Absolute regularity and Brillinger-mixing of stationary point processes

Abstract

Abstract. We study the following problem: How to verify Brillinger-mixing of stationary point processes in 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 an explicit condition in terms of this coefficient which implies finite total variation of the kth-order reduced factorial cumulant measure of the point process for fixed k ≥ 2. To prove this, we introduce higher-order covariance measures and use Statulevičius' 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.

On the strong Brillinger-mixing property of α-determinantal point processes and some applications

“…The proof of Corollary 1, which holds true for any strongly Brillinger-mixing PP, can be found in [8].…”

Weak dependence of point processes and application to second-order statistics†

“…Define the point process Z on the positive halfline with one point in each unit interval: P n is the only point in [n, n + 1) and is located at n + X n . then this point process is not regularly mixing in the sense of theorem 2 of [12], because the location of P n determines the location of P 0 . We begin the study of the dependence of this process.…”

“…[16] showed that strong mixing may be stable under clustering transformations but φ-mixing is not except under very restrictive assumptions. [12] proved regular and Brillinger mixing for some Markov point processes. Our aim is to extend weak dependence as they extended mixing to point processes, to show that some classes of point processes (Cox and Neyman-Scott processes) are weakly dependent under mild assumptions.…”

On the Variance of the Area of Planar Cylinder Processes Driven by Brillinger-Mixing Point Processes

“…[7]. In [11,14] the relations between (strong) Brillinger-mixing and classical mixing conditions are studied. Strong Brillinger-mixing requires exponential moments of the number of points in bounded sets.…”