-2
$\begingroup$

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!

$\endgroup$
2
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.