相关论文: Relative log-concavity and a pair of triangle ineq…

In information theory, the well-known log-sum inequality is a fundamental tool which indicates the non-negativity for the relative entropy. In this article, we establish a set of inequalities which are similar to the log-sum inequality…

We investigate quantitative implications of the notion of log-concavity through a probabilistic interpretation. In particular, we derive concentration inequalities, moment and entropy bounds for random variables satisfying a precise degree…

We investigate geometric and functional inequalities for the class of log-concave probability sequences. We prove dilation inequalities for log-concave probability measures on the integers. A functional analogue of this geometric inequality…

Consider a function $F(X,Y)$ of pairs of positive matrices with values in the positive matrices such that whenever $X$ and $Y$ commute $F(X,Y)= X^pY^q.$ Our first main result gives conditions on $F$ such that ${\rm Tr}[ X \log (F(Z,Y))]…

We study the approximation of arbitrary distributions $P$ on $d$-dimensional space by distributions with log-concave density. Approximation means minimizing a Kullback--Leibler-type functional. We show that such an approximation exists if…

This paper develops systematic approaches to obtain $f$-divergence inequalities, dealing with pairs of probability measures defined on arbitrary alphabets. Functional domination is one such approach, where special emphasis is placed on…

Log-concave distributions include some important distributions such as normal distribution, exponential distribution and so on. In this note, we show inequalities between two Lp-norms for log-concave distributions on the Euclidean space.…

Divergences are quantities that measure discrepancy between two probability distributions and play an important role in various fields such as statistics and machine learning. Divergences are non-negative and are equal to zero if and only…

We show that $h_\infty(X+Y)\leq h_\infty(Z+W)$, where $X, Y$ are independent log-concave random variables, and $Z, W$ are exponential random variables having the same respective $\infty$-R\'enyi entropies. Analogs for integer-valued…

We give counterexamples and a few positive results related to several conjectures of R. Pemantle and D. Wagner concerning negative correlation and log-concavity properties for probability measures and relations between them. Most of the…

We compare weighted sums of i.i.d. positive random variables according to the usual stochastic order. The main inequalities are derived using majorization techniques under certain log-concavity assumptions. Specifically, let $Y_i$ be i.i.d.…

A triangle $\{a(n,k)\}_{0\le k\le n}$ of nonnegative numbers is LC-positive if for each $r$, the sequence of polynomials $\sum_{k=r}^{n}a(n,k)q^k$ is $q$-log-concave. It is double LC-positive if both triangles $\{a(n,k)\}$ and…

We investigate stochastic comparisons between exponential family distributions and their mixtures with respect to the usual stochastic order, the hazard rate order, the reversed hazard rate order, and the likelihood ratio order. A general…

Let $f,g:\Bbb{R}^{N}\rightarrow (-\infty ,\infty ]$ be Borel measurable, bounded below and such that $\inf f+\inf g\geq 0.$ We prove that with $ m_{f,g}:=(\inf f-\inf g)/2,$ the inequality $||(f-m_{f,g})^{-1}||_{\phi…

A loss function measures the discrepancy between the true values (observations) and their estimated fits, for a given instance of data. A loss function is said to be proper (unbiased, Fisher consistent) if the fits are defined over a unit…

Mixed $f$-divergences, a concept from information theory and statistics, measure the difference between multiple pairs of distributions. We introduce them for log concave functions and establish some of their properties. Among them are…

We present theoretical properties of the log-concave maximum likelihood estimator of a density based on an independent and identically distributed sample in $\mathbb{R}^d$. Our study covers both the case where the true underlying density is…

In this note prove the following Berwald-type inequality, showing that for any integrable log-concave function $f:\mathbb R^n\rightarrow[0,\infty)$ and any concave function $h:L\rightarrow\mathbb [0,\infty)$, where $L$ is the epigraph of…

We discuss three convolution inequalities that are connected to additive combinatorics. Cloninger and the second author showed that for nonnegative $f \in L^1(-1/4, 1/4)$, $$ \max_{-1/2 \leq t \leq 1/2} \int_{\mathbb{R}}{f(t-x) f(x) dx}…

Integer programs defined by two equations with two free integer variables and nonnegative continuous variables have three types of nontrivial facets: split, triangle or quadrilateral inequalities. In this paper, we compare the strength of…