On the Heyde theorem for discrete Abelian groups

G. M. Feldman. "On the Heyde theorem for discrete Abelian groups." Studia Mathematica 177.1 (2006): 67-79. <http://eudml.org/doc/284985>.

@article{G2006,
abstract = {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.},
author = {G. M. Feldman},
journal = {Studia Mathematica},
keywords = {discrete Abelian group; characterization of probability distributions; idempotent distribution; torsion-free groups},
language = {eng},
number = {1},
pages = {67-79},
title = {On the Heyde theorem for discrete Abelian groups},
url = {http://eudml.org/doc/284985},
volume = {177},
year = {2006},
}

TY - JOUR
AU - G. M. Feldman
TI - On the Heyde theorem for discrete Abelian groups
JO - Studia Mathematica
PY - 2006
VL - 177
IS - 1
SP - 67
EP - 79
AB - 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.
LA - eng
KW - discrete Abelian group; characterization of probability distributions; idempotent distribution; torsion-free groups
UR - http://eudml.org/doc/284985
ER -