Normal Distribution Existence Non-affine Invariant Transformation?

Question

Let $X\sim \mathcal{N(\mu,\Sigma)}$ then an affine transformation $AX+b$ will also be a gaussian random vector. Is there any non-affine transformation $f$ such that $f(X)$ is also a gaussian random vector?

Answer 1

There are a huge number of non-affine transformations that will preserve normality of a variable.

For ease of exposition, let's focus on the one-dimensional case with a normal random variable $X$ with distribution function $F$. Note that the question can be reduced to the analogous situation for a uniform variate by means of the probability integral transform

$$U = F^{-1}(X).$$

Any measurable function $\phi:[0,1]\to [0,1]$ that preserves probability induces a probability-preserving transformation of $X$ via

$$\phi^F: X \to F(\phi(F^{-1}(X))).$$

Such functions $\phi$ may be incredibly complex. To illustrate, I will focus on those that are linear, with slope $1$, at all but finitely many points. Examples can be constructed by choosing any positive integer $n$, carving the interval $[0,1)$ into $n$ equal intervals $I_i = [(i-1)/n, i/n)$ for $i=1, 2, \ldots, n$, and applying an arbitrary permutation $\sigma$ to those intervals. Any $t\in[0,1)$ lies uniquely in the interval with index $i = 1 + \lfloor t/n \rfloor$. For $t \in[0, 1)$, define

$$\phi_\sigma(t) = t - \frac{i-1}{n} + \frac{\sigma(i)-1}{n} = t + \frac{\sigma(i) - i}{n}$$

and otherwise let $\phi_\sigma(t)=t$.

$\phi$ may be discontinuous at the values $0, 1/n, 2/n, \ldots, (n-1)/n, 1$, but otherwise obviously is linear with unit slope on $[0,1]$. That implies it preserves probabilities: that is, for any measurable set $E \subset [0, 1)$,

$$\Pr(U \in E) = \int_E du = \int_{\phi(E)} |D\phi(u)|^{-1} du = \int_{\phi(E)} du = \Pr(U \in \phi(E)).$$

Moreover, $\phi$ is invertible, with inverse $\phi_\sigma^{-1} = \phi_{\sigma^{-1}}$.

Here is the graph of a $\phi_\sigma$ for $n=7$:

Here are examples of $\phi^F$ for $n=3, 10, 31, 100$. Each example shows the graph of a one-to-one function for which $\phi(X)$ also has a standard Normal distribution. Obviously none of these are affine.

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The only special property of $F$ that was used in these examples was that it is the distribution function of a continuous variable. Thus, this same construction gives many non-affine transformations that will map a variable $X$ with any given continuous distribution into another variable that is identically distributed.

These are by no means the only examples of such probability-preserving non-affine transformations, but they do form a family that is sufficiently rich to construct examples of arbitrary complexity.