Histograms and associated point processes

Abstract

Non parametric inference for point processes is approached using histograms, which provide a nice tool for the analysis of on-line data. The construction of histograms depend on a sequence of partitions, which we take to be non embedded. This is quite natural in what regards applications, but presents some theoretical problems. On another direction, we drop the usual independence assumption on the sample, replacing it by an association hypothesis. Under this setting, we study the convergence of the histogram, in probability and almost surely, finding conditions on the covariance structure, which is well known to be the determinant factor under association, to ensure the convergence. On the final section we look at the similar question regarding the finite dimensional distributions, proving a convergence in distribution to a gaussian centered vector with a covariance we can describe. The main tool of analysis will be a decomposition of second order moment measures.