Let X be a countable discrete Abelian group, Aut(X) the set of automorphisms of X, and I(X) the set of idempotent distributions on X. Assume that α₁, α₂, β₁, β₂ ∈ Aut(X) satisfy . Let ξ₁, ξ₂ be independent random variables with values in X and distributions μ₁, μ₂. We prove that the symmetry of the conditional distribution of L₂ = β₁ξ₁ + β₂ξ₂ given L₁ = α₁ξ₁ + α₂ξ₂ implies that μ₁, μ₂ ∈ I(X) if and only if the group X contains no elements of order two. This theorem can be considered as an analogue for discrete Abelian groups of the well-known Heyde theorem where the Gaussian distribution on the real line is characterized by the symmetry of the conditional distribution of one linear form given another.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm177-1-5,
author = {G. M. Feldman},
title = {On the Heyde theorem for discrete Abelian groups},
journal = {Studia Mathematica},
volume = {173},
year = {2006},
pages = {67-79},
zbl = {1111.62013},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm177-1-5}
}
G. M. Feldman. On the Heyde theorem for discrete Abelian groups. Studia Mathematica, Tome 173 (2006) pp. 67-79. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm177-1-5/