  The 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, Cenac and Zitt. 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 rate of convergence in quadratic mean of the averaged algorithm is also
given.
