Related papers: Limit theorems for mixed-norm sequence spaces with…
We introduce the mixed-norm amalgam spaces $(L^{\vec{p}},L^{\vec{s}})(\mathbb{R}^n)$ and $(L^{\vec{p}},L^{\vec{s}})^{\alpha}(\mathbb{R}^n)$, and show their some basic properties. In addition, we find the predual…
The mixed area of a Reuleaux polygon and its symmetric with respect to the origin is expressed in terms of the mixed area of two explicit polygons. This gives a geometric explanation of a classical proof due to Chakerian. Mixed areas and…
The classic central limit theorem and $\alpha$-stable distributions play a key role in probability theory, and also in Boltzmann-Gibbs (BG) statistical mechanics. They both concern the paradigmatic case of probabilistic independence of the…
For normalized sums $Z_n$ of i.i.d. random variables, we explore necessary and sufficient conditions which guarantee the normal approximation with respect to the R\'enyi divergence of infinite order. In terms of densities $p_n$ of $Z_n$,…
We investigate the problem of testing the equivalence between two discrete histograms. A {\em $k$-histogram} over $[n]$ is a probability distribution that is piecewise constant over some set of $k$ intervals over $[n]$. Histograms have been…
The Minkowski mixed volume of $n$ subpolytopes $D_1, \dots, D_n$ of a polytope $P \subset {\mathbb R}^n$ clearly does not exceed the normalized volume $n! \text{Vol}(P)$. Equality holds if and only if the subpolytopes are interlaced, i.e.,…
If $x_1,\dots,x_m$ are finitely many points in $\mathbb{R}^d$, let $E_\epsilon=\cup_{i=1}^m\,x_i+Q_\epsilon$, where $Q_\epsilon=\{x\in \mathbb{R}^d,\,\,|x_i|\le \epsilon/2, \, i=1,...,d\}$ and let $\hat f$ denote the Fourier transform of…
The Generalized Central Limit Theorem is a remarkable generalization of the Central Limit Theorem, showing that the sum of a large number of independent, identically-distributed (i.i.d) random variables with infinite variance may converge…
We use bounds of mixed character sum to study the distribution of solutions to certain polynomial systems of congruences modulo a prime $p$. In particular, we obtain nontrivial results about the number of solution in boxes with the side…
In this paper we present a surprisingly general extension of the main result of a paper that appeared in this journal: I. Montes et al., Sklar's theorem in an imprecise setting, Fuzzy Sets and Systems, 278 (2015), 48--66. The main tools we…
We prove a general transfer theorem for multivariate random sequences with independent random indexes in the double array limit setting. We also prove its partial inverse providing necessary and sufficient conditions for the convergence of…
Consider the set of all sequences of $n$ outcomes, each taking one of $m$ values, that satisfy a number of linear constraints. If $m$ is fixed while $n$ increases, most sequences that satisfy the constraints result in frequency vectors…
The classic problems of testing uniformity of and learning a discrete distribution, given access to independent samples from it, are examined under general $\ell_p$ metrics. The intuitions and results often contrast with the classic…
In nature or societies, the power-law is present ubiquitously, and then it is important to investigate the mathematical characteristics of power-laws in the recent era of big data. In this paper we prove the superposition of non-identical…
In this paper we investigate the probability distribution of the sum $Y$ of $\ell$ independent identically distributed random variables taking values in $\mathbb{Z}_p$. Our main focus is the regime of small values of $\ell$, which is less…
We study asymptotics of $q$-distributed random lozenge tilings of sawtooth domains (equivalently, of random interlacing integer arrays with fixed top row). Under the distribution we consider each tiling is weighted proportionally to…
We obtain the analogue of the classical result by Erd\"os and Kac on the limiting distribution of the maximum of partial sums for exchangeable random variables with zero mean and variance one. We show that, if the conditions of the central…
The purpose of this paper is twofold. In one direction, we extend the spectral method for random piecewise expanding and hyperbolic dynamics developed by the first author \textit{et al}. to establish quenched versions of the large deviation…
Let $\mathbb{X}=\{X_{ij}: 1\le i,j\le n\}$ be an $n\times n$ array of independent random variables where $n\ge2$. Let $\pi$ be a uniform random permutation of $\{1,2,\dots,n\}$, independent of $\mathbb{X}$, and let…
We study a $q$-deformed random unitary ensemble associated with the little-$q$ Laguerre weight, which provides a discrete analogue of the classical Laguerre unitary ensemble. In the double scaling regime $q=e^{-\lambda/N}$, where $N$ is the…