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On a characterization theorem for connected locally compact abelian groups. (English) Zbl 1486.43002

This paper is devoted to prove that, if a locally compact abelian group \(X\) contains nonzero compact elements, then the condition \[ \operatorname{ker}(\alpha)= \{0\}\tag{1} \] is necessary and sufficient for characterization of the Gaussian distribution on \(X\), where \(I\) is the identity automorphism of the group and \(\alpha\) is a topological automorphism of \(X\). Moreover, the author proves that the characteristic functions of independent random variables \(\xi_1\) and \(\xi_2\) do not vanish and proves that if a locally compact abelian group contains no elements of order 2, then condition (1) is necessary and sufficient for characterizing the Gaussian distribution on \(X\) by the symmetry of the conditional distribution of the linear form given \(L_2= \xi_1+ \alpha\xi_2\), \(L_1= \xi_1+ \xi_2\).

MSC:

43A25 Fourier and Fourier-Stieltjes transforms on locally compact and other abelian groups
43A35 Positive definite functions on groups, semigroups, etc.
60B15 Probability measures on groups or semigroups, Fourier transforms, factorization
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