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Abstract <p>This paper explores the intricacies of the general regression functional where the explanatory variables are defined within a functional space. We are particularly interested in the Robbins-Monro-type estimator of this regression functional, especially when the data is drawn from an underlying weakly stationary process. To facilitate this study, we revisit the concept of weak dependence, initially introduced by [<CitationRef CitationID="CR22">22</CitationRef>] for real-valued random variables, and adapt it to accommodate functional data residing in a normed space. We also present several examples of functional processes that satisfy this weak dependence criterion. In our analysis, we rigorously establish the almost sure convergence of the estimator, along with its rate and the asymptotic distribution. These results are obtained under a set of relatively general conditions concerning the classes of functions and the distributions that underpin the data. The contributions of our research are twofold. Firstly, they provide deep insights that significantly enhance the existing statistical methodologies used in the analysis of functional data. Secondly, they lay the groundwork for further applications in various statistical functions. These applications include enhancing the understanding of regression functions, refining the estimation of conditional distribution functions. Through these applications, our findings have the potential to substantially advance the field of functional data analysis.</p>

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Limit Theorems for General Recursive Regression Models Involving Weakly Dependent Functional Data

  • 期刊论文

  • 发表日期：

    2025年12月1日

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  • Lahcen Douge,
  • Salim Bouzebda,
  • Noureddine Berrahou,
  • Kaouthar El Fassi

摘要

Abstract

This paper explores the intricacies of the general regression functional where the explanatory variables are defined within a functional space. We are particularly interested in the Robbins-Monro-type estimator of this regression functional, especially when the data is drawn from an underlying weakly stationary process. To facilitate this study, we revisit the concept of weak dependence, initially introduced by [22] for real-valued random variables, and adapt it to accommodate functional data residing in a normed space. We also present several examples of functional processes that satisfy this weak dependence criterion. In our analysis, we rigorously establish the almost sure convergence of the estimator, along with its rate and the asymptotic distribution. These results are obtained under a set of relatively general conditions concerning the classes of functions and the distributions that underpin the data. The contributions of our research are twofold. Firstly, they provide deep insights that significantly enhance the existing statistical methodologies used in the analysis of functional data. Secondly, they lay the groundwork for further applications in various statistical functions. These applications include enhancing the understanding of regression functions, refining the estimation of conditional distribution functions. Through these applications, our findings have the potential to substantially advance the field of functional data analysis.