Conditional Independence under Infinite Measures and Poisson Point Processes

TL;DR

This paper explores a novel notion of conditional independence for infinite measures on punctured spaces, linking it to classical independence in Poisson point processes and extending the framework to broader settings.

Contribution

It introduces a new concept of conditional independence for infinite measures, characterizes it via Poisson processes, and generalizes the framework to abstract spaces.

Findings

Conditional independence is equivalent to classical independence in Poisson processes.

Provides a functional characterization at the level of Poisson points.

Extends the framework to more general abstract spaces.

Abstract

We study conditional independence under infinite measures on punctured product spaces, a notion recently introduced for graphical modeling in multivariate extremes and L\'evy processes. In contrast to classical probabilistic conditional independence, this concept is formulated through normalized restrictions of an infinite measure that reflects the non-product structure of the punctured space. We show that this non-standard notion admits a natural probabilistic characterization: it is equivalent to classical conditional independence between coordinate projections of a Poisson point process defined on the punctured space with the given infinite measure as its mean measure. In addition, we provide a functional characterization of the conditional independence concept at the level of the enumerated points of the Poisson point process. We further extend the framework from punctured Euclidean product spaces to a more general abstract setting, thereby broadening its scope of potential applications.