Related papers: Levy stable two-sided distributions: exact and exp…
The functional method to derive the fractional Fokker-Planck equation for probability distribution from the Langevin equation with Levy stable noise is proposed. For the Cauchy stable noise we obtain the exact stationary probability density…
We consider goodness-of-fit tests of symmetric stable distributions based on weighted integrals of the squared distance between the empirical characteristic function of the standardized data and the characteristic function of the standard…
The $\beta$-generalized quasi-geostrophic equation is studied in the range of $\alpha \in (0, 1), \beta \in (1/2, 1), 1/2 < \alpha + \beta < 3/2$. When $\alpha \in (1/2, 1), \beta \in (1/2, 1)$ such that $1 \leq \alpha + \beta < 3/2$, using…
We study the behavior near the origin of $C^2$ positive solutions $u(x)$ and $v(x)$ of the system $0\le -\Delta u \le (\frac{1}{|x|^\alpha}* v)^\lambda$ $0\le -\Delta v \le (\frac{1}{|x|^\beta}* u)^\sigma$ in $B_2(0)\setminus\{0\} \subset…
We outline a relation between the densities for the $\beta$-ensembles with respect to the Jacobi weight $(1-x)^a(1+x)^b$ supported on the interval $(-1,1)$ and the Cauchy weight $(1-\mathrm{i}x)^{\eta}(1+\mathrm{i}x)^{\bar{\eta}}$ by…
Testing mutual independence among multiple random variables is a fundamental problem in statistics, with wide applications in genomics, finance, and neuroscience. In this paper, we propose a new class of tests for high-dimensional mutual…
The absorption of acoustic wave propagation in a broad variety of lossy media is characterized by an empirical power law function of frequency, w^y. It has long been noted that exponent y ranges from 0 to 2 for diverse media. Recently, the…
In this paper, we provide the sufficient and necessary conditions for the symmetry of the following stable L\'evy-type operator $\mathcal{L}$ on $\mathbb{R}$: $$\mathcal{L}=a(x){\Delta^{\alpha/2}}+b(x)\frac{\d}{\d x},$$ where $a,b$ are the…
Chebyshev was the first to observe a bias in the distribution of primes in residue classes. The general phenomenon is that if $a$ is a nonsquare\mod q and $b$ is a square\mod q, then there tend to be more primes congruent to $a\mod q$ than…
In this study, we focus on applying L-statistics to the high-dimensional one-sample location test problem. Intuitively, an L-statistic with $k$ parameters tends to perform optimally when the sparsity level of the alternative hypothesis…
Asymptotics of maximum likelihood estimation for $\alpha$-stable law are analytically investigated with a continuous parameterization. The consistency and asymptotic normality are shown on the interior of the whole parameter space. Although…
We study a one-dimensional stochastic differential equation driven by a stable L\'evy process of order $\alpha$ with drift and diffusion coefficients $b,\sigma$. When $\alpha\in (1,2)$, we investigate pathwise uniqueness for this equation.…
Let $X=\{X_{t},t\in R_{+}\}$ be a symmetric L\'evy process with local time $\{L^{x}_{t} ; (x,t)\in R^{1}\times R^{1}_{+}\}$. When the L\'evy exponent $\psi(\la)$ is regularly varying at infinity with index $1<\beta\leq 2$ and satisfies some…
We classify the asymptotic densities of the $\Delta^0_2$ sets according to their level in the Ershov hierarchy. In particular, it is shown that for $n \geq 2$, a real $r \in [0,1]$ is the density of an $n$-c.e.\ set if and only if it is a…
We study the density of the supremum of a strictly stable L\'evy process. We prove that for almost all values of the index $\alpha$ -- except for a dense set of Lebesgue measure zero -- the asymptotic series which were obtained in A.…
In this paper we consider the existence of weakly c\`adl\`ag versions of a solution to a linear equation in a Hilbert space $H$, driven by a Levy process taking values in a Hilbert space $U$. In particular we are interested in diagonal type…
The periodic homogenization problem of integro-differential equations of the alpha stable L{\'e}vy operators is studied in this paper. Thanking to the symmetry of the L{\'e}vy density, we can use the method of the formal asymptotic…
Many axiomatic definitions of entropy, such as the R\'enyi entropy, of a random variable are closely related to the $\ell_{\alpha}$-norm of its probability distribution. This study considers probability distributions on finite sets, and…
Let $n\geq 3$, $\alpha$, $\beta\in\mathbb{R}$, and let $v$ be a solution $\Delta v+\alpha e^v+\beta x\cdot\nabla e^v=0$ in $\mathbb{R}^n$, which satisfies the conditions $\lim_{R\to\infty}\frac{1}{\log R}\int_{1}^{R}\rho^{1-n}…
A new three-parameter cumulative distribution function defined on $(\alpha,\infty)$, for some $\alpha\geq0$, with asymmetric probability density function and showing exponential decays at its both tails, is introduced. The new distribution…