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 {θ _j  − θ ₀, 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 ε _j  > − 1 for every j and {u _j , 1 ≤ j ≤ n } is a basis of unit vectors in ℝⁿ with $\mathbf{u}_j^{\prime}$ denoting the transpose of the column vector u _j . 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 { E _s } is a collection of 2ⁿ Borel subsets of ℝⁿ such that { E _s } and {V s U (E _s )} 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 E _s is positive. The converse of this result also holds.

Our results constitute a sharpening of the results of Nabeya and Kariya (J. Multivariate Anal. 20 (1986) 251–264) and part of Khatri (Sankhyā Ser. A 49 (1987) 395–404).