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Heyde characterization theorem for some classes of locally compact Abelian groups

Gennadiy Feldman

TL;DR

The paper extends Heyde-type characterization theorems from real-valued variables to classes of locally compact Abelian groups by translating conditional symmetry into functional equations on the dual group. It reduces the problem to canonical forms $L_1 = \xi_1 + \xi_2$ and $L_2 = \xi_1 + \alpha\xi_2$ and analyzes the resulting equations using abstract harmonic analysis and finite-difference techniques, identifying the role of the kernel $K = \mathrm{Ker}(I+\alpha)$. It proves a comprehensive decomposition result for totally disconnected LCAGs without elements of order $2$, showing that the distributions are shifts of a common kernel-supported measure or a Gaussian component on an invariant subspace; it extends these ideas to $X = \mathbb{R}^n \times G$ with $G$ totally disconnected and compact elements, and clarifies the structure in groups containing an element of order $2$. The work also outlines open problems, guiding future research on broader classes of LCAGs and the limits of Gaussian-plus-kernel decompositions in Heyde-type settings.

Abstract

Let $L_1$ and $L_2$ be linear forms of real-valued independent random variables. By Heyde's theorem, if the conditional distribution of $L_2$ given $L_1$ 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 $\mathbb{R}^n\times G$, 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.

Heyde characterization theorem for some classes of locally compact Abelian groups

TL;DR

The paper extends Heyde-type characterization theorems from real-valued variables to classes of locally compact Abelian groups by translating conditional symmetry into functional equations on the dual group. It reduces the problem to canonical forms and and analyzes the resulting equations using abstract harmonic analysis and finite-difference techniques, identifying the role of the kernel . It proves a comprehensive decomposition result for totally disconnected LCAGs without elements of order , showing that the distributions are shifts of a common kernel-supported measure or a Gaussian component on an invariant subspace; it extends these ideas to with totally disconnected and compact elements, and clarifies the structure in groups containing an element of order . The work also outlines open problems, guiding future research on broader classes of LCAGs and the limits of Gaussian-plus-kernel decompositions in Heyde-type settings.

Abstract

Let and be linear forms of real-valued independent random variables. By Heyde's theorem, if the conditional distribution of given 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 . The article continues these studies. We consider the case, where is either a totally disconnected group or is of the form , where 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.
Paper Structure (5 sections, 15 theorems, 51 equations)

This paper contains 5 sections, 15 theorems, 51 equations.

Table of Contents

  1. Introduction and basic notation
  2. Heyde theorem for totally disconnected locally compact Abelian groups
  3. Heyde theorem for the groups $\mathbb{R}^n\times G$, where $G$ is a totally disconnected locally compact Abelian group consisting of compact elements
  4. Heyde theorem for some locally compact Abelian groups containing an element of order 2
  5. Some problems

Key Result

Theorem 2.1

Let $X$ be a totally disconnected locally compact Abelian group containing no elements of order $2$ and let $\alpha$ be a topological automorphism of the group $X$. Let $\xi_1$ and $\xi_2$ be independent random variables with values in $X$ and distributions $\mu_1$ and $\mu_2$. Assume that the chara

Theorems & Definitions (17)

  • Theorem 2.1
  • Lemma 2.2: M2013
  • Lemma 2.3: Febooknew
  • Lemma 2.4: Hewitt-Ross
  • Lemma 2.5: see, e.g., Febooknew
  • Lemma 2.6: H1, see also Febooknew
  • Lemma 2.7: Febooknew
  • Lemma 2.8: Febooknew
  • Proposition 2.9
  • Theorem 3.1
  • ...and 7 more