I'm working on trying to prove asymptotic independence of two series of discrete random variables. Suppose I have two sequences of random variables $X_n,Y_n$ with probability generating function $G_{X_n}(z),G_{Y_n}(z)$ respectively. When $X_n,Y_n$ converge in distribution to $X,Y$ respectively, then I would like to prove that
$G_{X_n,Y_n}(z_X,z_Y) \xrightarrow{n \rightarrow \infty} G_{X}(z_X) \cdot G_{Y}(z_Y)$.
Now suppose that for example $Y_n$ does not converge in distribution to some $Y$, for example because $Y_n \sim Poi(\lambda = n)$. Would it be sufficient to prove asymptotic independence by showing that either
$\frac{G_{X_n,Y_n}(z_X,z_Y)}{G_{X_n}(z_X) \cdot G_{Y_n}(z_Y)} \xrightarrow{n \rightarrow \infty} 1$
or
$G_{X_n,Y_n}(z_X,z_Y) - G_{X_n}(z_X) \cdot G_{Y_n}(z_Y) \xrightarrow{n \rightarrow \infty} 0$