median covariation matrix. Then an averaging step ([@PolyakJud92]) of the two initial recursive estimators of the median and the MCM permits to improve the accuracy of the initial stochastic gradient algorithms. A simple modification of the stochastic gradient algorithm is proposed in order to ensure that the median covariance estimator is non negative. We also explain how the eigenelements of the estimator of the MCM can be updated online without being obliged to perform a new spectral decomposition at each new observation.

The paper is organized as follows. The median covariation matrix as well as the recursive estimators are defined in Section 2. In Section 3, almost sure and quadratic mean consistency results are given for variables taking values in general separable Hilbert spaces. The proofs, which are based on new induction steps compared to [@CCZ11], allow to get better convergence rates in quadratic mean even if this new framework is much more complicated because two averaged non linear algorithms are running simultaneously. One can also note that the techniques generally employed to deal with two time scale Robbins Monro algorithms (see [@MR2260078] for the multivariate case) require assumptions on the rest of the Taylor expansion and the finite dimension 