$X_n \rightarrow X$ in distribution does not imply $(Y,X_n) \rightarrow (Y,X)$ in distribution

Question

Let $X,Y,X_1,X_2,...$ be r.v. on a polish space $E$ s.t. $X_n \rightarrow X$ in distribution. Does anybody have a counter example for $(Y,X_n) \rightarrow (Y,X)$ in distribution?

I can not think of one and intuitively I find it hard to believe that this implication does not hold.

Answer 1

Take $X_n=Y$ and $X=-Y$, where $Y\overset{d}{=}-Y$.