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I was looking for a simple way to find the number of samples $n$ needed to get a decent approximation to the covariance matrix $\boldsymbol{\Sigma}$. Given a random sample $\{ \mathbf{X}_1,\mathbf{X}_2, \dots ,\mathbf{X}_n\}$, the sample covariance matrix is

$$ \boldsymbol{\Sigma}_{n} = \frac{1}{n-1} \sum^{n}_{k=1} \left( \boldsymbol{\mathbf{X}}_k - \overline{\boldsymbol{\mathbf{X}}} \right)\left(\boldsymbol{\mathbf{X}}_k - \overline{\boldsymbol{\mathbf{X}}} \right)^T $$

I want to plot the number of samples $n$ (on the x-axis) vs some scalar value computed from the estimate $\boldsymbol{\Sigma}_{n}$ (on the y-axis). What quantity can I compute in a simple way to plot such a convergence graph?. What about the matrix norm $\left \| \cdot \right \|_2$, i.e. the maximum singular value of $\boldsymbol{\Sigma}_{n}$?.

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  • $\begingroup$ The near-duplicate at stats.stackexchange.com/questions/59478/… focuses on the multivariate Normal distribution, but its answers can be generalized. $\endgroup$
    – whuber ♦
    Commented May 8, 2019 at 14:51
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    $\begingroup$ Did you look at Wishart distribution already? $\endgroup$
    – Aksakal
    Commented May 8, 2019 at 15:26

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