Some characterization results on classical and free Poisson thinning

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.