Is there a concept of asymptotically independent random variables? To prove some results using a standard theorem I need my random variables to be i.i.d. 
However, my random variables are discrete uniforms emerging from a rank statistics, i.e. not independent: for two $u_1$,$u_2$ knowing $u_1$ gives me $u_2$.
Yet, my interest is in the asymptotics for ($u_i$) growing infinitely large, and thus these r.v. are becoming more and more independent.
Is there a concept to formalize this idea? Is this standard? Can I have an entry point in the literature?   

If $(X_n, Y_n)$ are dependent for all finite $n$, but converges in distribution (or "weakly") to some random variable $(X,Y)$ where $X$ and $Y$ is independent, then we would say that $(X_n, Y_n)$ are asymptotically independent. 
