Lithuanian Mathematical Journal, Vol. 53, No. 3, July, 2013, pp. 293–310 Absolute regularity and Brillinger-mixing of stationary point processes Lothar Heinrich a and Zbynˇ ek Pawlas b, 1 a Institut für Mathematik, Universität Augsburg, Universitätsstr. 14, 86135 Augsburg, Germany b Department of Probability and Mathematical Statistics, Faculty of Mathematics and Physics, Charles University in Prague, Sokolovská 83, 18675 Praha 8, Czech Republic (e-mail: heinrich@math.uni-augsburg.de; pawlas@karlin.mff.cuni.cz) Received January 19, 2013; revised March 13, 2013 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, 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  2. To prove this, we introduce higher-order covariance measures and use Statuleviˇ cius’ 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. MSC: primary 60G55, 37A25; secondary 60D05, 60F05 Keywords: Palm distribution, (reduced) factorial cumulant measure, Brillinger-mixing, higher-order covariance measure, β-mixing coefficient, germ-grain model, dependently thinned point process 1 Introduction and basic definitions Point processes (briefly PPs) are adequate models to describe randomly or irregularly scattered points in some Euclidean space R d (often d =1, 2, 3 in applications). Statistics of PPs is mostly based on a single observation of a point pattern in some large sampling window, which is assumed to expand unboundedly in all directions; see [18, Chap. 4]. Provided that the underlying PP model is homogeneous (i.e., stationary), the asymptotic behavior of parameter estimators and other empirical characteristics can only be determined under ergodicity and (strong) mixing assumptions, respectively. We encounter a similar situation in statistical physics, where stationary PPs are used to describe limits of configurations of interacting particles given in a “large (expanding) container” (see [13, 16]). Throughout, let Ψ := ∑ i1 δ Xi ∼ P denote a simple stationary PP on R d with distribution P defined on the σ-algebra N generated by the sets of the form {ψ ∈ N : ψ(B)= n} for any n ∈ N ∪{0} and B ∈B d b (= bounded sets of the Borel-σ-algebra B d in R d ), where N denotes the family of locally finite 1 The research of the author was supported in part by the Grant Agency of the Czech Republic, project No. P201/10/0472. 293 0363-1672/13/5303-0293 c  2013 Springer Science+Business Media New York