1-3(b), there is a positive constant $K'$ such that for all $n \geq 1$, $$\begin{aligned}
\mathbb{E}\left[ \left\| \overline{V}_{n} - \Gamma_{m} \right\|_{F}^{2}\right] &\leq \frac{K'}{n}.\end{aligned}$$

Assuming the eigenvalues of $\Gamma_m$ are of multiplicity one, it can be deduced from Theorem \[th:cvgeqm\] and Lemma 4.3 in [@Bosq], the convergence in quadratic mean of the eigenvectors of $\overline{V}_{n}$ towards the corresponding (up to sign) eigenvector of $\Gamma_m$ .

An illustration on simulated and real data
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A small comparison with other classical robust PCA techniques is performed in this section considering data in relatively high dimension but samples with moderate sizes. This permits to compare our approach with classical robust PCA techniques, which are generally not designed to deal with large samples of high dimensional data. In our comparison, we have employed the following well known robust techniques: robust projection pursuit (see [@CR-G2005] and [@CFO2007]), minimum covariance determinant (MCD, see [@RvD99]) and spherical PCA (see [@LMSTZC1999]). The computations were made in the R language ([@R10]), with the help of packages `pcaPP` and `rrcov`. For reproductible research, our codes for computing the MCM have been posted on CRAN in the `Gmedian` package. We will denote by MCM(R) the recursive estimator $\overline{V}_{n}$ defined in (\[def:Gammamedaver\]) and 