Some characterization results on classical and free Poisson thinning

TL;DR

This paper explores classical and free probability versions of Poisson thinning, providing new characterization results and connecting them to multivariate statistical theorems like Cochran's and Craig's.

Contribution

It introduces free probability analogues of Poisson thinning and establishes their characterization, linking high-dimensional asymptotics to classical statistical theorems.

Findings

Characterization results for classical Poisson thinning.

Introduction of free Poisson thinning and its properties.

Connections to Cochran's and Craig's theorems in multivariate statistics.

Abstract

Poisson thinning is an elementary result in probability, which is of great importance in the theory of Poisson point processes. In this article, we record a couple of characterization results on Poisson thinning. We also consider several free probability analogues of Poisson thinning, which we collectively dub as \emph{free Poisson thinning}, and prove characterization results for them, similar to the classical case. One of these free Poisson thinning procedures arises naturally as a high-dimensional asymptotic analogue of Cochran's theorem from multivariate statistics on the "Wishart-ness" of quadratic functions of Gaussian random matrices. We note the implications of our characterization results in the context of Cochran's theorem. We also prove a free probability analogue of Craig's theorem, another well-known result in multivariate statistics on the independence of quadratic functions of Gaussian random matrices.