arXiv subjects

Abdulmajeed Alqasem

Publications and source records attributed to Abdulmajeed Alqasem.

2 recordsLinked to original sources

arXiv2026

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

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.

math.PR↗

arXiv2023

On a Conjecture of Feige for Discrete Log-Concave Distributions

A remarkable conjecture of Feige (2006) asserts that for any collection of $n$ independent non-negative random variables $X_1, X_2, \dots, X_n$, each with expectation at most $1$, $$ \mathbb{P}(X < \mathbb{E}[X] + 1) \geq \frac{1}{e}, $$ where $X = \sum_{i=1}^n X_i$. In this paper, we investigate this conjecture for the class of discrete log-concave probability distributions and we prove a strengthened version. More specifically, we show that the conjectured bound $1/e$ holds when $X_i$'s are independent discrete log-concave with arbitrary expectation.

math.PR↗