Heyde characterization theorem for some classes of locally compact Abelian groups

2025/05/19 by Gennadiy Feldman, Feldman, Gennadiy
Mathematics · #advanced mathematical theories #Functional Equations Stability Results #Nonlinear Differential Equations Analysis

paper · pdf · doi:10.48550/arxiv.2505.13221

Abstract

Let L₁ and L₂ be linear forms of real-valued independent random variables. By Heyde's theorem, if the conditional distribution of L₂ given L₁ is symmetric, then the random variables are Gaussian. A number of papers are devoted to generalisation of Heyde's theorem to the case, where independent random variables take values in a locally compact Abelian group X. The article continues these studies. We consider the case, where X is either a totally disconnected group or is of the form ℝⁿ× G, where G is a totally disconnected group consisting of compact elements. The proof is based on the study of solutions of the Heyde functional equation on the character group of the original group. In so doing, we use methods of abstract harmonic analysis.

Citations

• Characterization of Probability Distributions on Locally Compact Abelian Groups by the Property of Identical Distribution of Linear Forms with Random Coefficients
• Characterization of probability distributions on some locally compact Abelian groups containing an element of order 2
• Using Aichinger's equation to characterize polynomial functions
• Characterization of Polynomials as solutions of certain functional equations
• On certain generalizations of the Levi-Civita and Wilson functional equations
• The Heyde Theorem on a-adic Solenoids

Related

• The Heyde characterization theorem on compact totally disconnected and connected Abelian groups
• The Heyde characterization theorem on some locally compact Abelian groups
• Generalization of the Heyde theorem to finite Abelian groups and groups of the form RxG, where G is a finite Abelian group
• An Analogue of Heyde's Theorem for a Certain Class of Compact Totally Disconnected Abelian Groups and p-quasicyclic Groups
• Heyde theorem on locally compact Abelian groups with the connected component of zero of dimension 1