Orderings of optimal stopping values and prophet inequalities for certain multivariate distributions

Abstract

Suppose you observe a finite sequence of random variables from some known joint distribution F, you can stop the process at any time and your profit is the last observed value. If an optimal stopping rule is used, denote the expected profit by V F . What kind of ordering on multivariate distributions F and G will guarantee V F ≤ V G ? The comparison depends on two combined aspects: the actual size of the observations and the structure of their dependence. Closely related to these comparisons is the prophet inequality which compares V F to the expectation of the maximum of the sequence which is a prophet's value. We study monotonicity properties of V F and related prophet inequalities in two classes of distributions: one consisting of negatively dependent PF 2 variables and the other of samples with partial replacement from a finite population.