# What is Asymptotic Independence

What does it mean if two random variables are asymptotically independent? And how would you prove that they are?

Assume you have a probability measure $$P$$. Two events $$A$$ and $$B$$ are independent iff $$P(A\cap B) = P(A)P(B)$$. Two random variables $$X$$ and $$Y$$ are independent iff for any measurable sets $$A$$ and $$B$$, $$P(\{X\in A\}\cap\{ Y \in B\}) = P(X\in A)P(Y \in B)$$.
If we have sequences of random variables $$X_n$$ and $$Y_n$$ and consider the behavior of the sequences as $$n\to\infty$$, we would say the random variables in these sequences are asymptotically independent if $$P(\{X_n\in A\}\cap\{ Y_n \in B\})$$ resembles $$P(X_n\in A)P(Y_n \in B)$$ in the limit, or more precisely, $$\left|P(\{X_n\in A\}\cap\{ Y_n \in B\}) - P(X_n\in A)P(Y_n \in B)\right|\to 0$$ as $$n\to \infty$$. This is what we would say if we were referring to sequences of random variables, though it's possible to be more general.