# The Heyde theorem on a-adic solenoids

Colloquium Mathematicae (2013)

• Volume: 132, Issue: 2, page 195-210
• ISSN: 0010-1354

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## Abstract

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We prove the following analogue of the Heyde theorem for a-adic solenoids. Let ξ₁, ξ₂ be independent random variables with values in an a-adic solenoid ${\Sigma }_{a}$ and with distributions μ₁, μ₂. Let ${\alpha }_{j},{\beta }_{j}$ be topological automorphisms of ${\Sigma }_{a}$ such that $\beta ₁{\alpha }^{-1}₁±\beta ₂{\alpha }^{-1}₂$ are topological automorphisms of ${\Sigma }_{a}$ too. Assuming that the conditional distribution of the linear form L₂ = β₁ξ₁ + β₂ξ₂ given L₁ = α₁ξ₁ + α₂ξ₂ is symmetric, we describe the possible distributions μ₁, μ₂.

## How to cite

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Margaryta Myronyuk. "The Heyde theorem on a-adic solenoids." Colloquium Mathematicae 132.2 (2013): 195-210. <http://eudml.org/doc/284269>.

@article{MargarytaMyronyuk2013,
abstract = {We prove the following analogue of the Heyde theorem for a-adic solenoids. Let ξ₁, ξ₂ be independent random variables with values in an a-adic solenoid $Σ_\{a\}$ and with distributions μ₁, μ₂. Let $α_j, β_j$ be topological automorphisms of $Σ_\{a\}$ such that $β₁α^\{-1\}₁ ± β₂α^\{-1\}₂$ are topological automorphisms of $Σ_\{a\}$ too. Assuming that the conditional distribution of the linear form L₂ = β₁ξ₁ + β₂ξ₂ given L₁ = α₁ξ₁ + α₂ξ₂ is symmetric, we describe the possible distributions μ₁, μ₂.},
author = {Margaryta Myronyuk},
journal = {Colloquium Mathematicae},
keywords = {Gaussian distribution; idempotent distribution; Heyde theorem; -adic solenoid},
language = {eng},
number = {2},
pages = {195-210},
title = {The Heyde theorem on a-adic solenoids},
url = {http://eudml.org/doc/284269},
volume = {132},
year = {2013},
}

TY - JOUR
AU - Margaryta Myronyuk
TI - The Heyde theorem on a-adic solenoids
JO - Colloquium Mathematicae
PY - 2013
VL - 132
IS - 2
SP - 195
EP - 210
AB - We prove the following analogue of the Heyde theorem for a-adic solenoids. Let ξ₁, ξ₂ be independent random variables with values in an a-adic solenoid $Σ_{a}$ and with distributions μ₁, μ₂. Let $α_j, β_j$ be topological automorphisms of $Σ_{a}$ such that $β₁α^{-1}₁ ± β₂α^{-1}₂$ are topological automorphisms of $Σ_{a}$ too. Assuming that the conditional distribution of the linear form L₂ = β₁ξ₁ + β₂ξ₂ given L₁ = α₁ξ₁ + α₂ξ₂ is symmetric, we describe the possible distributions μ₁, μ₂.
LA - eng
KW - Gaussian distribution; idempotent distribution; Heyde theorem; -adic solenoid
UR - http://eudml.org/doc/284269
ER -

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