# Convergence of covariance matrix

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 $$\boldsymbol{\Sigma}_{n}$$?.

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