[Paper Review] Brillinger mixing of determinantal point processes and statistical applications

Christophe A. N. Biscio, Frédéric Lavancier|arXiv (Cornell University)|Jul 23, 2015

Point processes and geometric inequalities36 references17 citations

TL;DR

This paper establishes that stationary determinantal point processes (DPPs) are Brillinger mixing, a key ergodicity property enabling asymptotic statistical inference. The authors prove a central limit theorem for a broad class of DPP functionals, yielding asymptotic normality for the intensity estimator and, newly, for the kernel estimator of the pair correlation function.

ABSTRACT

Stationary determinantal point processes are proved to be Brillinger mixing. This property is an important step towards asymptotic statistics for these processes. As an important example, a central limit theorem for a wide class of functionals of determinantal point processes is established. This result yields in particular the asymptotic normality of the estimator of the intensity of a stationary determinantal point process and of the kernel estimator of its pair correlation.

Motivation & Objective

• To establish the Brillinger mixing property for stationary determinantal point processes (DPPs), a critical step toward asymptotic statistics.
• To extend general mixing-based asymptotic results to the specific case of DPPs, leveraging their known moment structures.
• To derive the asymptotic normality of the intensity estimator for DPPs, confirming known results via a new approach.
• To establish the novel result of asymptotic normality for the kernel estimator of the pair correlation function of a DPP.
• To provide a theoretical foundation for minimum contrast estimation and other statistical inference procedures in parametric DPP models.

Proposed method

• Prove that the total variation of the reduced factorial cumulant measure of order $k$ is finite for all $k \geq 2$, satisfying the definition of Brillinger mixing.
• Use the known closed-form expressions for factorial moment measures and cumulants of DPPs, particularly the determinant-based structure of their joint intensities.
• Apply general asymptotic results from [16], [9], and [10] on Brillinger mixing processes to DPPs, simplifying and extending them in the stationary DPP context.
• Derive the asymptotic variance of functionals via explicit formulas involving the reduced second-order cumulant measure $c^{\text{red}}_{[2]}(y)$ and the intensity $\rho$.
• Apply the central limit theorem for Brillinger mixing processes to functionals of the form $\sum_{x \in \mathbf{X}} f(x)$ and $\sum_{(x,y) \in \mathbf{X}^2}^{\neq} f(x,y)$, leading to asymptotic normality results.
• Use the symmetry and compact support of test functions $f$ and $h$ to simplify covariance and variance expressions in the limit theorems.

Experimental results

Research questions

• RQ1 Are stationary determinantal point processes (DPPs) Brillinger mixing?
• RQ2 Does the Brillinger mixing property of DPPs imply asymptotic normality for general functionals of the process?
• RQ3 Can the kernel estimator of the pair correlation function of a DPP be shown to be asymptotically normal?
• RQ4 What is the asymptotic variance of the intensity estimator in a stationary DPP?
• RQ5 Can the general framework of asymptotic statistics for mixing processes be applied and simplified in the context of DPPs?

Key findings

• Stationary DPPs are proven to be Brillinger mixing, as the total variation of their reduced factorial cumulant measures of all orders $k \geq 2$ is finite.
• A central limit theorem is established for a wide class of functionals of DPPs, including the sum of a bounded, compactly supported function over the point process.
• The intensity estimator of a stationary DPP is asymptotically normal, confirming a known result through a new mixing-based proof.
• The kernel estimator of the pair correlation function of a DPP is shown to be asymptotically normal for the first time in this paper.
• The asymptotic variance of the intensity estimator is expressed in terms of the reduced second-order cumulant measure and the intensity $\rho$.
• The results provide a theoretical foundation for the asymptotic normality of minimum contrast estimators in parametric DPP models, as used in ongoing work [2].