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- W2238354502 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 $T(mathbf{x}) = sumlimits_{mathbf{s}} 1_{E_{mathbf{s}}} V mathbf{s} U mathbf{x}$ 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" @default.
- W2238354502 created "2016-06-24" @default.
- W2238354502 creator A5060963010 @default.
- W2238354502 date "2011-11-21" @default.
- W2238354502 modified "2023-09-27" @default.
- W2238354502 title "Two remarks on Normality Preserving Borel Automorphisms of R^n" @default.
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