Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Sharp bounds on $p$-norms for sums of independent uniform random variables, $0 < p < 1$

Giorgos Chasapis, Keerthana Gurushankar, Tomasz Tkocz

Abstract

We provide a sharp lower bound on the $p$-norm of a sum of independent uniform random variables in terms of its variance when $0 < p < 1$. We address an analogous question for $p$-Rényi entropy for $p$ in the same range.

Sharp bounds on $p$-norms for sums of independent uniform random variables, $0 < p < 1$

Abstract

We provide a sharp lower bound on the -norm of a sum of independent uniform random variables in terms of its variance when . We address an analogous question for -Rényi entropy for in the same range.

Paper Structure

This paper contains 14 sections, 21 theorems, 113 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction and results
  2. Moments
  3. Rényi entropy
  4. Organisation of the paper
  5. Proof of the main result
  6. Overview
  7. Details
  8. Auxiliary lemmas
  9. Lemmas concerning the sinc function
  10. Lemmas concerning sums of p-th powers
  11. Lemmas concerning the gamma function
  12. Integral inequality: proofs of Theorems \ref{['thm:s>1']} and \ref{['thm:s>2']}
  13. Inductive argument
  14. Rényi entropy: Proof of Theorem \ref{['thm:ent']}

Key Result

Theorem 1

For $0 < p < 1$, $c_ p = \|Z\|_p/\sqrt{3}$ is the best constant in eq:khin.

Figures (2)

  • Figure 1: Functions $f$, $g$ and the set $\{t > 0, f(t) < y\}$. Here $m=3$, i.e. $y_3 < y < y_4$.
  • Figure 2: The slope of the segment $AB$ is not smaller than the slope of either $AC$ or $BC$.

Theorems & Definitions (38)

  • Theorem 1
  • Theorem 2
  • Remark 3
  • Lemma 4
  • proof
  • Theorem 5
  • Theorem 6
  • Lemma 7
  • proof
  • Lemma 8
  • ...and 28 more