On independence of time and cause

TL;DR

This paper characterizes when the minimum of two independent random variables is independent of the event that one exceeds the other, revealing specific distributional conditions involving exponential or geometric distributions.

Contribution

It provides a complete characterization of independence between the minimum and the comparison event for two independent variables, extending to multivariate cases.

Findings

Independence occurs only when variables are related via the same increasing function.

Distributional conditions are limited to exponential and geometric distributions.

Results generalize to multivariate scenarios.

Abstract

For two independent, almost surely finite random variables, independence of their minimum (time) and the event that one of them is either greater, equal or less than the other (cause) is completely characterized. It is shown that, other than for trivial cases where, almost surely, one random variable is greater than or equal to the other, this happens if and only if both random variables are distributed like the same strictly increasing function of two independent random variables, where either both are exponentially distributed or both are geometrically distributed. This is then easily generalized to the multivariate case.