Bounty: 50
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 $kinmathbb{N}$ 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 $zinmathbb{R}^n$:
$$Phi_n(z)=[Phi(z_1),dots,Phi(z_n)],$$ 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 $iin{1,dots,n}$, $F_{X_{k,i}}$ is the CDF of the $i^mathrm{th}$ component of $F_{X_k}$.
It is easy to see that for all $kinmathbb{N}$, $X_k$ 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 $krightarrowinfty$, and (ideally) such that $tilde{X}_k$ also converges in distribution $X$ as $krightarrowinfty$.
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?
