arXiv subjects

Gennadiy Feldman

Publications and source records attributed to Gennadiy Feldman.

18 recordsLinked to original sources

arXiv2026

A Group Analogue of the Ghurye--Olkin--Ibragimov Theorem

According to the classical Skitovich--Darmois theorem, the Gaussian distribution on the real line is characterized by the independence of two linear forms of a finite number of independent random variables $ξ_j$. This theorem has been extended in various directions. In particular, S.G. Ghurye and I. Olkin established an analogous result for the case where $ξ_j$ are $n$-dimensional independent random vectors and the coefficients of the linear forms are invertible $n\times n$ matrices. Subsequently, A.A. Zinger and later I.A. Ibragimov investigated linear forms of an infinite sequence of $n$-dimensional independent random vectors. In the present paper, we investigate, for the first time, linear forms of a sequence of independent random variables taking values in a second-countable locally compact Abelian group $X$ under either of the following conditions: $X$ contains no nontrivial compact subgroups, or $X$ contains no subgroups topologically isomorphic to the circle group and has a finite topological automorphism group. Furthermore, we investigate the case where the independent random variables take values in an $\mathbf{a}$-adic solenoid $Σ_{\mathbf{a}}$. In all these settings, the coefficients of the linear forms are topological automorphisms of the corresponding group.

math.PR↗

arXiv2026

An Analogue of Heyde's Theorem for Discrete Torsion Abelian Groups with Cyclic $p$-Components

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 $n$ independent random variables given another. In the article, we prove an analogue of this theorem for two independent random variables taking values in a discrete torsion Abelian group $X$ with cyclic $p$-components. In doing so, we do not impose any restrictions on coefficients of the linear forms and the characteristic functions of random variables. The proof uses methods of abstract harmonic analysis and is based on the solution some functional equation on the character group of the group $X$.

math.PR↗

arXiv2025

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

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.

math.PR↗

arXiv2025

Heyde characterization theorem for some classes of locally compact Abelian groups

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.

math.PR↗

arXiv2025

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

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.

math.PR↗

arXiv2024

Characterization of probability distributions on some locally compact Abelian groups containing an element of order 2

The well-known Heyde theorem characterizes the Gaussian distributions on the real line by the symmetry of the conditional distribution of one linear form of independent random variables given another. We generalize this theorem to groups of the form $\mathbb{R}\times F$, where $F$ is a finite Abelian group such that its 2-component is isomorphic to the additive group of the integers modulo $2$. In so doing, coefficients of the linear forms are arbitrary topological automorphisms of the group. Previously, a similar result was proved in the case when the group $F$ contains no elements of order 2. The presence of an element of order 2 in $F$ leads to the fact that a new class of probability distributions is characterized

math.PR↗

arXiv2024

Heyde theorem for locally compact Abelian groups containing no subgroups topologically isomorphic to the 2-dimensional torus

We prove the following group analogue of the well-known Heyde theorem on a characterization of the Gaussian distribution on the real line. Let $X$ be a second countable locally compact Abelian group containing no subgroups topologically isomorphic to the 2-dimensional torus. Let $G$ be the subgroup of $X$ generated by all elements of $X$ of order $2$ and let $α$ be a topological automorphism of the group $X$ such that ${\rm Ker}(I+α)=\{0\}$. Let $ξ_1$ and $ξ_2$ be independent random variables with values in $X$ and distributions $μ_1$ and $μ_2$ with nonvanishing characteristic functions. If the conditional distribution of the linear form $L_2 = ξ_1 + αξ_2$ given $L_1 = ξ_1 +ξ_2$ is symmetric, then $μ_j$ are convolutions of Gaussian distributions on $X$ and distributions supported in $G$. We also prove that this theorem is false if $X$ is the 2-dimensional torus.

