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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)?

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What is the question? If $(X_n,Y_n)\to(X,Y)$ in distribution, then, for any continuous function $u$, $u(X_n,Y_n)\to u(X,Y)$. For example $u$ can be the projection on some coordinates. So yes, this implies that $X_n\to X$ and that $Y_n\to Y$. –  Did Oct 26 '11 at 6:54
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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 $F_{(X,Y)}(x,y)=P(X\le x, Y\le y)$.

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