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On the Heyde theorem for discrete Abelian groups

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Feldman G.

Studia Mathematica

2006

tom 177

nr 1

67-79

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 $β₁α₁^{-1} ± β₂α₂^{-1} ∈ Aut(X)$. 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.

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1 Studia Mathematica

1 Feldman G.

1 2006

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On the Heyde theorem for discrete Abelian groups