I want to show that if $X_n\to^w X$ and $Y_n\to^w Y$ which is 'weak convergence'
and the $X_n,Y_n$ are independent RV's on the same probability space,
Then we also have weak convergence of the random vector $(X_n,Y_n)\to^w (X,Y)$ Apparantly he independence condition is crucial here ..
I only know that the joint probability distribution is the product of both distributions. I'm not sure how this implies weak convergence of the random vector..