math.PR↗

arXiv2024

On the Li--Zheng theorem

By the well-known I.Kotlarski lemma, if $ξ_1$, $ξ_2$, and $ξ_3$ are independent real-valued random variables with nonvanishing characteristic functions, $L_1=ξ_1-ξ_3$ and $L_2=ξ_2-ξ_3$, then the distribution of the random vector $(L_1, L_2)$ determines the distributions of the random variables $ξ_j$ up to shift. Siran Li and Xunjie Zheng generalized this result for the linear forms $L_1=ξ_1+a_2ξ_2+a_3ξ_3$ and $L_2=b_2ξ_2+b_3ξ_3+ξ_4$ assuming that all $ξ_j$ have first and second moments, $ξ_2$ and $ξ_3$ are identically distributed, and $a_j$, $b_j$ satisfy some conditions. In the article, we give a simpler proof of this theorem. In doing so, we also prove that the condition of existence of moments can be omitted. Moreover, we prove an analogue of the Li--Zheng theorem for independent random variables with values in the field of $p$-adic numbers, in the field of integers modulo $p$, where $p\ne 2$, and in the discrete field of rational numbers.

math.PR↗

arXiv2023

Arithmetic of a certain semigroup of probability distributions on the group $\mathbb{R}\times \mathbb{Z}(2)$

We consider a certain convolution semigroup $Θ$ of probability distributions on the group $\mathbb{R}\times \mathbb{Z}(2)$, where $\mathbb{R}$ is the group of real numbers and $\mathbb{Z}(2)$ is the additive group of the integers modulo 2. This semigroup appeared in connection with the study of a characterization problem of mathematical statistics on $a$-adic solenoids containing an element of order 2. We answer the questions that arise in the study of arithmetic of the semigroup $Θ$. Namely, we describe the class of infinitely divisible distributions, the class of indecomposable distributions, and the class of distributions which have no indecomposable factors.

math.PR↗

arXiv2023

Heyde theorem on locally compact Abelian groups with the connected component of zero of dimension 1

Let $X$ be a locally compact Abelian group with the connected component of zero of dimension 1. Let $ξ_1$ and $ξ_2$ be independent random variables with values in $X$ with nonvanishing characteristic functions. We prove that if a topological automorphism $α$ of the group $X$ satisfies the condition ${{\rm Ker}(I+α)=\{0\}}$ and the conditional distribution of the linear form ${L_2 = ξ_1 + αξ_2}$ given ${L_1 = ξ_1 + ξ_2}$ is symmetric, then the distributions of $ξ_j$ are convolutions of Gaussian distributions on $X$ and distributions supported in the subgroup $\{x\in X:2x=0\}$. This result can be viewed as a generalization of the well-known Heyde theorem on the characterization of the Gaussian distribution on the real line.

math.PR↗

arXiv2023

On a characterization of shifts of Haar distributions on compact open subgroups of a compact Abelian group

Let X be a compact Abelian group. In the article we obtain a characterization of shifts of Haar distributions on compact open subgroups of the group X by the symmetry of the conditional distribution of one linear form of independent random variables taking values in X given another. Coefficients of the linear forms are topological automorphisms of the group X. This result can be viewed as an analogue for compact Abelian groups of the well-known Heyde theorem, where 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.

math.PR↗

arXiv2022

Generalization of the Heyde theorem to finite Abelian groups and groups of the form RxG, where G is a finite Abelian group

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. We study analogues of this theorem for some locally compact Abelian groups. We consider linear forms of two independent random variables with values in a locally compact Abelian group X. We assume that the characteristic functions of these independent random variables do not vanish. Unlike most previous works, we do not impose any restrictions on coefficients of the linear forms. They are arbitrary topological automorphisms of X.

math.PR↗

arXiv2021

The Heyde characterization theorem on compact totally disconnected and connected Abelian groups

By 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 case of two independent random variables we give a complete description of compact totally disconnected Abelian groups X, where an analogue of this theorem is valid. We also prove that even a weak analogue of the Heyde theorem fails on compact connected Abelian groups X. Coefficients of considered linear forms are topological automorphisms of X. The proofs are based on the study of solutions of a functional equation on the character group of the group X in the class of Fourier transforms of probability distributions.

