Limit of two uncorrelated random variables

Question

Let $\{X_n\}$ and $\{Y_n\}$ be two sequences of uncorrelated random variables with finite fourth moments. Also, let
$$
X_n \xrightarrow{d} X \text{ and } Y_n\xrightarrow{d} Y,
$$
where $X$ and $Y$ are standard normal random variables. Can we conclude $(X_n,Y_n)\xrightarrow{d} (X',Y')$ such that $X'$ and $Y'$ are independent standard normal random variables? If not, can you please provide some sufficient conditions? Thanks for the help.

Answer 1

Since you're declining to assume the pairs $(X_i,Y_i)$ are independent of each other for different values of $i,$ the answer is no, because you can take $(X_i,Y_i)=(X_1,Y_1)$ for every i, and $X_1\sim\operatorname N(0,1)$ and $Y_1=\pm X_1$ where the choice between $\pm$ is independent of the value of $X_1$ and plus and minus are equally probable.