Let $\tilde{X}_0$ be some random variable on $\mathbb{R}^n$, with a strictly positive p.d.f..
Define: $$X_0:=(\operatorname{var}{\tilde{X}_0})^{-\frac{1}{2}}(\tilde{X}_0-\mathbb{E}\tilde{X}_0),$$ where we take the unique positive definite matrix square root.
Further, for all $k\in\mathbb{N}$ define: $$\tilde{X}_{k+1}:=\Phi^{-1}_n(F_{X_k}(X_k)),$$ and: $$X_{k+1}:=(\operatorname{var}{\tilde{X}_k})^{-\frac{1}{2}}\tilde{X}_k,$$ where for all $z\in\mathbb{R}^n$: $$\Phi_n(z)=[\Phi(z_1),\dots,\Phi(z_n)],$$ and:$$F_{X_k}(z)=[F_{X_{k,1}}(z_1),\dots,F_{X_{k,n}}(z_n)],$$ where $\Phi$ is the CDF of a standard Normal distribution, and for $i\in\{1,\dots,n\}$, $F_{X_{k,i}}$ is the CDF of the $i^\mathrm{th}$ component of $F_{X_k}$.
It is easy to see that for all $k\in\mathbb{N}$, $X_k$ is mean zero and has covariance given by the identity matrix, and that $\tilde{X}_{k+1}$ has standard Normal marginals.
I would like to prove that there is some $X$ such that $X_k$ converges in distribution to $X$ as $k\rightarrow\infty$, and (ideally) such that $\tilde{X}_k$ also converges in distribution $X$ as $k\rightarrow\infty$.
The Banach fixed point theorem cannot be applicable as the mapping has more than one fixed point. E.g. with $n=2$, both the bivariate standard Normal distribution and the distribution with p.d.f. $(x,y)\mapsto \frac{1}{2}(1-\Phi(\max(|x|,|y|)))$ are fixed points.
Can you prove the convergence to a fixed point?
