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I hope this question is not too general, but I am not completely sure yet how to phrase it.

So we all know that when we have two sequences of random variables $X_n$ and $Y_n$ for $n \ge 1$, that converge to $X$ and $Y$ (using some measure, almost surely, or with probability, in distribution, etc.) then we have all kind of nice properties such as $X_n + Y_n \rightarrow X + Y$ and $X_n Y_n \rightarrow X Y$ so on (with some regularity conditions).

My question is: what if we have a general operator on $X$ and $Y$? Are there some mild/not mild conditions on an operator $g(X_n,Y_n)$ that returns a random variable $Z_n$ such that $g(X_n,Y_n) \rightarrow g(X,Y)$?

I think the continuous mapping theorem would state that if $g$ is continuous, then we have that this holds. Are there milder conditions?

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The mode of convergence could make a difference. Note for example that $X_n\to X$ in distribution and $Y_n\to Y$ in distribution does not imply that $X_n+Y_n\to X+Y$ in distribution. –  Did May 20 '11 at 20:05

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