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

Salim Bouzebda, Fethi Madani, Youssouf Souddi

Abstract



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.

Keywords


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

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