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

Let $P$ be a simple,stationary point process 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 $P_n:= P \cap W_n$ be its restriction to windows $W_n:= [-{1 \over 2}n^{1/d},{1 \over 2}n^{1/d}]^d \subset \mathbb{R}^d$. We consider the statistic $H_n^\xi:= \sum_{x \in P_n}\xi(x,P_n)$ where $\xi(x,P_n)$ denotes a score function representing the interaction of $x$ with respect to $P_n$. When $\xi$ depends on local data in the sense that its radius of stabilization has an exponential tail, we establish expectation asymptotics, variance asymptotics, and CLT for $H_n^{\xi}$ and, more generally, for statistics of the re-scaled, possibly signed, $\xi$-weighted point measures $\mu_n^{\xi} := \sum_{x \in P_n} \xi(x,P_n) \delta_{n^{-1/d}x}$, as $W_n \uparrow \mathbb{R}^d$. This gives the limit theory for non-linear geometric statistics (such as clique counts, intrinsic volumes of the Boolean model, and total edge length of the $k$-nearest neighbors graph) of $\alpha$-determinantal point processes having fast decreasing kernels extending the CLTs of Soshnikov (2002) to non-linear statistics. It also gives the limit theory for geometric U-statistics of $\alpha$-permanental point processes and the zero set of Gaussian entire functions, extending the CLTs of Nazarov and Sodin (2012) and Shirai and Takahashi (2003), which are also confined to linear statistics. The proof of the central limit theorem relies on a factorial moment expansion originating in Blaszczyszyn (1995), Blaszczyszyn, Merzbach, Schmidt (1997) to show the fast decay of the correlations of $\xi$-weighted point measures. The latter property is shown to imply a condition equivalent to Brillinger mixing and consequently yields the CLT for $\mu_n^\xi$ via an extension of the cumulant method.Comment: 62 pages. Fundamental changes to the terminology including the title. The earlier 'clustering' condition is now introduced as a notion of mixing and its connection to Brillinger mixing is remarked. Newer results for superposition of independent point processes have been adde

Limit theorems for weighted functionals of cyclical long-range dependent random fields

The paper investigates isotropic random fields for which the spectral density is unbounded at some frequencies. Limit theorems for weighted functionals of these random fields are established. It is shown that for a wide class of functionals, which includes the Donsker scheme, the limit is not affected by singularities at non-zero frequencies. For general schemes, in contrast to the Donsker line, we demonstrate that the singularities at non-zero frequencies play a role even for linear functionals.Comment: 19 pages, 2 figures. This is an Author's Accepted Manuscript of an article in the Stochastic Analysis and Applications, Vol. 31, No. 2. (2013), 199--213. [copyright Taylor \& Francis], available online at: http://www.tandfonline.com/ [DOI:10.1080/07362994.2013.741410

Linearization of randomly weighted empiricals under long range dependence with application to nonlinear regression quantiles

This paper discusses some asymptotic uniform linearity results of randomly weighted empirical processes based on long range dependent random variables+ These results are subsequently used to linearize nonlinear regression quantiles in a nonlinear regression model with long range dependent errors, where the design variables can be either random or nonrandom+ These, in turn, yield the limiting behavior of the nonlinear regression quantiles+ As a corollary, we obtain the limiting behavior of the least absolute deviation estimator and the trimmed mean estimator of the parameters of the nonlinear regression model+ Some of the limiting properties are in striking contrast with the corresponding properties of a nonlinear regression model under independent and identically distributed error random variables+ The paper also discusses an extension of rank score statistic in a nonlinear regression model

Non-Gaussian Geostatistical Modeling using (skew) t Processes

We propose a new model for regression and dependence analysis when addressing spatial data with possibly heavy tails and an asymmetric marginal distribution. We first propose a stationary process with $t$ marginals obtained through scale mixing of a Gaussian process with an inverse square root process with Gamma marginals. We then generalize this construction by considering a skew-Gaussian process, thus obtaining a process with skew-t marginal distributions. For the proposed (skew) $t$ process we study the second-order and geometrical properties and in the $t$ case, we provide analytic expressions for the bivariate distribution. In an extensive simulation study, we investigate the use of the weighted pairwise likelihood as a method of estimation for the $t$ process. Moreover we compare the performance of the optimal linear predictor of the $t$ process versus the optimal Gaussian predictor. Finally, the effectiveness of our methodology is illustrated by analyzing a georeferenced dataset on maximum temperatures in Australi