Convergence in distribution of product of two sequences of random variables

Question

For random variables $\{X_n\}$, $\{Y_n\}$ that converge in distribution to X and Y, what would it mean if $\{(X_n,Y_n)\}$ converges in distribution to (X,Y)?

Answer 1

I hope you want to know the meaning of $(X_n,Y_n)$ converges in distribution to $(X,Y)$. As in the one dimensional case, $(X_n,Y_n)$ is said to converge in distribution to $(X,Y)$ if $F_{(X_n,Y_n)}(x,y) \to F_{(X,Y)}(x,y)$ at all points $(x,y)$ where $F_{(X,Y)}$ is continuous, where $F_{(X,Y)}(x,y)=P(X\le x, Y\le y)$.