Just a short question about joint convergence. Assume random variables $(X_n)_{n \geq 1}$ and $(Y_n)_{n \geq 1}$ and the following convergences in distribution $X_n \to X$, $Y_n \to Y$. Furthermore, assume $Cov(X_n,Y_n)=c$. What does this imply for the joint convergence in distribution of $(X_n,Y_n)$ (and the convergence of the sum and the product of $X_n$ and $Y_n$)?
Thanks!
Pretty much nothing. The covariance is a very weak measure for the dependence of two random variables.
To see an example: Let $X \sim \mathcal{N}(0, 1)$ and $Z$ be Rademacher distributed, so that $X$ and $Z$ are independent. Set $X_n = X$ and $Y_n = Y = ZX$. Then $X$ and $Y$ are uncorrelated standard Gaussians, but their sum isn't Gaussian. If $X$ and $Y$ were indepedent Gaussians, they would still be uncorrelated, but their sum would be Gaussian aswell.