Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Two remarks on Normality Preserving Borel Automorphisms of R^n   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

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The categories of this paper are: dg-ga
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The categories of this paper are: gr-qc
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The categories of this paper are: cs
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The categories of this paper are: comp-gas
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The categories of this paper are: econ
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The categories of this paper are: nucl-ex
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The categories of this paper are: ao-sci
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The categories of this paper are: cmp-lg
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The categories of this paper are: q-bio
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The categories of this paper are: adap-org
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The categories of this paper are: q-alg
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The categories of this paper are: nlin
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The categories of this paper are: funct-an
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The categories of this paper are: physics
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The categories of this paper are: bayes-an
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The categories of this paper are: nucl-th