Related papers: Concentration of the information in data with log-…

In this note, we show that the convolution of a discrete symmetric log-concave distribution and a discrete symmetric bimodal distribution can have any strictly positive number of modes. A similar result is proved for smooth distributions.

We study ergodic properties of compositions of holomorphic endomorphisms of the complex projective space chosen independently at random according to some probability distribution. Along the way, we construct positive closed currents which…

We investigate concentration properties of functions of random vectors with values in the discrete cube, satisfying the stochastic covering property (SCP) or the strong Rayleigh property (SRP). Our result for SCP measures include…

In this paper, we study the asymptotic behavior of a fully-coupled slow-fast McKean-Vlasov stochastic system. Using the non-linear Poisson equation on Wasserstein space, we first establish the strong convergence in the averaging principle…

A connection between fractional calculus and statistical distribution theory has been established by the authors recently. Some extensions of the results to matrix-variate functions were also considered. In the present article, more results…

We establish concentration inequalities in the class of ultra log-concave distributions. In particular, we show that ultra log-concave distributions satisfy Poisson concentration bounds. As an application, we derive concentration bounds for…

Let $X,X_1,...,X_n$ be independent identically distributed random variables. In this paper we study the behavior of the concentration functions of the weighted sums $\sum\limits_{k=1}^{n}a_k X_k$ with respect to the arithmetic structure of…

We provide a sharp rate of convergence in the central limit theorem for random vectors with an unconditional, log-concave density. The argument relies on analysis of the Neumann laplacian on convex domains and on the theory of optimal…

We study the problem of estimating multivariate log-concave probability density functions. We prove the first sample complexity upper bound for learning log-concave densities on $\mathbb{R}^d$, for all $d \geq 1$. Prior to our work, no…

We present an extension of the famous Littlewood-Offord problem when Bernoulli distributions are replaced with discrete log-concave distributions. A variant of the Littlewood-Offord problem for arithmetic progressions, as well as an…

In this paper, we investigate the properties of sequences and series under the action of the log-concave operator \(\mathcal{L}\). We explore the relationship between the convergence of a sequence \((a_k)\) and the convergence of sequences…

We present a class of positive discrete random variables extending the Conway--Maxwell-Poisson distribution. This class emerges in a natural way from an application in queueing theory and contains distributions exhibiting quite different…

In this paper we present the concept of description of random processes in complex systems with the discrete time. It involves the description of kinetics of discrete processes by means of the chain of finite-difference non-Markov equations…

This paper extends the classic theory of convex optimization to the minimization of functions that are equal to the negated logarithm of what we term as a sum-log-concave function, i.e., a sum of log-concave functions. In particular, we…

We propose a new approach to deriving quantitative mean field approximations for any probability measure $P$ on $\mathbb{R}^n$ with density proportional to $e^{f(x)}$, for $f$ strongly concave. We bound the mean field approximation for the…

This paper studies estimation of and inference on a distribution function $F$ that is concave on the nonnegative half line and admits a density function $f$ with potentially unbounded support. When $F$ is strictly concave, we show that the…

We give two different definitions of what it means for a matrix-valued function to be log concave, guided by similar notions in complex differential geometry. After discussing a few simple examples, we proceed to develop some of the basic…

This paper introduces max-characteristic functions (max-CFs), which are an offspring of multivariate extreme-value theory. A max-CF characterizes the distribution of a random vector in R^d , whose components are nonnegative and have finite…

Let $m$ and $k \geq 2$ be positive integers. We show that polynomial $P = (1+x)^m(1+x^k)$ is strongly unimodal (frequently known as {\it log concave\/}) if and only if $m \geq k^2 -3$; this is also the criterion for $P$ to be merely…

We prove stability estimates for the Shannon-Stam inequality (also known as the entropy-power inequality) for log-concave random vectors in terms of entropy and transportation distance. In particular, we give the first stability estimate…