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G. M. Feldman

Publications and source records attributed to G. M. Feldman.

12 recordsLinked to original sources
arXiv2023

On a characterization theorem in the space $\mathbb{R}^n$

By Heyde's theorem, the class of Gaussian distributions on the real line is characterized by the symmetry of the conditional distribution of one linear form of independent random variables given another. We prove an analogue of this theorem for two independent random vectors taking values in the space $\mathbb{R}^n$. The obtained class of distributions consists of convolutions of Gaussian distributions and a distribution supported in a subspace, which is determined by coefficients of the linear forms.

math.PR↗
arXiv2021

Generalized Polya's theorem on connected locally compact Abelian groups of dimension 1

According to the generalized Polya theorem, the Gaussian distribution on the real line is characterized by the property of equidistribution of a monomial and a linear form of independent identically distributed random variables. We give a complete description of a-adic solenoids for which an analog of this theorem is true. The proof of the main theorem is reduced to solving some functional equation in the class of continuous positive definite functions on the character group of an a-adic solenoid

math.PR↗
arXiv2020

On Heyde's theorem for locally compact Abelian groups containing elements of order 2

According to the well-known Heyde theorem the class of Gaussian distributions on the real line is characterized by the symmetry of the conditional distribution of one linear form of independent random variables given the other. We study analogues of this theorem for some locally compact Abelian groups X containing an element of order 2. We prove that if X contains an element of order 2, this can lead to the fact that a wide class of non-Gaussian distributions on X is characterized by the symmetry of the conditional distribution of one linear form given the other. In so doing coefficients of linear forms are topological automorphisms of X.

math.PR↗
arXiv2020

The Skitovich--Darmois and Heyde theorems for complex and quaternion random variables

We prove the following analogue of the classical Skitovich--Darmois theorem for complex random variables. Let $α=a+ib$ be a nonzero complex number. Then the following statements hold. $1$. Let either $b\ne 0$, or $b=0$ and $a>0$. Let $ξ_1$ and $ξ_2$ be independent complex random variables. Assume that the linear forms $L_1=ξ_1+ξ_2$ and $L_2=ξ_1+αξ_2$ are independent. Then $ξ_j$ are degenerate random variables. $2$. Let $b=0$ and $a<0$. Then there exist complex Gaussian random variables in the wide sense $ξ_1$ and $ξ_2$ such that they are not complex Gaussian random variables in the narrow sense, whereas the linear forms $L_1=ξ_1+ξ_2$ and $L_2=ξ_1+αξ_2$ are independent. We also study an analogue of the Heyde theorem for complex random variables.

math.PR↗
arXiv2019

On a characterisation theorem for $a$-adic solenoids

According to the 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 prove an analogue of this theorem for linear forms of two independent random variables taking values in an $a$-adic solenoid $Σ_a$ without elements of order 2, assuming that the characteristic functions of the random variables do not vanish, and coefficients of the linear forms are topological automorphisms of $Σ_a$.

math.PR↗
arXiv2018

On analogues of C.R.Rao's theorems for locally compact Abelian groups

Let $ξ_1$, $ξ_2$, $ξ_3$ be independent random variables with nonvanishing characteristic functions, and $a_j$, $b_j$ be real numbers such that $a_i/b_i\ne a_j/b_j$ for $i\ne j$. Let $L_1=a_1ξ_1+a_2ξ_2+a_3ξ_3$, $L_2=b_1ξ_1+b_2ξ_2+b_3ξ_3$. By C.R.Rao's theorem the distribution of the random vector $(L_1, L_2)$ determines the distributions of the random variables $ξ_j$ up to a change of location. We prove an analogue of this theorem for independent random variables with values in a locally compact Abelian group. We also prove an analogue for independent random variables with values in an $a$-adic solenoid of similar C.R.Rao's theorem. In so doing coefficients of linear forms are continuous endomorphisms of the group.

math.PR↗
arXiv2018

On a characterisation theorem for probability distributions on discrete Abelian groups

Let $X$ be a countable discrete Abelian group containing no elements of order 2, $α$ be an automorphism of $X$, $ξ_1$ and $ξ_2$ be independent random variables with values in the group $X$ and distributions $μ_1$ and $μ_2$. The main result of the article is the following statement. The symmetry of the conditional distribution of the linear form $L_2 = ξ_1 + αξ_2$ given $L_1 = ξ_1 + ξ_2$ implies that $μ_j$ are shifts of the Haar distribution of a finite subgroup of $X$ if and only if the automorphism $α$ satisfies the condition ${\rm Ker}(I+α)=\{0\}$. This theorem is an analogue for discrete Abelian groups the well-known Heyde theorem where 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 also prove some generalisations of this theorem.

math.GR↗
arXiv2017

The Heyde characterization theorem on some locally compact Abelian groups

By the Heyde theorem, the Gaussian distribution on the real line is characterized by the symmetry of the conditional distribution of one linear form of of $n$ independent random variables given another. When $n=2$ we prove analogues of this theorem in the case when independent random variables take values in a locally compact Abelian group $X$ and coefficients of the linear forms are topological automorphisms of $X$.

math.PR↗
arXiv2015

On a characterization of idempotent distributions on discrete fields and on the field of p-adic numbers

We prove the following theorem. Let $X$ be a discrete field, $ξ$ and $η$ be independent identically distributed random variables with values in $X$ and distribution $μ$. The random variables $S=ξ+η$ and $D=(ξ-η)^2$ are independent if and only if $μ$ is an idempotent distribution. A similar result is also proved in the case when $ξ$ and $η$ are independent identically distributed random variables with values in the field of $p$-adic numbers $\mathbf{Q}_p$, where $p>2$, assuming that the distribution $μ$ has a continuous density.

math.PR↗
arXiv2015

Independent random variables on Abelian groups with independent the sum and difference

Let X be a second countable locally compact Abelian group. Let $ξ_1, ξ_2$ be independent random variables with values in the group X and distributions $μ_1, μ_2$ such that the sum $ξ_1+ξ_2$ and the difference $ξ_1-ξ_2$ are independent. Assuming that the connected component of zero of the group $X$ contains a finite number elements of order 2 we describe the possible distributions $μ_k$.

math.PR↗
arXiv2013

Independent linear statistics on the cylinders

Let either $X=\mathbf{R}\times\mathbf{T}$ or $X=Σ_\text{\boldmath $a$}\times\mathbf{T}$, where $\mathbf{R}$ is the additive group of real number, $\mathbf{T}$ is the cycle group and $Σ_\text{\boldmath $a$}$ is an $\text{\boldmath $a$}$-adic solenoid . Let $α_{ij}$, where $i, j=1,2,3,$ be topological automorphisms of the group $X$. We prove the following analogue of the well-known Skitovich--Darmois theorem for the group $X$. Let $ξ_j$, where $j=1, 2, 3$, be independent random variables with values in the group $X$ and distributions $μ_j$ such that their characteristic functions do not vanish. If the linear statistics $L_1=α_{11}ξ_1+α_{12}ξ_2+α_{13}ξ_3$, $L_2=α_{21}ξ_1+α_{22}ξ_2+α_{23}ξ_3$, and $L_3=α_{31}ξ_1+α_{32}ξ_2+α_{33}ξ_3$ are independent, then all $μ_j$ are Gaussian distributions.

math.PR↗