052, Australia.
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Abstract

The paper is concerned with the relationship between various modes of convergence for stochastically monotone sequences of random variables. A necessary and sufficient condition, as well as a sufficient condition, for convergence in probability of a vaguely convergent sequence are given. If, in addition, the sequence is assumed Markovian the same conditions are shown to pertain to almost sure convergence. A counterexample in the case when stochastic monotonicity fails is presented and some applications to branching processes are discussed.

Keywords

VAGUE CONVERGENCECONVERGENCE IN PROBABILITYALMOST SURE CONVERGENCESTOCHASTIC MONOTONICITYMARKOV CHAINBRANCHING PROCESSES
Type
Research Papers
Information
Journal of Applied Probability , Volume 18 , Issue 3 , September 1981 , pp. 592 - 605
Copyright
Copyright © Applied Probability Trust 

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