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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 $1$.

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  • $\begingroup$ Do you know the strong law of large numbers? $\endgroup$ – kimchi lover Oct 2 '17 at 21:56
  • $\begingroup$ I see, the strong law of large numbers state that this matrix will convergenge to the expected value, and the expected value is the same as $C$, since it is unbiased? $\endgroup$ – Atvin Oct 3 '17 at 9:48

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