相关论文: An unexpected encounter with Cauchy and L\'evy
The Gaussian product inequality is a long-standing conjecture. In this paper, we investigate the three-dimensional inequality $E[X_1^{2}X_2^{2m_2}X_3^{2m_3}]\ge E[X_1^{2}]E[X_2^{2m_2}]E[X_3^{2m_3}]$ for any centered Gaussian random vector…
A famous conjecture of Erd\H os and Straus is that for every integer $n\ge2$, $4/n$ can be represented as $1/x+1/y+1/z$, where $x,y,z$ are positive integers. This conjecture was generalized to $5/n$ by Sierpi\'nski, and then Schinzel…
We apply the method of determinants to study the distribution of the largest singular values of large $ m \times n $ real rectangular random matrices with independent Cauchy entries. We show that statistical properties of the (rescaled by a…
Heavy-tailed distributions, such as the Cauchy distribution, are acknowledged for providing more accurate models for financial returns, as the normal distribution is deemed insufficient for capturing the significant fluctuations observed in…
We consider the probability that a weighted sum of $n$ i.i.d. random variables $X_j$, $j = 1, . . ., n$, with stretched exponential tails is larger than its expectation and determine the rate of its decay, under suitable conditions on the…
We examine the properties of distributions with the density of the form: $% \frac{2A_{n}c^{n-2}\sqrt{c^{2}-x^{2}}}{\pi \prod_{j=1}^{n}(c(1+a_{j}^{2})-2a_{j}x)},$ where $c,a_{1},\ldots ,a_{n}$ are some parameters and $A_{n}$ a suitable…
We consider the defocusing nonlinear wave equation of power-type on $\mathbb{R}^3$. We establish an almost sure global existence result with respect to a suitable randomization of the initial data. In particular, this provides examples of…
In this paper, we consider the Cauchy problem {align*} \{{array}{ll}&i u_t+\Delta u=\lambda_1|u|^{p_1}u+\lambda_2|u|^{p_2}u, \quad t\in\mathbb{R}, \quad x\in\mathbb{R}^N &u(0,x)=\phi(x)\in \Sigma, \quad x\in\mathbb{R}^N, {array}. {align*}…
An infinitely divisible distribution on $\mathbb{R}$ is a probability measure $\mu$ such that the characteristic function $\hat{\mu}$ has a L\'{e}vy-Khintchine representation with characteristic triplet $(a,\gamma, \nu)$, where $\nu$ is a…
Cauchy's formula was originally established for random straight paths crossing a body $B \subset \mathbb{R}^{n}$ and basically relates the average chord length through $B$ to the ratio between the volume and the surface of the body itself.…
In this paper we consider the general fractional equation \sum_{j=1}^m \lambda_j \frac{\partial^{\nu_j}}{\partial t^{\nu_j}} w(x_1,..., x_n ; t) = -c^2 (-\Delta)^\beta w(x_1,..., x_n ; t), for \nu_j \in (0,1], \beta \in (0,1] with initial…
The paper discusses a test for the hypothesis that a random sample comes from the Cauchy distribution. The test statistics is derived from a characterization and is based on the characteristic function. Properties of the test are discussed…
An interesting result of Veech more than 50 years ago is a parity, or mod $2$, version of the Kronecker--Weyl equidistribution theorem concerning the irrational rotation sequence $\{q\alpha\}$, $q=0,1,2,3,\ldots.$ If $\alpha$ is badly…
Consider a graph with a set of vertices and oriented edges connecting pairs of vertices. Each vertex is associated with a random variable and these are assumed to be independent. In this setting, suppose we wish to solve the following…
In this paper, we are concerned with the Cauchy problem for the generalized KdV equation with random data and rough data. Firstly, when $s\in\mathbf{R}$, by using the initial value randomization technique introduced by Shen et al.…
In this paper, a new and convenient $\chi^2$ wald test based on MCMC outputs is proposed for hypothesis testing. The new statistic can be explained as MCMC version of Wald test and has several important advantages that make it very…
The class of $\alpha$-stable distributions received much interest for modelling impulsive phenomena occur in engineering, economics, insurance, and physics. The lack of non-analytical form for probability density function is considered as…
We study the distribution of the maximum likelihood estimate (MLE) in high-dimensional logistic models, extending the recent results from Sur (2019) to the case where the Gaussian covariates may have an arbitrary covariance structure. We…
We study the local existence of strong solutions for the cubic nonlinear wave equation with data in $H^s(M)$, $s<1/2$, where $M$ is a three dimensional compact riemannian manifold. This problem is supercritical and can be shown to be…
Let $\mathbf{F}_q$ be a finite field of $q$ elements. For multiplicative characters $\chi_1,\dots, \chi_m$ of $\mathbf{F}_q^\times$, we let $J(\chi_1,\dots, \chi_m)$ denote the Jacobi sum. Nicholas Katz and Zhiyong Zheng showed that for…