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?