math.PR↗

arXiv2019

On a generalisation of the Skitovich--Darmois theorem for several linear forms on Abelian groups

A.M. Kagan introduced a class of distributions $\mathcal{D}_{m, k}$ in $\mathbb{R}^m$ and proved that if the joint distribution of $m$ linear forms of $n$ independent random variables belongs to the class $\mathcal{D}_{m, m-1}$, then the random variables are Gaussian. A.M. Kagan's theorem implies, in particular, the well-known Skitovich--Darmois theorem, where the Gaussian distribution on the real line is characterized by independence of two linear forms of $n$ independent random variables. In the note we describe a wide class of locally compact Abelian groups where A.M. Kagan's theorem is valid.

math.PR↗

arXiv2018

On the Skitovich-Darmois theorem for some locally compact Abelian groups

Let $X$ be a locally compact Abelian group, $α_{j}, β_j$ be topological automorphisms of $X$. Let $ξ_1, ξ_2$ be independent random variables with values in $X$ and distributions $μ_j$ with non-vanishing characteristic functions. It is known that if $X$ contains no subgroup topologically isomorphic to the circle group $\mathbb{T}$, then the independence of the linear forms $L_1=α_1ξ_1+α_2ξ_2$ and $L_2=β_1ξ_1+β_2ξ_2$ implies that $μ_j$ are Gaussian distributions. We prove that if $X$ contains no subgroup topologically isomorphic to $\mathbb{T}^2$, then the independence of $L_1$ and $L_2$ implies that $μ_j$ are either Gaussian distributions or convolutions of Gaussian distributions and signed measures supported in a subgroup of $X$ generated by an element of order 2. The proof is based on solving the Skitovich-Darmois functional equation on some locally compact Abelian groups.

math.GR↗

arXiv2017

Characterization theorems for $Q$-independent random variables with values in a locally compact Abelian group

Let $X$ be a locally compact Abelian group, $Y$ be its character group. Following A. Kagan and G. Székely we introduce a notion of $Q$-independence for random variables with values in $X$. We prove group analogues of the Cramér, Kac-Bernstein, Skitovich-Darmois and Heyde theorems for $Q$-independent random variables with values in $X$. The proofs of these theorems are reduced to solving some functional equations on the group $Y$.

math.PR↗

arXiv2014

On a characterization theorem for the group of p-adic numbers

It is well known Heyde's characterization of the Gaussian distribution on the real line: Let $ξ_1, ξ_2,\dots, ξ_n$, $n\ge 2,$ be independent random variables, let $α_j, β_j$ be nonzero constants such that $β_iα_i^{-1} + β_jα_j^{-1} \ne 0$ for all $i \ne j$. If the conditional distribution of the linear form $L_2 = β_1ξ_1 + β_2ξ_2+ \cdots + β_nξ_n$ given $L_1 = α_1ξ_1 + α_2ξ_2+\cdots + α_nξ_n$ is symmetric, then all random variables $ξ_j$ are Gaussian. We prove an analogue of this theorem for two independent random variables in the case when they take values in the group of $p$-adic numbers $Ω_p$, and coefficients of linear forms are topological automorphisms of $Ω_p$.

math.PR↗

arXiv2013

On the Skitovich--Darmois theorem for the group of p-adic numbers

Let $Ω_p$ be the group of $p$-adic numbers, $ ξ_1$ and $ξ_2$ be independent random variables with values in $Ω_p$ and distributions $μ_1$ and $μ_2$. Let $α_j, β_j$ be topological automorphisms of $Ω_p$. Assuming that the linear forms $L_1=α_1ξ_1 + α_2ξ_2$ and $L_2=β_1ξ_1 + β_2ξ_2$ are independent, we describe possible distributions $μ_1$ and $μ_2$ depending on the automorphisms $α_j, β_j$. This theorem is an analogue for the group $Ω_p$ of the well-known Skitovich--Darmois theorem, where a Gaussian distribution on the real line is characterized by the independence of two linear forms.

math.PR↗