{{\mathbb{E}}\left[(X-m)(X-m)^T\right]}$, which is well defined in this case, and $\Gamma_m$ share the same eigenvectors: if $e_j$ is an eigenvector of $\Sigma$ with corresponding eigenvalue $\lambda_j$, then $\Gamma_m e_j = \tilde{\lambda}_j e_j$, for some non negative value $\tilde{\lambda}_j$. This important result means that for Gaussian and more generally symmetric distribution (with finite second order moments), the covariance operator and the median covariation operator have the same eigenspaces. Note that it is also conjectured in [@KrausPanaretos2012] that the order of the eigenfunctions is also the same.

Efficient recursive algorithms
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We suppose now that we have i.i.d. copies $X_1, \ldots, X_n, \ldots$ of random variables with the same law as $X$.

For simplicity, we temporarily suppose that the median $m$ of $X$ is known. We consider a sequence of (learning) weights $\gamma_n = c_\gamma / n^{\alpha}$, with $c_\gamma>0$ and $1/2 <\alpha <1$ and we define the recursive estimation procedure as follows $$\begin{aligned}
W_{n+1} &= W_n + \gamma_n \frac{ (X_{n+1}-m)(X_{n+1}-m)^T - W_n}{ {\left\| (X_{n+1}-m)(X_{n+1}-m)^T - W_n \right\|}_F} \label{def:algoRMcov}\\
\overline{W}_{n+1} &= \overline{W}_{n} - \frac{1}{n+1} \left( \overline{W}_{n}  - W_{n+1} \right).\end{aligned}$$ This algorithm can be seen as a particular case of the averaged stochastic gradient algorithm studied in [@CCZ11]. Indeed, the first recursive algorithm (\[def:algoRMcov\]) is 