Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2610.05439

Majorization, Entropy and Concentration Inequalities for Discrete $α$-Concave Random Variables

Abdulmajeed Alqasem·Heshan Aravinda·Arnaud Marsiglietti
View original source ↗Read source document ↗Export metadata ↓

Abstract

We establish a maximum-variance inequality for discrete $α$-concave random variables in the regime $-1/3<α<0$, extending corresponding results beyond the discrete log-concave setting. As applications, we obtain a reverse entropy power inequality and bounds for the Lévy concentration function. We further establish polynomial-type concentration inequalities, reflecting the heavy-tailed nature of the class. A main tool in our approach is a convex majorization principle that reduces the relevant extremal problems to suitable $α$-affine distributions.

Explore related subjects

math.PR

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Abdulmajeed Alqasem, Heshan Aravinda, Arnaud Marsiglietti. 2026-10-04. Majorization, Entropy and Concentration Inequalities for Discrete $α$-Concave Random Variables. https://arxiv.org/abs/2610.05439

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Highest Trees of Random Mappings

We study the heights of the trees of a random mapping of $n$ elements. Using singularity analysis of exact generating functions, we prove that the mapping has a unique highest tree with probability $1 - \sqrt{\fracπ{8}}\, n^{-1/2} + o(n^{-1/2})$. The property of having a unique highest tree plays a crucial role in the solution of the Road Coloring Problem [Trahtman, 2009]. More generally, we consider $c$-branches, the subtrees rooted at distance $c$ from the cycles. For all fixed $c \ge 0$ and $j \ge 1$ we show that the highest $c$-branch exceeds every other $c$-branch in height by at least $j$, except with probability asymptotic to $(2j-1) \sqrt{\fracπ{8}}\, n^{-1/2}$. We also show that, for any fixed $α> 1$, with probability $1 - Θ(n^{-1/2})$ the part of the highest $c$-branch that lies above the second highest one (its crown) has more than $α$ times as many vertices as roots. The last result is used in the author's proof that a random $2$-letter automaton with $n$ states is synchronizing with probability $1 - Θ(1/n)$.

math.PR↗
arXiv2026

Rapid mixing in unimodal landscapes and efficient simulatedannealing for multimodal distributions

We consider nearest neighbor weighted random walks on the $d$-dimensional box $[n]^d$ that are governed by some function $g:[0,1] \ra [0,\iy)$, by which we mean that standing at $x$, a neighbor $y$ of $x$ is picked at random and the walk then moves there with probability $(1/2)g(n^{-1}y)/(g(n^{-1}y)+g(n^{-1}x))$. We do this for $g$ of the form $f^{m_n}$ for some function $f$ which assumed to be analytically well-behaved and where $m_n \ra \iy$ as $n \ra \iy$. This class of walks covers an abundance of interesting special cases, e.g., the mean-field Potts model, posterior collapsed Gibbs sampling for Latent Dirichlet allocation and certain Bayesian posteriors for models in nuclear physics. The following are among the results of this paper: \begin{itemize} \item If $f$ is unimodal with negative definite Hessian at its global maximum, then the mixing time of the random walk is $O(n\log n)$. \item If $f$ is multimodal, then the mixing time is exponential in $n$, but we show that there is a simulated annealing scheme governed by $f^K$ for an increasing sequence of $K$ that mixes in time $O(n^2)$. Using a varying step size that decreases with $K$, this can be taken down to $O(n\log n)$. \item If the process is studied on a general graph rather than the $d$-dimensional box, a simulated annealing scheme expressed in terms of conductances of the underlying network, works similarly. \end{itemize} Several examples are given, including the ones mentioned above.

math.PR↗
arXiv2026

Picard Approximation and Martingale Integrand Estimation for Path-Dependent FBSDEs

This paper develops a Picard iteration method for path-dependent forward-backward stochastic differential equations (FBSDEs). We construct a Picard iteration scheme, prove its convergence to the exact solution, and derive an explicit convergence rate. Although the underlying problem is conceptually straightforward, establishing rigorous convergence results in the path-dependent setting is technically demanding. Path dependence introduces a logarithmic correction: the time-discretization error bound is of order $|π|\log(1/|π|)$, compared with the order $|π|$ obtained in the Markovian setting, where $|π|$ denotes the mesh size of the time partition. We also introduce an estimator for the martingale integrand of the FBSDE that is tailored to path dependence and establish, within a Monte Carlo framework, a concentration inequality that quantifies its statistical error. The applicability of the proposed method is demonstrated in two non-Markovian derivative-pricing settings. The primary objective of this paper is to provide a theoretical convergence analysis of the Picard iteration method in the path-dependent setting and illustrate its application to option pricing, rather than to develop a faster or more computationally efficient algorithm.

math.PR↗