$\overline{W}_n$ is efficient in the sense that it has the same asymptotic distribution as the empirical risk minimizer related to $G_m(V)$ (see for the derivation of its asymptotic normality in [@MNO2010] in the multivariate case and [@ChaCha2014] in a more general functional framework).

Using the delta method for weak convergence in Hilbert spaces (see [@DauxoisPousseRomain82] or [@CGER2007]), one can deduce, from Theorem \[theo:asymptnorm\], the asymptotic normality of the estimated eigenvectors of $\overline{W}_n$. It can also be proven (see [@godichon2015]), under Assumptions 1-3, that there is a positive constant $K$ such that for all $n \geq 1$, $$\mathbb{E}\left[ \left\| \overline{W}_{n} - \Gamma_{m} \right\|_{F}^{2} \right] \leq \frac{K}{n}.$$ Note finally that non asymptotic bounds for the deviation of $\overline{W}_n$ around $\Gamma_m$ can be derived readily with the general results given in [@CCG2015].

The more realistic case in which $m$ must also be estimated is more complicated because $\overline{V}_n$ depends on $\overline{m}_n$ which is also estimated recursively with the same data. We first state the strong consistency of the estimators $V_n$ and $\overline{V}_n$.

\[theops\] If assumptions 1-3(b) hold, we have $$\begin{aligned}
\lim_{n \rightarrow \infty}\left\|V_{n} -\Gamma_{m}\right\|_{F}=0 \quad a.s.\end{aligned}$$ and $$\begin{aligned}
\lim_{n \rightarrow \infty}\left\| \overline{V}_{n} - \Gamma_{m}\right\|_{F} &=0 \quad a.s.\end{aligned}$$

The obtention of the rate convergence 