Histogram estimation of Radon-Nikodym derivatives for strong mixing data

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 to allow partitions with sets of equal measure. This presents some theoretical problems, which are addressed with an assumption on the decomposition of second order moments. In another direction, we drop the usual independence assumption on the sample, replacing it by a strong mixing assumption. Under this setting, we study the convergence of the histogram in probability, which depends on approximation conditions between the distributions of random pairs and the product of their marginal distributions, and almost completely, which is based on the decomposition of the second order moments. This last convergence is stated on two versions according to the assumption of Laplace transforms or the Cramer moment conditions. These are somewhat stronger, bet enable us to recover the usual condition on the decrease rate of sets on each partition. In the final section we prove that the finite dimensional distributions converge in distribution to a gaussian centered vector with a specified covariance