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.