I came across the following problem.
Convergence in the following always means weak convergence, i.e. $X_n \rightarrow X$ if and only if $Ef(X_n) \rightarrow Ef(X)$ for all $f$ bounded, continuous and real functions.
Assume $(X_n, Y_n)$ converges jointly to $(X, Y)$.
1) Does it hold, that $f(X_n, Y_n)$ converges to $f(X, Y)$ for all continuous $f$?
I.e. does the continuous mapping theorem hold in the multivariate case?
And if it does not, is it sufficient to additionally assume independence of $X$ and $Y$?
2) Furthermore, f still continuous, is it true, that $(X_n, Y_n, f(X_n, Y_n))$ converges jointly to $(X, Y, f(X, Y))$? Again if not, does it hold for $X$ and $Y$ independent? Respectively what does $f$ have to fulfill?
I came across a paper, where the author is implicitly using both statements. But the only thing I can find is the continuous mapping theorem in the univariate case.
And obviously I know, that if $X_n$ and $Y_n$ are independent and converge to $X$ and $Y$, then they also converge jointly.
I'd appreciate any help or maybe also some keywords to google myself.
Thanks a lot!
Best Answer
For (1), you need to verify that for each continuous and bounded $h$, the sequence $Eh(f(X_n,Y_n)))$ converges to $Eh(f(X,Y))$. But the composition $g(x,y)=h(f(x,y))$ is continuous and bounded if $h$ is c&b and if $f$ is continuous. So the weak convergence of $(X_n,Y_n)$ implies the weak convergence of $f(X_n,Y_n)$.
Similarly for (2), except now you need to check that for each continuous and bounded $h(x,y,z)$ that $Eh(X_n,Y_n,f(X_n,Y_n))$ converges to $Eh(X,Y,f(X,Y))$. In this case you mentioned no hypothesis on $f$, but continuity will allow the above argument to go through.
As follows. For given bounded continuous $h$ define the function $g$ of two variables by $g(x,y)=h(x,y,f(x,y))$. Assuming $f$ is continuous, so is $g$; clearly $g$ is also bounded. By assumption $(X_n,Y_n)$ converge in distribution to $(X,Y)$, so $ Eg(X_n,Y_n)\to Eg(X,Y)$. But this means $ Eh(X_n,Y_n,f(X_n,Y_n)) \to E h(X,Y,f(X,Y))$, by the way $g$ is defined in terms of $h$. Since this holds for all continuous and bounded $h$ we see that $(X_n,Y_n,f(X_n,Y_n))$ converges in distribution to $(X,Y,f(X,Y))$.