the averaging step, estimator $\overline{m}_n$ performs well and is not too sensitive to the choice of $c_m$, provided that the value of $c_m$ is not too small. An intuitive explanation could be that here the recursive process is in some sense “self-normalized” since the deviations at each iteration in (\[def:algoRMcov\]) have unit norm and finding some universal values for $c_m$ is possible. Usual values for $c_m$ and $c_\gamma$ are in the interval $[2,20]$. When $n$ is fixed, this averaged recursive algorithm is about 30 times faster than the Weiszfeld’s approach (see [@CCZ11]).

Asymptotic properties
=====================

When $m$ is known, $\overline{W}_n$ can be seen as an averaged stochastic gradient estimator of the geometric median in a particular Hilbert space and the asymptotic weak convergence of such estimator has been studied in [@CCZ11]. They have shown that:

([@CCZ11], Theorem 3.4). \[theo:asymptnorm\]\
If assumptions 1-3(a) hold, then as $n$ tends to infinity, $$\sqrt{n} \left(\overline{W}_n - \Gamma_m \right)   \rightsquigarrow \mathcal{N}(0, \Delta)$$ where $\rightsquigarrow$ stands for convergence in distribution and $\Delta = \left(\nabla _m^2 (\Gamma_m)\right)^{-1} \Psi  \left(\nabla _m^2 (\Gamma_m)\right)^{-1}$ is the limiting covariance operator, with $\Psi = {{\mathbb{E}}\left[\frac{(Y(m) - \Gamma_m) \otimes_F  (Y(m) - \Gamma_m) }{{{\left\| Y(m) - \Gamma_m \right\|}_F}^2}\right]}.$

As explained in [@CCZ11], the estimator 