Indian J. Pure Appl. Math., 51(2): 465-481, June 2020 c  Indian National Science Academy DOI: 10.1007/s13226-020-0411-y ASYMPTOTIC NORMALITY OF CONDITIONAL MODE ESTIMATION FOR FUNCTIONAL DEPENDENT DATA Oussama Bouanani * , Saˆ adia Rahmani * , Ali Laksaci ** and Mustapha Rachdi *** * Laboratoire de Mod` eles Stochastiques, Statistique et Applications, Universit´ e Dr Tahar Moulay, Saida, Algeria ** Department of Mathematics, College of Science, King Khalid University, Abha, 61413, Saudi Arabia *** Univ. Grenoble Alpes, CNRS, Grenoble INP, TIMA, UFR SHS, BP 47, F38040 Grenoble Cedex 09, France e-mails: oussamaproba@gmail.com, saadia.rahmani@gmail.com; alilak@yahoo.fr; mustapha.rachdi@univ-grenoble-alpes.fr (Received 25 May 2018; after final revision 8 February 2019; accepted 20 February 2019) Based on the local polynomial smoother idea, we construct a local linear estimator of the con- ditional mode for dependent functional covariables. Precisely, observations are assumed to be a sequence of stationary α-mixing random variables. Then, we establish the asymptotic normality of the constructed estimator. Key words : Functional data analysis; local linear method; kernel method; conditional mode; asymptotic normality. 2010 Mathematics Subject Classification : 62G05, 62G08, 62G20. 1. I NTRODUCTION AND MOTIVATIONS Nonparametric estimation in models containing functional data has been the subject of many studies over the last decade. This area of statistical research, called non-parametric functional data analysis (NFDA), is concerned with non-parametric modeling of data in the form of curves, images or objects. For a general overview on this subject, one can refer to the monographs by Ferraty and Romain [20] and Ferraty and Vieu [19] as well as to the references therein.