Some characterisation results on classical and free Poisson thinning

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 characterisation results on Poisson thinning. We also consider free Poisson thinning, the free probability analogue of Poisson thinning, which 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. The main difference between classical and free Poisson thinning is that, in the former, the involved Poisson random variable can have an arbitrary mean, whereas, in the free version, the "mean" of the relevant free Poisson variable must be 1. We prove similar characterisation results for free Poisson thinning and note their implications in the context of Cochran's theorem.