Limit theory for geometric statistics of point processes having fast decay of correlations

Abstract

Let P be a simple, stationary point process on R d having fast decay of correlations, i.e., its correlation functions factorize up to an additive error decaying faster than any power of the separation distance. Let Pn := P ∩ Wn be its restriction to windows Wn := [− 1 2 n 1/d , 1 2 n 1/d ] d ⊂ R d . We consider the statistic H ξ n := x∈Pn ξ(x, Pn) where ξ(x, Pn) denotes a score function representing the interaction of x with respect to Pn. When ξ depends on local data in the sense that its radius of stabilization has an exponential tail, we establish expectation asymptotics, variance asymptotics, and central limit theorems for H ξ n and, more generally, for statistics of the re-scaled, possibly signed, ξ-weighted point measures µ ξ n := x∈Pn ξ(x, Pn)δ n −1/d x , as Wn ↑ R d . This gives the limit theory for non-linear geometric statistics (such as clique counts, the number of Morse critical points, intrinsic volumes of the Boolean model, and total edge length of the k-nearest neighbors graph) of α-determinantal point processes (for −1/α ∈ N) having fast decreasing kernels, including the β-Ginibre ensembles, extending the Gaussian fluctuation results of Soshnikov [72] to non-linear statistics. It also gives the limit theory for geometric U-statistics of α-permanental point processes (for 1/α ∈ N) as well as the zero set of Gaussian entire functions, extending the central limit theorems of Nazarov and Sodin [53] and Shirai and Takahashi [71], which are also confined to linear statistics. The proof of the central limit theorem relies on a factorial moment expansion originating in [12,13] to show the fast decay of the correlations of ξ-weighted point measures. The latter property is shown to imply a condition equivalent to Brillinger mixing and consequently yields the asymptotic normality of µ ξ n via an extension of the cumulant method.

Testing goodness of fit for point processes via topological data analysis

“…To deal with the double integral in (13), we recall that ξ is a local score function and that P exhibits exponential decay of correlations. Hence, as in [8,Equation (3.26)], the factorial moment measure expansion shows that…”

“…, A p ⊂ R 2 , where P(A i ) denotes the number of points of P in A i . Moreover, as we rely on the framework of [8], we also require that P exhibits exponential decay of correlations. Loosely speaking this expresses an approximate factorization of the factorial moment densities and is made precise in Section 4 below.…”

“…For this purpose, we derived sufficient conditions for a large-domain functional CLT for the M -bounded persistent Betti numbers on point processes exhibiting exponential decay of correlations. Following the framework developed in [8], the main difficulty arose from a detailed analysis of geometric configurations when bounding higher-order cumulants.…”

Statistical learning of geometric characteristics of wireless networks

“…If this interaction is local in some sense, and the underlying point process exhibits some decay of correlations, then it is possible to establish asymptotic results (including central limit theorems) regarding the sums of the geometric marks observed in increasing window; cf e.g. [13], [14], [15]. These results can be used to study large-scale asymptotic of the scattering moments as in [16].…”

Central Limit Theorem for Exponentially Quasi-local Statistics of Spin Models on Cayley Graphs

“…While all the aforementioned results are important precursors to our article, to the best of our knowledge, we are not aware of a CLT with easy-to-use geometric conditions on ξ , and simple mixing conditions on X . Such conditions are more common in the literature on point processes in continuous settings such as Euclidean spaces or some nice compact manifolds (see [91,77,11,63]). In this article, we prove similar generic central limit theorems for 'nice' geometric statistics of the form (1.1) for suitably mixing or clustering random fields on Cayley graphs of finitely generated infinite groups.…”