Estimating the geometric median in Hilbert spaces with stochastic gradient algorithms: LpLp and almost sure rates of convergence

The geometric median, also called L1L1-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 LpLp 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.