$V_n$ and $\overline{V}_n$ are non negative.

Online estimation of the principal components
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It is also possible to approximate recursively the $q$ eigenvectors (unique up to sign) of $\Gamma_m$ associated to the $q$ largest eigenvalues without being obliged to perform a spectral decomposition of $\overline{V}_{n+1}$ at each new observation. Many recursive strategies can be employed (see [@CD2015] for a review on various recursive estimation procedures of the eigenelements of a covariance matrix). Because of its simplicity and its accuracy, we consider the following one: $$\begin{aligned}
 u_{j,n+1} &= u_{j,n} + \frac{1}{n+1} \left( \overline{V}_{n+1} \frac{u_{j,n}}{\| u_{j,n}\|} - u_{j,n} \right), \quad j=1, \ldots, q 
 \label{algo:vectp}\end{aligned}$$ combined with an orthogonalization by deflation of $u_{1,n+1}, \ldots u_{q,n+1}$. This recursive algorithm is based on ideas developed by [@Wengetal2003] that are related to the power method for extracting eigenvectors. If we assume that the $q$ first eigenvalues $\lambda_1 > \cdots > \lambda_q$ are distinct, the estimated eigenvectors $u_{1,n+1}, \ldots u_{q,n+1}$, which are uniquely determined up to sign change, tend to $\lambda_1 u_1, \ldots, \lambda_q u_q.$

Once the eigenvectors are computed, it is possible to compute the principal components as well as indices of outlyingness for each new observation (see [@HRVA2008] for a review of outliers detection with 