Let $X_1,...,X_n$ be i.i.d. sample from a multivariate distribution, and assume that their common covariance matrix $C$ exists. Prove that the empirical covariance matrix based on the sample is a strongly consistent estimator of $C$!
If I am right, the empirical covariance matrix is the following:
$$\frac{1}{n-1} \sum_{i=1}^n (X_i-\overline{X})(X_i-\overline{X})^T,$$
where $\overline{X}$ is the sample mean.
I have read that this matrix is an unbiased estimator of the actual covariance matrix. However, I can't seem to find the proof, why it is strongly consistent, meaning that as $n \to \infty$, the probability, that they are the same, will be $1$.