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Margaryta Myronyuk

Publications and source records attributed to Margaryta Myronyuk.

11 recordsLinked to original sources
arXiv2025

The Klebanov theorem for the group $\mathbb{R}\times \mathbb{Z}(2)$

L. Klebanov proved the following theorem. Let $ξ_1, \dots, ξ_n$ be independent random variables. Consider linear forms $L_1=a_1ξ_1+\cdots+a_nξ_n,$ $L_2=b_1ξ_1+\cdots+b_nξ_n,$ $L_3=c_1ξ_1+\cdots+c_nξ_n,$ $L_4=d_1ξ_1+\cdots+d_nξ_n,$ where the coefficients $a_j, b_j, c_j, d_j$ are real numbers. If the random vectors $(L_1,L_2)$ and $(L_3,L_4)$ are identically distributed, then all $ξ_i$ for which $a_id_j-b_ic_j\neq 0$ for all $j=\overline{1,n}$ are Gaussian random variables. The present article is devoted to an analogue of the Klebanov theorem in the case when random variables take values in the group $\mathbb{R}\times \mathbb{Z}(2)$ and the coefficients of the linear forms are topological endomorphisms of this group.

math.PR↗
arXiv2023

Identically distributed random vectors on locally compact Abelian groups

L. Klebanov proved the following theorem. Let $ξ_1, \dots, ξ_n$ be independent random variables. Consider linear forms $L_1=a_1ξ_1+\cdots+a_nξ_n,$ $L_2=b_1ξ_1+\cdots+b_nξ_n,$ $L_3=c_1ξ_1+\cdots+c_nξ_n,$ $L_4=d_1ξ_1+\cdots+d_nξ_n,$ where the coefficients $a_j, b_j, c_j, d_j$ are real numbers. If the random vectors $(L_1,L_2)$ and $(L_3,L_4)$ are identically distributed, then all $ξ_i$ for which $a_id_j-b_ic_j\neq 0$ for all $j=\overline{1,n}$ are Gaussian random variables. The present article is devoted to an analog of the Klebanov theorem in the case when random variables take values in a locally compact Abelian group and the coefficients of the linear forms are integers.

math.PR↗
arXiv2022

The Kagan characterization theorem on Banach spaces

A. Kagan introduced classes of distributions $\mathcal{D}_{m,k}$ in $m$-dimensional space $\mathbb{R}^m$. He proved that if the joint distribution of $m$ linear forms of $n$ independent random variables belong to the class $\mathcal{D}_{m,m-1}$ then the random variables are Gaussian. If $m=2$ then the Kagan theorem implies the well-known Darmois-Skitovich theorem, where the Gaussian distribution is characterized by the independence of two linear forms of $n$ independent random variables. In the paper we describe Banach spaces where the analogue of the Kagan theorem is valid.

math.PR↗
arXiv2020

The Heyde theorem on a group $\mathbb{R}^n\times D$, where $D$ is a discrete Abelian group

Heyde proved that a Gaussian distribution on the real line is characterized by the symmetry of the conditional distribution of one linear statistic given another. The present article is devoted to a group analogue of the Heyde theorem. We describe distributions of independent random variables $ξ_1$, $ξ_2$ with values in a group $X=\mathbb{R}^n\times D$, where $D$ is a discrete Abelian group, which are characterized by the symmetry of the conditional distribution of the linear statistic $L_2 = ξ_1 + δξ_2$ given $L_1 = ξ_1 + ξ_2$, where $δ$ is a topological automorphism of $X$ such that ${Ker}(I+δ)=\{0\}$.

math.FA↗
arXiv2019

On a group analogue of the Heyde theorem

Heyde proved that a Gaussian distribution on a real line is characterized by the symmetry of the conditional distribution of one linear form given another. The present article is devoted to an analog of the Heyde theorem in the case when random variables take values in a locally compact Abelian group and the coefficients of the linear forms are integers.

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

Independent linear forms on the group $Ω_p$

Let $Ω_p$ be the group of $p$-adic numbers, $ ξ_1$, $ξ_2$, $ξ_3$ be independent random variables with values in $Ω_p$ and distributions $μ_1$, $μ_2$, $μ_3$. Let $α_j, β_j, γ_j$ be topological automorphisms of $Ω_p$. We consider linear forms $L_1 = α_1ξ_1 + α_2 ξ_2+α_3 ξ_3$, $L_2=β_1ξ_1 + β_2 ξ_2+ β_3 ξ_3$ and $L_3=γ_1ξ_1 + γ_2 ξ_2+ γ_3 ξ_3$. Assuming that the linear forms $L_1$, $L_2$ and $L_3$ are independent, we describe possible distributions $μ_1$, $μ_2$, $μ_3$. This theorem is an analogue 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.NT↗
arXiv2014

Random walks on discrete Abelian groups

In the present paper we find necessary and sufficient conditions for recurrence of random walks on arbitrary subgroups of the group of rational numbers $\mathbb{Q}$.

math.PR↗