Stochastic relations of random variables and processes

TL;DR

This paper extends stochastic order concepts to arbitrary measurable spaces, providing a functional characterization, conditions for preservation, and algorithms, with applications to various stochastic processes.

Contribution

It introduces a generalized notion of stochastic relations, including criteria and algorithms for their preservation, applicable to complex stochastic systems.

Findings

Characterization of stochastic relations via functions

Necessary and sufficient conditions for preservation

Algorithm for identifying preserved subrelations

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.