Let $(X_n)_{n \in \mathbb{N}}$ and $(Y_n)_{n \in \mathbb{N}}$ be two sequences of random variables such that the sequences $(X_n)_{n \in \mathbb{N}}$ and $(Y_n)_{n \in \mathbb{N}}$ converge in law to $X$ and $Y$, where $X$ and $Y$ are also random variable, respectively.
We say then that $X_n$ and $Y_n$ are asymptotically independent if and only if $$P(\{X_n\in A\}\cap\{ Y_n \in B\}) \rightarrow P(X\in A)P(Y \in B)$$ as $n\to \infty$ for every Borel sets $A$ and $B$.
Is it correct?
Best Answer
No, it's not. You need that $P(X_n\in A) \to P(X\in A)$ and $P(Y_n\in B) \to P(Y\in B)$, $n\to\infty$. It is e.g. sufficient that $P(X\in \partial A) = P(Y\in \partial B)=0$.