A rigidity property of superpositions involving determinantal processes

TL;DR

This paper proves that the superposition of a Poisson process and a determinantal process uniquely determines their individual distributions, highlighting a rigidity property in their combined structure.

Contribution

It establishes a novel rigidity property showing that the superposition of independent Poisson and determinantal processes uniquely identifies each process's distribution.

Findings

Superposition of Poisson and determinantal processes is uniquely identifiable.

The result applies to non-self-adjoint correlation kernels.

Provides new insights into the structure of combined point processes.

Abstract

The main result of this paper states that if $(N, \Pi)$ is a pair of independent point processes on a common ground space with $N$ Poisson and $\Pi$ determinantal induced by a locally trace class (not necessarily self-adjoint) correlation kernel, then their independent superposition $N + \Pi$ determines uniquely the distributions of $N$ and $\Pi$.