Probability Seminar: Deterministic Poisson thinning on finite and infinite volumes

Given a homogeneous Poisson point process it is well known that selecting each point independently with some fixed probability gives a homogeneous Poisson process of lower intensity. This is often referred to as thinning. Can thinning be achieved without additional randomization; that is, is it possible to choose a subset of the Poisson points as a deterministic function of the Poisson process so that the chosen points form a Poisson process of any given lower intensity? On an infinite volume, it is always possible. Furthermore, on R^d , it is possible to define the deterministic function to be a translation-equivariant factor (that is, if a translation is applied to the original process, the chosen points are translated by the same vector).

On a finite volume, the answer depends on both the intensities of the original and resulting Poisson processes. We will discuss joint work with Omer Angel, Alexander Holroyd, and Russell Lyons.