相关论文: On the Li--Zheng theorem

The Lee-Yang property of certain moment generating functions having only pure imaginary zeros is valid for Ising type models with one-component spins and XY models with two-component spins. Villain models and complex Gaussian multiplicative…

We investigate time-dependent properties of a single particle model in which a random walker moves on a triangle and is subjected to non-local boundary conditions. This model exhibits spontaneous breaking of a Z_2 symmetry. The reduced size…

The aim of the dissertation is threefold: the first two parts are devoted to explore the Fourth Moment Theorem and universality properties for homogeneous sums, while the last part approaches the classical theory of orthogonal polynomials…

Let $X$ be a centered random variable with unit variance, zero third moment, and such that $E[X^4] \ge 3$. Let $\{F_n : n\geq 1\}$ denote a normalized sequence of homogeneous sums of fixed degree $d\geq 2$, built from independent copies of…

Contextual situations are those in which seemingly "the same" random variable changes its identity depending on the conditions under which it is recorded. Such a change of identity is observed whenever the assumption that the variable is…

Under certain conditions, a symmetric unimodal continuous random variable $\xi$ can be represented as a scale mixture of the standard Normal distribution $Z$, i.e., $\xi = \sqrt{W} Z$, where the mixing distribution $W$ is independent of…

We propose results of the investigation of properties of the random sums of random variables. We consider the case, where the number of summands is the first moment of an event occurrence. An integral equation is presented that determines…

Let $X$ and $Y$ be two independent random walks on $\Z^2$ with zero mean and finite variances, and let $L_t(X,Y)$ be the local time of $X-Y$ at the origin at time $t$. We show that almost surely with respect to $Y$, $L_t(X,Y)/\log t$…

We consider the probability theory, and in particular the moment problem and universality theorems, for random groups of the sort of that arise or are conjectured to arise in number theory, and in related situations in topology and…

We consider the invariance principle without the classical condition of asymptotic negligibility of individual terms. More precisely, we explore the difference of the following two distributions in the space C (of continuous functions on…

It is known that if X is uniformly distributed modulo 1 and Y is an arbitrary random variable independent of X then Y+X is also uniformly distributed modulo 1. We prove a converse for any continuous random variable Y (or a reasonable…

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…

We consider the distribution of spacings between consecutive elements in subsets of Z/qZ where q is highly composite and the subsets are defined via the Chinese remainder theorem. We give a sufficient criterion for the spacing distribution…

This note concerns a well-known result which we term the ``spread lemma,'' which establishes the existence (with high probability) of a desired structure in a random set. The spread lemma was central to two recent celebrated results: (a)…

A rearrangement of $n$ independent uniform $[0,1]$ random variables is a sequence of $n$ random variables $Y_1,...,Y_n$ whose vector of order statistics has the same distribution as that for the $n$ uniforms. We consider rearrangements…

We propose new statistical tests, in high-dimensional settings, for testing the independence of two random vectors and their conditional independence given a third random vector. The key idea is simple, i.e., we first transform each…

Lin's condition is used to establish the moment determinacy/indeterminacy of absolutely continuous probability distributions. Recently, a number of papers related to Lin's condition for functions of random variables have emerged. In this…

We study a class of non-reversible, continuous-time random walks in random environments on $\mathbb{Z}^d$ that admit a cycle representation with finite cycle length. The law of the transition rates, taking values in $[0, \infty)$, is…

We introduce two new measures for the dependence of $n \ge 2$ random variables: distance multivariance and total distance multivariance. Both measures are based on the weighted $L^2$-distance of quantities related to the characteristic…

Assuming the Riemann hypothesis (RH) and the linear independence conjecture (LI), we show that the weighted count of primes in multiple short intervals follows a multivariate Gaussian distribution with weak negative correlations. As an…