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Absolute regularity and Brillinger-mixing of stationary point processes

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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 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.

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Correspondence to Lothar Heinrich.

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1The research of the author was supported in part by the Grant Agency of the Czech Republic, project No. P201/10/0472.

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Heinrich, L., Pawlas¹, Z. Absolute regularity and Brillinger-mixing of stationary point processes. Lith Math J 53, 293–310 (2013). https://doi.org/10.1007/s10986-013-9209-5

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  • DOI: https://doi.org/10.1007/s10986-013-9209-5

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