Characterization of Probability Distributions on Locally Compact Abelian Groups by the Property of Identical Distribution of Linear Forms with Random Coefficients

TL;DR

The paper characterizes probability distributions on locally compact Abelian groups by requiring two linear forms of independent $X$-valued random variables, one with a Bernoulli coefficient, to be identically distributed. It reduces the problem to solving functional equations on the dual group and derives explicit structure theorems for key group classes, showing that in connected 1-D cases the distribution has the form $\mu=\tau(M)*m_K$ or $\mu=\tau(M)$ (with a hyperbolic secant $M$ on $\mathbb{R}$), while in totally disconnected cases the form is $\mu=m_K$ or $\mu=m_K*E_x$ with $x$ of order $2$. For groups with no nonzero compact subgroups, a symmetric two-point distribution at opposites emerges as the unique solution. The work extends Klebanov’s real-line results to a broad harmonic-analysis framework, providing concrete descriptions and highlighting open questions about general nonconnected settings.

Abstract

Let X be a locally compact Abelian group. We consider linear forms of independent random variables with values in X. In doing so, one of the coefficients of the linear forms is a random variable with a Bernoulli distribution. For some classes of groups we describe possible distributions of random variables provided that the linear forms are identically distributed. The proof of the theorems is reduced to solving some functional equations on the character group of the group X, and to solve functional equations, methods of abstract harmonic analysis are used.

Theorems & Definitions (23)

• Lemma 2.1
• proof
• Lemma 2.2
• Lemma 2.3
• Theorem 2.4
• proof
• Corollary 2.5
• Proposition 2.6
• proof
• Proposition 2.7