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The infimum values of three probability functions for the Laplace distribution and the student's $t$ distribution

Rong-Sheng Hu, Ze-Chun Hu, Zhen Huang, Mu-Xuan Li

TL;DR

The paper addresses the problem of computing the infimums of concentration-related probabilities $C(y)$, $T(y)$, and anti-concentration $H(y)$ over parameter families for the Laplace and Student’s $t$ distributions. It derives exact closed-form results for the Laplace family and provides a detailed finite-$v$ analysis for the $t$-distribution, showing convergence to Gaussian limits as $v\to\infty$. The work confirms concentration and anti-concentration properties for these families and connects to classical conjectures in probabilistic combinatorics, offering practical computation methods via hypergeometric representations. Overall, the results enhance understanding of how concentration and anti-concentration phenomena behave across distributional families and parameter regimes.

Abstract

Let $\{X_α\}$ be a family of random variables satisfying some distribution with a parameter $α$, $E(X_α)$ be the expectation, and $Var(X_α)$ be the variance. In this paper, we study the infimum values of three probability functions: $P(X_α\leq y E(X_α))$, $P\left(|X_α-E(X_α)|\leq y\sqrt{Var(X_α)}\right)$ and $P\left(|X_α-E(X_α)|\geq y\sqrt{Var(X_α)}\right), \forall y>0$, with respect to the parameter $α$ for the Laplace distribution and the student's $t$ distribution. Our motivation comes from three former conjectures: Chvátal's conjecture, Tomaszewski's conjecture and Hitczenko-Kwapień's conjecture.

The infimum values of three probability functions for the Laplace distribution and the student's $t$ distribution

TL;DR

The paper addresses the problem of computing the infimums of concentration-related probabilities , , and anti-concentration over parameter families for the Laplace and Student’s distributions. It derives exact closed-form results for the Laplace family and provides a detailed finite- analysis for the -distribution, showing convergence to Gaussian limits as . The work confirms concentration and anti-concentration properties for these families and connects to classical conjectures in probabilistic combinatorics, offering practical computation methods via hypergeometric representations. Overall, the results enhance understanding of how concentration and anti-concentration phenomena behave across distributional families and parameter regimes.

Abstract

Let be a family of random variables satisfying some distribution with a parameter , be the expectation, and be the variance. In this paper, we study the infimum values of three probability functions: , and , with respect to the parameter for the Laplace distribution and the student's distribution. Our motivation comes from three former conjectures: Chvátal's conjecture, Tomaszewski's conjecture and Hitczenko-Kwapień's conjecture.
Paper Structure (4 sections, 7 theorems, 121 equations)

This paper contains 4 sections, 7 theorems, 121 equations.

Table of Contents

  1. Introduction and main results
  2. Proof of Theorem \ref{['main-thm1']}
  3. The case $y \in (0, 1]$
  4. Proof of Lemma \ref{['lemma1']}

Key Result

Theorem 1.1

Let $X_{\mu,b}$ be a Laplace random variable with the location parameter $\mu$ and the scale parameter $b$. Then (i) $C(y)=0$ if $y>0$ and $y\neq 1$, and $C(1)=\frac{1}{2}$; (ii) $T(y)=1-e^{-\sqrt{2}y},\forall y>0$; (iii) $H(y)=e^{-\sqrt{2}y},\forall y>0$.

Theorems & Definitions (10)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Remark 1.4
  • Remark 1.5
  • Lemma 3.1
  • Lemma 3.2
  • Lemma 3.3
  • Lemma 3.4
  • Lemma 3.5