Asymptotic results for recursive multivariate associated-kernel estimators of the probability density mass function of a data stream

In this article, our central focus is to investigate the non parametric estimator of the probability density or mass function of a data stream by using the general family of kernels, also called multivariate associated kernels. Being able to estimate categorial, count, discrete, and (semi)continuous distributions, these multivariate associated kernel estimators unify several particular cases. Within this framework, we first introduce a recursive estimator for non classical associated kernels. Under reasonable assumptions, we subsequently exhibit their asymptotic results, such as the local uniform Lq-consistency, the pointwise asymptotic normality and finally the strong global consistency. Illustrative examples of associated kernels are revisited.