https://hal-univ-bourgogne.archives-ouvertes.fr/hal-01413440Godichon-Baggioni, AntoineAntoineGodichon-BaggioniIMB - Institut de Mathématiques de Bourgogne [Dijon] - UB - Université de Bourgogne - CNRS - Centre National de la Recherche Scientifique - UFC - Université de Franche-Comté - UBFC - Université Bourgogne Franche-Comté [COMUE]Estimating the geometric median in Hilbert spaces with stochastic gradient algorithms: $L^p$ and almost sure rates of convergenceHAL CCSD2016Functional data analysisLaw of large numbersMartingales in Hilbert spaceRecursive estimationRobust statisticsSpatial medianStochastic gradientAlgorithms[MATH.MATH-ST] Mathematics [math]/Statistics [math.ST]université de Bourgogne, IMB -2016-12-09 17:35:542022-08-04 17:08:012016-12-09 17:35:54enJournal articles10.1016/j.jmva.2015.09.0131The geometric median, also called $L^1$-median, is often used in robust statistics. Moreover, it is more and more usual to deal with large samples taking values in high dimensional spaces. In this context, a fast recursive estimator has been introduced by Cardot et al. (2013). This work aims at studying more precisely the asymptotic behavior of the estimators of the geometric median based on such non linear stochastic gradient algorithms. The $L^p$ rates of convergence as well as almost sure rates of convergence of these estimators are derived in general separable Hilbert spaces. Moreover, the optimal rates of convergence in quadratic mean of the averaged algorithm are also given.