Convergence in distribution under equivalent measures Say we have two probability measures $P$ and $Q$ on the same probability space and that they are equivalent (absolutely continuous with respect to each other). Then, one can show that convergence in probability under $P$ implies convergence under $Q$ and vice-versa (for example, see here).
But does the same hold for convergence in distribution? Specifically, if we have a sequence of random variables $X_n$ and a limit $X$, is it true that $X_n \xrightarrow{D} X$ under $P$ if and only if $X_n \xrightarrow{D} X$ under $Q$? I would think it is since convergence in probability is generally stronger, but I can't seem to find a result like this anywhere.

Hint: On any probability space $X,-X,X,-X,\cdots$ converges in distribution if and only if $X$ and $-X$ have the same distribution. For a counter-example all you need is two probability measures $P$ and $Q$ which are absolutely continuous w.r.t. each  other such $X$ had a symmetric distribution w.r.t $P$ but not w.r.t  $Q$. I will leave the construction to  you.
