A rigidity property of superpositions involving determinantal processes

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