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.
翻译:Poisson 瘦化是概率的一个基本结果, 这在 Poisson 点进程理论中非常重要 。 在文章中, 我们记录了几个关于 Poisson 瘦化的特征化结果 。 我们还考虑自由 Poisson 瘦化的免费概率比喻Poisson 瘦化的免费概率比喻, 这自然产生为Cochran 理论的高维非湿性比喻, 它来自关于 Gaussian 随机矩阵的二次函数“ Wishart-ness” 的多变量统计。 古典和免费 Poisson 瘦化的主要区别在于, 在前者中, 所涉及的Poisson 随机变量可能具有任意的平均值, 而在自由版本中, 相关的自由 Poisson 变异的“ 意思” 必须是 1 。 我们证明自由 Poisson 瘦化的相似的特征化结果, 并注意其在 Cochran 的理论中的影响 。