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An Analogue of Heyde's Theorem for a Certain Class of Compact Totally Disconnected Abelian Groups and p-quasicyclic Groups

Gennadiy Feldman

TL;DR

This work extends Heyde's characterisation of Gaussian distributions to a broad class of compact totally disconnected Abelian groups, including finite cyclic groups and $p$-adic integers, and to $p$-quasicyclic groups. The main contribution is a Heyde-type theorem asserting that symmetry of the conditional distribution of the linear form $L_2=\xi_1+\alpha\xi_2$ given $L_1=\xi_1+\xi_2$ forces the marginal distributions to be shifts of a single distribution supported on a compact subgroup $G$ with $(I-\alpha)(G)=G$, along with minimality and factorization properties for the corresponding Haar measure. The results are established without restrictions on the automorphisms or the characteristic functions, and are extended to corollaries that recover classical finite-group and $p$-adic cases; a parallel theorem is proved for $p$-quasicyclic groups, with a dichotomy depending on whether $\alpha=-I$. The proofs utilize abstract harmonic analysis, functional equations on character groups, and reductions to finite subgroups to obtain precise structural descriptions of the distributions.

Abstract

According to the well-known Heyde theorem, the Gaussian distribution on the real line is characterized by the symmetry of the conditional distribution of one linear form of independent random variables given another. In the article, we study an analogue of this theorem for two independent random variables taking values either in a compact totally disconnected Abelian group of a certain class, which includes finite cyclic groups and groups of p-adic integers, or in a p-quasicyclic group. In contrast to previous works devoted to group analogues of Heyde's theorem, we do not impose any restrictions on either coefficients of linear forms (they can be arbitrary topological automorphisms of the group) or the characteristic functions of random variables. For the proof we use methods of abstract harmonic analysis.

An Analogue of Heyde's Theorem for a Certain Class of Compact Totally Disconnected Abelian Groups and p-quasicyclic Groups

TL;DR

This work extends Heyde's characterisation of Gaussian distributions to a broad class of compact totally disconnected Abelian groups, including finite cyclic groups and -adic integers, and to -quasicyclic groups. The main contribution is a Heyde-type theorem asserting that symmetry of the conditional distribution of the linear form given forces the marginal distributions to be shifts of a single distribution supported on a compact subgroup with , along with minimality and factorization properties for the corresponding Haar measure. The results are established without restrictions on the automorphisms or the characteristic functions, and are extended to corollaries that recover classical finite-group and -adic cases; a parallel theorem is proved for -quasicyclic groups, with a dichotomy depending on whether . The proofs utilize abstract harmonic analysis, functional equations on character groups, and reductions to finite subgroups to obtain precise structural descriptions of the distributions.

Abstract

According to the well-known Heyde theorem, the Gaussian distribution on the real line is characterized by the symmetry of the conditional distribution of one linear form of independent random variables given another. In the article, we study an analogue of this theorem for two independent random variables taking values either in a compact totally disconnected Abelian group of a certain class, which includes finite cyclic groups and groups of p-adic integers, or in a p-quasicyclic group. In contrast to previous works devoted to group analogues of Heyde's theorem, we do not impose any restrictions on either coefficients of linear forms (they can be arbitrary topological automorphisms of the group) or the characteristic functions of random variables. For the proof we use methods of abstract harmonic analysis.

Paper Structure

This paper contains 3 sections, 15 theorems, 67 equations.

Table of Contents

  1. Introduction
  2. Main theorem
  3. Heyde theorem for $p$-quasicyclic groups

Key Result

Theorem 2.1

Let $X$ be a compact totally disconnected Abelian group of the form where ${\mathcal{P}}$ is a set of pairwise distinct prime numbers such that $2\notin {\mathcal{P}}$ and $X_{p_j}$ is either the cyclic $p_j$-group $\mathbb{Z}(p_j^{k_j})$ or the group of $p_j$-adic integers $\mathbb{Z}_{p_j}$. Let $\alpha$ be a topological automorphism of the group $X$. Let $\xi_1$

Theorems & Definitions (23)

  • Theorem 2.1
  • Lemma 2.2: Febooknew
  • Lemma 2.3
  • Lemma 2.4: Rima, see also Febooknew
  • Lemma 2.5
  • proof
  • Corollary 2.6
  • Corollary 2.7
  • Corollary 2.8
  • proof
  • ...and 13 more