# What is the covariance matrix of the unbiased sample covariance matrix?

I am interested in calculating the covariance matrix of the unbiased sample covariance matrix.

Let $x_1, ..., x_n$ be independent samples of a random vector $x$ with mean $\mathbb E[x]=\mu$ and covariance matrix $\operatorname{Var}[x]=\Sigma$. The sample covariance matrix is given by

$$\mathbf{S}=\frac{1}{n-1}\sum(x_i-\bar{x})(x_i-\bar{x})^\top.$$

I have shown that $\mathbb E[S]=\Sigma$ but am having trouble finding out what $\operatorname{Var}[S]$ is.

• This is the multivariate generalization of the results at stats.stackexchange.com/questions/79808 and stats.stackexchange.com/questions/29905. It can be addressed in the same way described in the nice explanation on the Math site at math.stackexchange.com/a/73080/1489. – whuber Aug 28 '15 at 14:05
• Thank you, I managed to derive it with the help of what you linked me. – Syltherien Aug 31 '15 at 14:01
• +1. I edited your question to improve the formatting. I change the formulations to refer to the "covariance matrix of the sample covariance matrix" and not to the "variance of the sample covariance matrix" because that's presumably what you meant. Apart from that, consider writing up your solution (and maybe even the derivation) as an answer! This will be very useful for future readers. – amoeba says Reinstate Monica Dec 5 '15 at 22:13