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

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.

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 $X$.

关于不含拓扑同构于二维环面的子群的局部紧阿贝尔群的 Heyde 定理

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

本文将实直线高斯分布刻画定理 Heyde 定理推广到不含二维环面子群的局部紧阿贝尔群，利用条件对称性证明分布可分解为高斯分布与支撑在 2 阶元子群上的分布的卷积，并指出二维环面情形不成立。

Characterization of distributions of $Q$-independent random variables on locally compact Abelian groups

Let $X$ be a second countable locally compact Abelian group. We prove some group analogues of the Skitovich--Darmois, Heyde and Kac--Bernstein characterisation theorems for $Q$-independent random variables taking values in the group $X$. The proofs of these theorems are reduced to solving some functional equations on the character group of the group $X$.

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 $Y$.

关于紧阿贝尔群的紧开子群上 Haar 分布平移的刻画

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

本文通过线性形式的条件分布对称性，刻画了紧阿贝尔群紧开子群上 Haar 分布的平移，将其作为 Heyde 定理在紧阿贝尔群上的类比。

包含二阶元素的某些局部紧Abelian群上概率分布的刻画

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

本文将著名的Heyde定理推广至包含二阶元素的局部紧Abelian群$\mathbb{R}\times F$上，利用独立随机变量线性形式的条件分布对称性刻画了一类新的概率分布。

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 Ab…