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When we have two independent sequences or random variables $\{X_{n}\}$ and $\{Y_{n}\}$ for which $X_{n}$ converges weakly to $X$ ( $X_n \overset{w}{\rightarrow}X$) and $Y_n \overset{w}{\rightarrow}Y$ I want to show that holds $$(X_n,Y_n) \overset{w}{\rightarrow}(X,Y)$$

I know that holds for $f\in C_{b}$: $\mathbb{E}_{X}f(X_n)\rightarrow \mathbb{E}f(X)$ and the same for $Y$. I want to show that $\mathbb{E}_{X\times Y}f(X_n,Y_n)\rightarrow \mathbb{E}_{X\times Y}f(X,Y)$

As $X_n$ and $Y_n$ are independent I can write $\mathbb{E}_{X\times Y} = \mathbb{E}_X \times \mathbb{E}_Y$, the product measure.

So $$\mathbb{E}_{X\times Y}f(X_n,Y_n)=\mathbb{E}_X (X_n,Y_n)\times \mathbb{E}_Y (X_n,Y_n)=\mathbb{E}_{X}(X,Y_{n})\times \mathbb{E}_Y (X_n,Y)$$

I now need some help making the last step to get $\mathbb{E}_{X\times Y}(X,Y)$. Could anyone help me with this? I'd really prefer to do it without using characteristic functions if possible.

  • 1
    Another approach is to look at characteristic functions along with [Lévy's continuity theorem](http://en.wikipedia.org/wiki/L%C3%A9vy's_continuity_theorem).2012-11-14
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    As I've not had characteristic functions yet I would really prefer not to..2012-11-14
  • 0
    What would work is the following: $$ \lim_{n\rightarrow \infty} P(X_n \leq x, Y_n \leq y) = \lim_{n\rightarrow \infty} P(X_n \leq x)P(Y_n \leq y) = P(X \leq x)P(Y \leq y)=P(X \leq x,Y \leq y)$$ For all points(x,y) of continuity of the cdf. We just need to show that the discontinuity set has measure 0.2012-11-14
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    The following might help: $\partial(A\times B) = (\partial B\times \overline{A}) \cup (\partial A\times \overline{B})$. $\partial$ means boundary. I am urging u to use the portmanteau theorem.2012-11-14
  • 0
    I'm sorry I've never heard of this theorem.. I can't see how this is relevant. I am trying to keep it simple. So again I'd prefer to do it my way if that is possible without using theorems I havent encountered yet.2012-11-14

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