Open Access Open Access  Restricted Access Subscription or Fee Access

Some Asymptotic Properties of the Conditional Set-Indexed Empirical Process Based on Dependent Functional Data

Salim Bouzebda, Fethi Madani, Youssouf Souddi


The purpose of this paper is to establish the invariance principle for the conditional setindexed empirical process formed by strong mixing random variables when the covariates are functional. We establish our results under some assumptions on the richness of the index class C of sets in terms of metric entropy with bracketing. We apply our main result for testing the conditional independence, that is, testing whether whether two random vectors Y1 and Y2 are independent, given X. The theoretical results of the present paper are (or will be) key tools for many further developments in functional data analysis.


Conditional distribution; Nadaraya-Watson regression estimator; Empirical process; Strong mixing; Functional data; Semi-metric space; Covering number; Small ball probability.

Full Text:


Disclaimer/Regarding indexing issue:

We have provided the online access of all issues and papers to the indexing agencies (as given on journal web site). It’s depend on indexing agencies when, how and what manner they can index or not. Hence, we like to inform that on the basis of earlier indexing, we can’t predict the today or future indexing policy of third party (i.e. indexing agencies) as they have right to discontinue any journal at any time without prior information to the journal. So, please neither sends any question nor expects any answer from us on the behalf of third party i.e. indexing agencies.Hence, we will not issue any certificate or letter for indexing issue. Our role is just to provide the online access to them. So we do properly this and one can visit indexing agencies website to get the authentic information. Also: DOI is paid service which provided by a third party. We never mentioned that we go for this for our any journal. However, journal have no objection if author go directly for this paid DOI service.