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Stochastic Relations of Random Variables and Processes

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  • Lasse Leskelä

    (Helsinki University of Technology)

Abstract

This paper generalizes the notion of stochastic order to a relation between probability measures over arbitrary measurable spaces. This generalization is motivated by the observation that for the stochastic ordering of two stationary Markov processes, it suffices that the generators of the processes preserve some, not necessarily reflexive or transitive, subrelation of the order relation. The main contributions of the paper are: a functional characterization of stochastic relations, necessary and sufficient conditions for the preservation of stochastic relations, and an algorithm for finding subrelations preserved by probability kernels. The theory is illustrated with applications to hidden Markov processes, population processes, and queueing systems.

Suggested Citation

  • Lasse Leskelä, 2010. "Stochastic Relations of Random Variables and Processes," Journal of Theoretical Probability, Springer, vol. 23(2), pages 523-546, June.
    • Handle: RePEc:spr:jotpro:v:23:y:2010:i:2:d:10.1007_s10959-009-0216-8
    DOI: 10.1007/s10959-009-0216-8
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    References listed on IDEAS

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    1. William A. Massey, 1987. "Stochastic Orderings for Markov Processes on Partially Ordered Spaces," Mathematics of Operations Research, INFORMS, vol. 12(2), pages 350-367, May.
    2. Ward Whitt, 1986. "Stochastic Comparisons for Non-Markov Processes," Mathematics of Operations Research, INFORMS, vol. 11(4), pages 608-618, November.
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