The infimum values of the probability functions for some infinitely divisible distributions motivated by Chv\'{a}tal's theorem

TL;DR

This paper investigates the minimum probability that certain infinitely divisible distributions, such as inverse Gaussian and log-normal, assign to the event that the variable is less than a multiple of its expectation, inspired by Chvátal's theorem.

Contribution

It extends Chvátal's theorem to a broader class of infinitely divisible distributions by analyzing their infimum probability bounds.

Findings

Identifies infimum probability values for inverse Gaussian, log-normal, Gumbel, and logistic distributions.

Provides theoretical bounds for probabilities of the form P(X ≤ κE[X]) for these distributions.

Connects classical binomial probability results to a wider class of distributions.

Abstract

Let $B(n,p)$ denote a binomial random variable with parameters $n$ and $p$. Chv\'{a}tal's theorem says that for any fixed $n\geq 2$, as $m$ ranges over $\{0,\ldots,n\}$, the probability $q_m:=P(B(n,m/n)\leq m)$ is the smallest when $m$ is closest to $\frac{2n}{3}$. Motivated by this theorem, in this paper we consider the infimum value of the probability $P(X\leq \kappa E[X])$, where $\kappa$ is a positive real number, and $X$ is a random variable whose distribution belongs to some infinitely divisible distributions including the inverse Gaussian, log-normal, Gumbel and logistic distributions.