With the development of automatic sensors, it is more and more usual to study large samples of observations taking values in high dimensional spaces such as functional spaces. In this framework, the geometric median, which is a generalization of the real median for metric spaces, is an interesting robust location parameter. Two recursive estimators will be considered. Some results were given on the almost sure convergence of the stochastic gradient estimator and on the asymptotic normality of the averaged version. First, some new results on the L2 rates of convergence of the algorithms are given. Finally, using some exponential inequalities for the martingale differences, asymptotic confidence intervals will be given.