On the Heyde theorem for discrete Abelian groups

• Autorzy: G. M. Feldman
• Języki publikacji: EN

Abstrakty

EN

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.

Kategorie tematyczne

• 62E10: Characterization and structure theory
• 60B15: Probability measures on groups or semigroups, Fourier transforms, factorization
• 39B52: Equations for functions with more general domains and/or ranges

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 2006
• Tom: 177
• Numer: 1
• Strony: 67-79

Daty

wydano

2006

Twórcy

autor

G. M. Feldman

• Mathematical Division, B. Verkin Institute for Low Temperature Physics and Engineering, National Academy of Sciences of Ukraine, 47, Lenin Ave., Kharkov, 61103, Ukraine

Identyfikatory

DOI

10.4064/sm177-1-5