How can I find the limiting / asymptotic distribution of
$\sqrt{n}(\hat{\sigma}_{XY}-\sigma_{XY})$,
provided
$\sigma_{XY}=E[(X-\mu_X)(Y-\mu_Y)]$ and $\hat{\sigma}_{XY} = n^{-1}\sum(X_i-\bar{X})(Y_i-\bar{Y})$ by definition?
If this is related to matrix since we talk about joint distribution, I would like to ask you how should I build up the step.
Plus, if there is a condition that $E[X^4]$ and $E[Y^4]$ - in other words, kurtoses - are finite, does it change my answer?