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Heyde's characterization theorem for some locally compact Abelian groups

G. M. Feldman

B. Verkin Institute for Low Temperature Physics and Engineering, National Academy of Sciences of Ukraine, Khar'kov

Abstract: By Heyde's theorem, the Gaussian distribution on the real line is characterized by the symmetry of the conditional distribution of one linear form of $n$ independent random variables with the other fixed. When $n=2$ we prove analogues of this theorem in the case when independent random variables take values in a locally compact Abelian group $X$ and coefficients of the linear forms are topological automorphisms of $X$.

Keywords: locally compact Abelian group, Gaussian distribution, conditional distribution.

Received: 15.02.2016
Revised: 01.02.2017
Accepted: 20.02.2017

DOI: 10.4213/tvp5127

 English version: Theory of Probability and its Applications, 2018, 62:3, 399–412

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