Related papers: Identically distributed random vectors on locally …

Let $\Omega_p$ be the group of $p$-adic numbers, $ \xi_1$, $\xi_2$, $\xi_3$ be independent random variables with values in $\Omega_p$ and distributions $\mu_1$, $\mu_2$, $\mu_3$. Let $\alpha_j, \beta_j, \gamma_j$ be topological…

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…

Let $X$ be a locally compact Abelian group, $Y$ be its character group. Following A. Kagan and G. Sz\'ekely we introduce a notion of $Q$-independence for random variables with values in $X$. We prove group analogues of the Cram\'er,…

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…

Let $X$ be a compact connected Abelian group. It is well-known that then there exist topological automorphisms $\alpha_j, \beta_j $ of $X$ and independent random variables $\xi_1$ and $\xi_2$ with values in $X$ and distributions $\mu_1,…

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…

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…

We prove the following analogue of the classical Skitovich--Darmois theorem for complex random variables. Let $\alpha=a+ib$ be a nonzero complex number. Then the following statements hold. $1$. Let either $b\ne 0$, or $b=0$ and $a>0$. Let…

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…

Let $\Omega_p$ be the group of $p$-adic numbers, $ \xi_1$ and $\xi_2$ be independent random variables with values in $\Omega_p$ and distributions $\mu_1$ and $\mu_2$. Let $\alpha_j, \beta_j$ be topological automorphisms of $\Omega_p$.…

Let either $X=\mathbf{R}\times\mathbf{T}$ or $X=\Sigma_\text{\boldmath $a$}\times\mathbf{T}$, where $\mathbf{R}$ is the additive group of real number, $\mathbf{T}$ is the cycle group and $\Sigma_\text{\boldmath $a$}$ is an $\text{\boldmath…

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 main result of this paper states that for independent random variables $X, Y$ taking values in a compact metrisable abelian group, $X + Y$ has the same distribution as $X$, if and only if there exists a compact subgroup $A$ such that…

Let $X_1,\ldots,X_n$ be independent identically distributed random vectors in $\mathbb{R}^d$. We consider upper bounds on $\max_x \mathbb{P}(a_1X_1+\cdots+a_nX_n=x)$ under various restrictions on $X_i$ and the weights $a_i$. When…

Suppose that A_1,\dots, A_N are independent random matrices whose atoms are iid copies of a random variable \xi of mean zero and variance one. It is known from the works of Newman et. al. in the late 80s that when \xi is gaussian then…

Kagan and Shalaevski 1967 have shown that if the random variables $X_1,\dots,X_n$ are independent and identically distributed and the distribution of $\sum_{i=1}^n(X_i+a_i)^2$ $a_i\in \mathbb{R}$ depends only on $\sum_{i=1}^na_i^2$ , then…

We analyze the distribution of $\sum_{i=1}^m v_i \bx_i$ where $\bx_1,...,\bx_m$ are fixed vectors from some lattice $\cL \subset \R^n$ (say $\Z^n$) and $v_1,...,v_m$ are chosen independently from a discrete Gaussian distribution over $\Z$.…

Motivated by an eigenvalue-eigenfunction problem posed in IR^n x {\Omega}, where {\Omega} is a probability space, we are concerned in this paper with the Sobolev space on groups. Hence it is established an equivalence between locally…

Let $\{X_i,i=1,2,...\}$ be i.i.d. standard gaussian variables. Let $S_n=X_1+...+X_n$ be the sequence of partial sums and $$ L_n=\max_{0\leq i<j\leq n}\frac{S_j-S_i}{\sqrt{j-i}}. $$ We show that the distribution of $L_n$, appropriately…

Let $X,X_1,\ldots,X_n$ be independent identically distributed random variables. The paper deals with the question about the behavior of the concentration function of the random variable $\sum\limits_{k=1}^{n}X_k a_k$ according to the…