Abstract
According to the well-known Heyde theorem, the Gaussian distribution on the real line is characterized by the symmetry of the conditional distribution of one linear form of independent random variables given another. In the article we study analogues of this theorem for some locally compact Abelian groups. We consider linear forms of two independent random variables with values in a locally compact Abelian group X, whose characteristic functions do not vanish. Unlike most previous works, we do not impose any restrictions on coefficients of the linear forms. They are arbitrary topological automorphisms of X.
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Acknowledgements
The final version of this article was written during my stay at the Department of Mathematics University of Toronto as a Visiting Professor. I am very grateful to Ilia Binder for his invitation and support. I also thank the referee for a careful reading of the paper and useful suggestions and comments.
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Feldman, G.M. Generalization of the Heyde Theorem to Some Locally Compact Abelian Groups. Results Math 77, 177 (2022). https://doi.org/10.1007/s00025-022-01719-z
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DOI: https://doi.org/10.1007/s00025-022-01719-z