Central limit theorems for empirical product densities of stationary point processes

Abstract

We prove the asymptotic normality of kernel estimators of second-and higher-order product densities (and of the pair correlation function) for spatially homogeneous (and isotropic) point processes observed on a sampling window W n which is assumed to expand unboundedly in all directions as n → ∞ . We first study the asymptotic behavior of the covariances of the empirical product densities under minimal moment and weak dependence assumptions. The proof of the main results is based on the Brillinger-mixing property of the underlying point process and certain smoothness conditions on the higher-order reduced cumulant measures. Finally, the obtained limit theorems allow to construct χ 2 -goodness-of-fit tests for hypothetical product densities.

Envelope tests for spatial point patterns with and without simulation

“…The authors often observed such a behavior in their statistical work and believe that it is a somewhat intuitive knowledge in the statistical community. Fortunately, there is theoretical work on central limit theorems which confirms these empirical findings and gives them a theoretical explanation for spatially homogeneous (and isotropic) point processes (Heinrich and Klein , Heinrich ).…”

A general central limit theorem and a subsampling variance estimator for α‐mixing point processes

“…In that case, we do not require further assumptions on . However, in applications dealing with kernel estimators depending on a bandwidth h n tending towards 0, we may have of the order (e.g., Heinrich & Klein, ). Then, ( ) can be fulfilled if s n =1/ h n and η >0 so that, by ( ), is also of the order .…”

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

“…For Brillinger-mixing PPs, various CLTs have been proved in [9], [10] for both the estimators (4.3) and (4.4). These results permit us to establish asymptotic χ 2tests to check hypotheses about the second product densities and pair correlation functions.…”

“…In our situation, these tests can be reformulated for testing hypotheses about the function |C 0 (x)| or its isotropic counterpart |c(r)| = |C 0 (x)| for r = x . The following limit theorem (formulated as Theorems 3.3 and 4.1 in [10] in a more general setting) provide the basis for these tests. and |W n |( λ n − C 0 (o)) is asymptotically normally distributed with mean zero and variance σ 2 = C 0 (o) + α R d |C 0 (x)| 2 dx (by Corollary 2), see also [25] for α = −1.…”

“…The evaluation of the logarithmic derivatives in (1.1) yields the formula (which goes back to the pioneering paper of Leonov & Shiryaev [20], see also [3], pp. 147-148, or [10])…”