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The expected squared deviation of a random variable from its mean; or, the average squared deviation of data about their mean.

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Second order statistics of sample statistics for random vectors

I have a set of iid random vectors $\{\boldsymbol{X}^i\}_{i=1}^N$, whose expected value is $\mathbb{E}[\boldsymbol{X}^i] = \boldsymbol{\mu}$ and whose variance - covariance matrix is $\boldsymbol{\sigma … frac{1}{N}\boldsymbol{\sigma}$$ Also, I know that $$\mathbb{E}[\boldsymbol{S}_N] = \boldsymbol{\sigma}$$ What is the general result (that is, without the normality or independence assumption) for the variance
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