Let $X_1, \ldots, X_T$ are sub-Gaussian random vectors in $\mathbb{R}^d$ coming from a common distribution with population covariance $\Sigma$. If they are independent, it is known that the sample covariance converges to $\Sigma$ with high-probability. In fact, one can establish strong concentration bounds. My question is, can we say similar concentration results when $X_1, \ldots, X_T$ are dependent. I believe we need some sort of $\alpha$- mixing or $\beta$-mixing condition on the dependence structure. I found some related results for random variables (not random vectors) under some mixing conditions (Rio 1993). But not sure whether this holds for random vectors or not? Any help will be appreciated.



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