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

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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.

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