A Central Limit Theorem for averaged stochastic gradient algorithms in Hilbert spaces and online estimation of the asymptotic variance. Application to the Geometric Median and Quantiles

Stochastic gradient algorithms are more and more studied since they can deal efficiently and online with large samples in high dimensional spaces. In this paper, we first establish a Central Limit Theorem for these estimates as well as for their averaged version in general Hilbert spaces. Moreover, since having the asymptotic normality of estimates is often unusable without an estimation of the asymptotic variance, we introduce a recursive algorithm of the asymptotic variance of the averaged estimator, and we establish its almost sure rate of convergence as well as its rate of convergence in quadratic mean. Finally, an example in robust statistics is given: the estimation of Geometric Quantiles and of the Geometric Median.