Can an arbitrary random vector be approximated by a normal random vector?

Question

Let $X$ be an $n$-dimensional gaussian random vector, and $Y$ be an $n$-dimensional random vector such that $\det(cov(X))\neq 0$ and $\det (cov(Y))\neq 0$.

Then, does there exists a sequence of invertible $C^1$ functions $f_k$ such that $f_k\circ X \Rightarrow Y$ (weak convergence)?

Is there any reference related to this kind of problem?