Background
Suppose that $X_n\to X$ and $Y_n\to Y$ in distribution. If $X_n$ and $Y_n$ are mutually uncorrelated but not independent, what we can say about $X$ and $Y$ knowing that $X_n\sim Y_n$, $\forall n\in\mathbb{N}$?
In my particular case I have that $X_n\to X\sim \Gamma(1/2,1)$ and similarly $Y_n\to Y\sim\Gamma(1/2,1)$, and experimentally I've seen (by using Monte Carlo simulation) that $X+Y\sim\Gamma(1,1)\sim Exp(1)$, as they were independent.
Question
There exists a concept of asymptotic independence that I can use to prove that indeed the limit distributions $X$ and $Y$ are independent? What conditions $X_n$ and $Y_n$ should satisfy? Thank you in advance.