$X_n \xrightarrow[]{d} X$, $Y_n \xrightarrow[]{d} Y$ where $X \sim N(\mu_x, \sigma_x)$ and $Y \sim N(\mu_y, \sigma_y)$, but $X_n \not\!\perp\!\!\!\perp Y_n$. What do we need to analyze $(X_n, Y_n) \xrightarrow[]{d} \: ?$.
In other words, we have two dependent random variables that converge in distribution and we are interested in the asymptotics of their joint distribution. Is there any work out there that addresses this topic?
My intuition says that under specific conditions on the dependence, this will converge to a joint normal, but what are these conditions?
Your intuition is correct. Suppose $X_n$ are iid $N(0,1)$ and $Y_n=(-1)^n X_n$. Clearly the sequences $X_n$ and $Y_n$ each converge in distribution, but $X_nY_n$ does not. So the sequence of pairs $(X_n,Y_n)$ does not, either.