Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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?

share|improve this question
    
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

1 Answer 1

Look up the continuous mapping theorem.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.