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If I have a random variable $X \in \mathbb{R}^n$, under which conditions is there a $C^1$ function $\varphi: \mathbb{R}^n \rightarrow \mathbb{R}^n$ such that $\varphi(X) \sim \mathcal{N}(0, I_n)$ (vector of $n$ independent normal variables)?

$X$ probably has to have a density, to begin with... But I'm kind of stuck there. I know that if $n=1$, inverse transform sampling can be used to solve this issue, but I'm having trouble finding a generalization to $R^n$.

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If $X \in U$ where $U$ is diffeomorphic to $\mathbb{R}^n$, and $X$ has a continuous and positive density on $U$, then it can be diffeomorphically mapped to $\mathcal{N}(0, I_n)$.

Indeed, let $\varphi_1: U \rightarrow \mathbb{R}^n$ be a diffeomorphism, and $X' = \varphi(X)$. Then $X'$ has a continuous positive density on $\mathbb{R}^n$. By Theorem 1. in Hyvärinen & Pajunen, 1999, there is a diffeomorphism $\varphi_2: \mathbb{R}^n \rightarrow (0, 1)^n$ such that $X'' = \varphi_2(X') \sim U((0,1)^n)$. The components of $X''$ are independent (see the paper for details) and thus can individually be mapped (via inverse transform sampling) to $\mathcal{N}(0, 1)$, which is the same as mapping $X''$ to $\mathcal{N}(0, I_n)$.

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