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Couplings and Poisson approximation for stabilizing functionals of determinantal point processes

Moritz Otto

Abstract

We prove a Poisson process approximation result for stabilizing functionals of a determinantal point process. Our results use concrete couplings of determinantal processes with different Palm measures and exploit their association properties. Second, we focus on the Ginibre process and show in the asymptotic scenario of an increasing window size that the process of points with a large nearest neighbor distance converges after a suitable scaling to a Poisson point process. As a corollary, we obtain the scaling of the maximum nearest neighbor distance in the Ginibre process, which turns out to be different from its analogue for independent points.

Couplings and Poisson approximation for stabilizing functionals of determinantal point processes

Abstract

We prove a Poisson process approximation result for stabilizing functionals of a determinantal point process. Our results use concrete couplings of determinantal processes with different Palm measures and exploit their association properties. Second, we focus on the Ginibre process and show in the asymptotic scenario of an increasing window size that the process of points with a large nearest neighbor distance converges after a suitable scaling to a Poisson point process. As a corollary, we obtain the scaling of the maximum nearest neighbor distance in the Ginibre process, which turns out to be different from its analogue for independent points.
Paper Structure (8 sections, 4 theorems, 75 equations)

This paper contains 8 sections, 4 theorems, 75 equations.

Table of Contents

  1. Introduction
  2. Model and main results
  3. Preliminaries
  4. Palm calculus and negative association
  5. Fast decay of correlation
  6. Poisson process approximation
  7. Proof of Theorem \ref{['th:decr']}
  8. Proof of Theorem \ref{['th:balls']}

Key Result

Theorem 1

Let $\xi$ be a stationary determinantal process with kernel $K$ satisfying fastdecay and intensity $\rho \in (0,\infty)$. Let $\Xi$ be the score sum defined in def:Xigen with intensity measure $\mathbf L$ and suppose that $g$ satisfies kiss and mono and is stabilizing with respect to the stopping se Let $\zeta$ be a finite Poisson process on $\mathbb R^d$ with intensity measure $\mathbf M$. Then,

Theorems & Definitions (7)

  • Theorem 1
  • Theorem 2
  • Corollary 3
  • Remark 4
  • Proposition 5
  • proof : Proof of Theorem \ref{['th:decr']}
  • proof : Proof of Theorem \ref{['th:balls']}