Consider two sequences $(X_n)$ and $(Y_n)$ of random elements in some nice (e.g. Polish) space s.t. $X_n\Rightarrow X$ and $Y_n\Rightarrow Y$ ("$\Rightarrow$" denotes weak convergence). Then we know that in general $(X_n,Y_n)\Rightarrow(X,Y)$ doesn't hold. However, if for instance $Y=c$ is deterministic, then $(X_n,Y_n)\Rightarrow(X,c)$ is true. One can find this in every basic book on proba theory.
Now are there any more interesting conditions apart from the one I gave where one still gets weak convergence of the pair $(X_n,Y_n)$? Like $(X_n,Y_n)$ independent, or somehow mixing. Or $X_n$ and $Y_n$ independent/mixing. Are there some results out there? The weaker the condition, the better ;)
Let $f,g$ be two positive Borelian functions.
If $X_n ,Y_n$ are independent and $X,Y$ as well, then $$ E[f(X_n) g(Y_n)] = E[f(X_n)] E[g(Y_n)] \to E[f(X)] E[g(Y)] = E[f(X) g(Y)] $$ hence $(X_n,Y_n)\Rightarrow(X,Y)$.