Absolute regularity and Brillinger-mixing of stationary point processes

Abstract

We study the following problem: How to verify Brillinger-mixing of stationary point processes in \( {{\mathbb{R}}^d} \) by imposing conditions on a suitable mixing coefficient? For this, we define an absolute regularity (or β-mixing) coefficient for point processes and derive, in terms of this coefficient, an explicit condition that implies finite total variation of the kth-order reduced factorial cumulant measure of the point process for fixed \( k\geqslant 2 \). To prove this, we introduce higher-order covariance measures and use Statulevicius’ representation formula for mixed cumulants in case of random (counting) measures. To illustrate our results, we consider some Brillinger-mixing point processes occurring in stochastic geometry.