Non-Parametric Estimation of the Renewal Function for Multidimensional Random Fields

Abstract

This paper addresses the almost sure convergence and the asymptotic normality of an estimator of the multidimensional renewal function based on random fields. The estimator is based on a sequence of non-negative independent and identically distributed (i.i.d.) multidimensional random fields and is expressed as infinite sums of k-folds convolutions of the empirical distribution function. It is an extension of the work from the case of the two-dimensional random fields to the case of the d-dimensional random fields where d>2. This is established by the definition of a “strict order relation”. Concrete applications are given.