Let $(X_n)_{n\in \mathbb N}$ and $(Y_n)_{n\in \mathbb N}$ be random sequences independent of each other. Further suppose both $X_n$ and $Y_n$ converge in distribution (to some random variable).
I know that if $X'$ and $Y'$ are weak limits of $X_n$ and $Y_n$, then they need not be independent.
My question is, can I (always) find "independent" random variables $X$ and $Y$ that are weak limits of $X_n$ and $Y_n$ respectively?
Thanks and regards.
Best Answer
Yes, you can.
Suppose you are given two limit distributions $X'$ and $Y'$ which are dependent. What you can always do is to take an $X$ (random variable) which is equal in distribution to $X'$ but which is independent of $Y'$. You can view this $X$ as a random sample of size $1$ from the distribution of $X'$.
I think this explanation is a bit vague, so don't hesitate to comment. And I hope this helps!