Poisson thinning is an elementary result in probability, which is of great importance in the theory of Poisson point processes. In this paper, 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 free Poisson, 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.

泊松细化是概率的基本结果，在泊松点过程理论中具有重要意义。在本文中，我们记录了一些关于泊松细化的表征结果。我们还考虑了几个泊松细化的自由概率类似物，我们统称为自由泊松，并证明它们的表征结果，类似于经典案例。这些自由泊松细化程序之一自然地作为科克伦定理的高维渐近类似物从关于高斯随机矩阵的二次函数的“Wishart-ness”的多元统计中产生。我们注意到我们的表征结果在科克伦定理的背景下的含义。我们还证明了 Craig 定理的自由概率类似物，这是关于高斯随机矩阵的二次函数独立性的多元统计中的另一个著名结果。