Suppose I have a sequence $\{x_n\}$ that I already know converges in the mean-square-sense ($\lim_n E |x_n|^2\to 0$).
Suppose further I know that the sequence $\{x_n\}$ converges in distribution to a Gaussian with zero-mean and covariance matrix $\Sigma$ at the rate of $1/n$ (usual CLT-type results).
Can I now say anything about the evolution of $E|x_n|^2$ in relationship to $\operatorname{Tr}(\Sigma)$ in the limit as $n\to\infty$? Would it be $E |x_n|^2 \sim \operatorname{Tr}(\Sigma)/n$? or do I have to study it explicitly?
Thanks.