Two remarks on Normality Preserving Borel Automorphisms of R^n

TL;DR

This paper characterizes Borel automorphisms of imensional space that preserve normal distributions, showing they are affine transformations or specific orthogonal transformations under certain conditions.

Contribution

It provides new characterizations of normality-preserving Borel automorphisms, extending previous results by Nabeya and Kariya with sharper conditions.

Findings

Borel automorphisms preserving certain Gaussian measures are affine linear.

Under specific covariance conditions, such automorphisms are compositions of orthogonal transformations and sign changes.

The results sharpen and extend earlier characterizations of normality-preserving transformations.

Abstract

Let $T$ be a bijective map on $\mathbb{R}^n$ such that both $T$ and $T^{-1}$ are Borel measurable. For any $\btheta \in \mathbb{R}^n$ and any real $n \times n$ positive definite matrix $\Sigma,$ let $N (\btheta, \Sigma)$ denote the $n$-variate normal (gaussian) probability measure on $\mathbb{R}^n$ with mean vector $\btheta$ and covariance matrix $\Sigma.$ Here we prove the following two results: (1) Suppose $N(\btheta_j, I)T^{-1}$ is gaussian for $0 \leq j \leq n$ where $I$ is the identity matrix and $\{\btheta_j - \btheta_0, 1 \leq j \leq n \}$ is a basis for $\mathbb{R}^n.$ Then $T$ is an affine linear transformation; (2) Let $\Sigma_j = I + \epsilon_j \mathbf{u}_j \mathbf{u}_j^{\prime},$ $1 \leq j \leq n$ where $\epsilon_j > -1$ for every $j$ and ${\mathbf{u}_j, 1 \leq j \leq n}$ is a basis of unit vectors in $\mathbb{R}^n$ with $\mathbf{u}_j^{\prime}$ denoting the transpose of the column vector $\mathbf{u}_j.$ Suppose $N(\mathbf{0}, I)T^{-1}$ and $N (\mathbf{0}, \Sigma_j)T^{-1},$ $1 \leq j \leq n$ are gaussian. Then $T(\mathbf{x}) = \sum\limits_{\mathbf{s}} 1_{E_{\mathbf{s}}} V \mathbf{s} U \mathbf{x}$ a.e. $\mathbf{x}$ where $\mathbf{s}$ runs over the set of $2^n$ diagonal matrices of order $n$ with diagonal entries $\pm 1,$ $U,\, V$ are $n \times n$ orthogonal matrices and $\{E_{\mathbf{s}}\}$ is a collection of $2^n$ Borel subsets of $\mathbb{R}^n$ such that $\{E_{\mathbf{s}}\}$ and $\{V \mathbf{s} U (E_{\mathbf{s}})\}$ are partitions of $\mathbb{R}^n$ modulo Lebesgue-null sets and for every $j,$ $V \mathbf{s} U \Sigma_j (V \mathbf{s} U)^{-1}$ is independent of all $\mathbf{s}$ for which the Lebesgue measure of $E_{\mathbf{s}}$ is positive. The converse of this result also holds. \vskip0.1in Our results constitute a sharpening of the results of S. Nabeya and T. Kariya