Histograms and Associated Point Processes

Abstract

Abstract Nonparametric inference for point processes is discussed by way histograms, which provide a nice tool for the analysis of on-line data. The construction of histograms depends on a sequence of partitions, which we take to be nonembedded. This is quite natural in what regards applications, but presents some theoretical problems. In another direction, we drop the usual independence assumption on the sample, replacing it by an association assumption. Under this setting, we study the convergence of the histogram, in probability and almost surely which, under association, depends on conditions on the covariance structure. In the final section we prove that the finite dimensional distributions converge in distribution to a Gaussian centered vector with a specified covariance. The main tool of analysis is a decomposition of second order moment measures.