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I know from this post, that Asymptotic distribution of sample variance of i.i.d non-normal sample that the sample variance $s^2$ is distributed as follows: $$\sqrt n(s^2 - \sigma^2) \rightarrow_d N\left(0,\mu_4 - \sigma^4\right)\;\; $$

My question is, does this still hold in the case where instead of having independent random variables, we have uncorrelated random variables?

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