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

零连通分支维数为1的局部紧Abel群上的Heyde定理

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

该研究在零连通分支维数为1的局部紧Abel群上推广了Heyde定理，证明满足特定拓扑自同构条件与线性形式条件分布对称性时，独立随机变量的分布为高斯分布与特定子群支撑分布的卷积。

Probabilistic Trace and Poisson Summation Formulae on Locally Compact Abelian Groups

We investigate convolution semigroups of probability measures with continuous densities on locally compact abelian groups, which have a discrete subgroup such that the factor group is compact. Two interesting examples of the quotient structure are the $d$--dimensional torus, and the adèlic circle. Our main result is to show that the Poisson summation formula for the density can be interpreted as a probabilistic trace formula, linking values of the density on the factor group to the trace of the associated semigroup on $L^{2}$-space. The Gaussian is a very important example. For rotationally in…

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 $X$.

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. T…

若干类局部紧阿贝尔群的Heyde表征定理

Heyde characterization theorem for some classes of locally compact Abelian groups

该研究将实值随机变量的Heyde定理推广至两类局部紧阿贝尔群，通过研究群特征上的Heyde泛函方程，结合抽象调和分析方法完成证明，拓展了群上概率表征理论。

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。

局部紧阿贝尔群子群上平稳核的最小范数延拓与高斯条件化

Minimal-Norm Extensions of Stationary Kernels on Subgroups of Locally Compact Abelian Groups and Gaussian Conditioning

该研究针对局部紧阿贝尔群子群的平稳核，推导最小范数延拓算子的傅里叶表示，确定算子范数等性质，在紧群时延拓为压缩映射并对应高斯条件期望，还通过实例阐释理论。

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…

某些局部紧阿贝尔群上的弱无穷可分测度

Weakly infinitely divisible measures on some locally compact Abelian groups

在环面群、p-adic整数群和p-adic螺线管上，利用值在实数子群（可能无穷多个）乘积中的随机元，给出了任意弱无限可分概率测度的构造.作为我们结果的一个特例，我们得到了p-adic螺线管上Haar测度的一个新的构造.

局部紧阿贝尔群上的极限定理

Limit theorems on locally compact Abelian groups

证明了值在局部紧阿贝尔群中的行独立无穷小随机变量阵列的行和的极限定理。首先，我们给出了盖泽定理的证明，因为它不容易获得，而且它是不完整的。这个定理给出了行和收敛的充分条件，但极限测度不能有非退化幂等因子。然后证明了行和收敛的充要条件，其中极限测度也可以是紧子群上的非退化Haar测度.最后，我们研究了特殊情况：环面群，群的p-adic整数和p-adic螺线管。

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。

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。

紧致阿贝尔群上非平稳随机遍历定理

Ergodic Theorem for nonstationary random walks on compact abelian groups

研究紧致阿贝尔群上非平稳随机游走，在严格非周期性假设下证明了弱-*收敛到Haar测度、遍历定理及大偏差型估计。

局部紧阿贝尔群上同分布的随机向量

Identically distributed random vectors on locally compact Abelian groups

本文将Klebanov关于独立随机变量线性形式同分布蕴含高斯性的经典定理推广到局部紧阿贝尔群上，并研究系数为整数时的对应结论。