Two remarks on normality preserving Borel automorphisms of $\boldsymbol{\mathbb{R}{\kern-8.3pt}\mathbb{R}}^{{\bf \textit{n}}}$

Abstract

Let T be a bijective map on ℝn such that both T and T − 1 are Borel measurable. For any θ ∈ ℝn and any real n ×n positive definite matrix Σ, let N (θ, Σ) denote the n-variate normal (Gaussian) probability measure on ℝn with mean vector θ and covariance matrix Σ. Here we prove the following two results: (1) Suppose \(N(\boldsymbol{\theta}_j, I)T^{-1}\) is gaussian for 0 ≤ j ≤ n, where I is the identity matrix and {θj − θ0, 1 ≤ j ≤ n } is a basis for ℝn. Then T is an affine linear transformation; (2) Let \(\Sigma_j = I + \varepsilon_j \mathbf{u}_j \mathbf{u}_j^{\prime},\) 1 ≤ j ≤ n where ej > − 1 for every j and {uj, 1 ≤ j ≤ n } is a basis of unit vectors in ℝn with \(\mathbf{u}_j^{\prime}\) denoting the transpose of the column vector uj. Suppose N(0, I)T − 1 and \(N (\mathbf{0}, \Sigma_j)T^{-1},\) 1 ≤ j ≤ n are gaussian. Then \(T(\mathbf{x}) = \sum\nolimits_{\mathbf{s}} 1_{E_{\mathbf{s}}}(\mathbf{x}) V \mathbf{s} U \mathbf{x}\) a.e. x, where s runs over the set of 2n diagonal matrices of order n with diagonal entries ±1, U, V are n ×n orthogonal matrices and { Es} is a collection of 2n Borel subsets of ℝn such that { Es} and {VsU (Es)} are partitions of ℝn modulo Lebesgue-null sets and for every j, \(V \mathbf{s} U \Sigma_j (V \mathbf{s} U)^{-1}\) is independent of all s for which the Lebesgue measure of Es is positive. The converse of this result also holds.