Functional estimation of a density under a new weak dependence condition

The purpose of this paper is to prove, through the analysis of the behaviou r of a standard kernel density estimator, that the notion of weak dependenc e defined in a previous paper (cf. Doukhan & Louhichi, 1999) has sufficient ly sharp properties to be used in various situations, More precisely we inv estigate the asymptotics of high order losses, asymptotic distributions and uniform almost sure behaviour of kernel density estimates. We prove that t hey are the same as for independent samples (with some restrictions for a.s . behaviours), Recall finally that this weak dependence condition extends o n the previously defined ones such as mixing, association and it allows con siderations of new classes such as weak shifts processes based on independe nt sequences as well as some non-mixing Markov processes.