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Convolution comparison measures

Otte Heinävaara

TL;DR

This work establishes a sharp, fourth-derivative–based ordering between classical and free convolutions for compactly supported measures. It introduces a convolution comparison measure on R^2 whose positivity governs the inequality ∫ f d(μ * ν) ≥ ∫ f d(μ ⊞ ν) for all f ∈ C^4 with f^{(4)} ≥ 0, and provides an explicit construction in the finitely supported case via a commutation-based density ω_{A,B}. The core argument combines Hermitian matrix identities, Gegenbauer polynomial expansions, and hypergeometric identities to express the comparison measure and prove its positivity, then extends the result to general μ, ν by approximation. The results imply leading-order and higher-moments inequalities m_{2k}(μ * ν) ≥ m_{2k}(μ ⊞ ν), and suggest a 4th-order notion of ordering among probability measures that is robust under convolution and of potential relevance to finite free and c-convolutions.

Abstract

We give a precise functional comparison between classical and free convolutions. If $μ$ and $ν$ are compactly supported probability measures, we show that the expectation of $f$ over the classical convolution $μ* ν$ is at least the expectation of $f$ over the free convolution $μ\boxplus ν$, as long as the fourth derivative of $f$ is non-negative. Conversely, the non-negativity of the fourth derivative is necessary for such a comparison. This comparison is based on the positivity of a related measure on $\mathbb{R}^{2}$, which we dub the convolution comparison measure. We give an expression for this measure using a curious identity involving Hermitian matrices.

Convolution comparison measures

TL;DR

This work establishes a sharp, fourth-derivative–based ordering between classical and free convolutions for compactly supported measures. It introduces a convolution comparison measure on R^2 whose positivity governs the inequality ∫ f d(μ * ν) ≥ ∫ f d(μ ⊞ ν) for all f ∈ C^4 with f^{(4)} ≥ 0, and provides an explicit construction in the finitely supported case via a commutation-based density ω_{A,B}. The core argument combines Hermitian matrix identities, Gegenbauer polynomial expansions, and hypergeometric identities to express the comparison measure and prove its positivity, then extends the result to general μ, ν by approximation. The results imply leading-order and higher-moments inequalities m_{2k}(μ * ν) ≥ m_{2k}(μ ⊞ ν), and suggest a 4th-order notion of ordering among probability measures that is robust under convolution and of potential relevance to finite free and c-convolutions.

Abstract

We give a precise functional comparison between classical and free convolutions. If and are compactly supported probability measures, we show that the expectation of over the classical convolution is at least the expectation of over the free convolution , as long as the fourth derivative of is non-negative. Conversely, the non-negativity of the fourth derivative is necessary for such a comparison. This comparison is based on the positivity of a related measure on , which we dub the convolution comparison measure. We give an expression for this measure using a curious identity involving Hermitian matrices.
Paper Structure (21 sections, 16 theorems, 102 equations)

This paper contains 21 sections, 16 theorems, 102 equations.

Table of Contents

  1. Introduction
  2. Comparing classical and free convolutions
  3. Convolution comparison measures
  4. Commutation measuring measure
  5. Overview of the proof and the paper
  6. Preliminaries and some identities
  7. Divided differences
  8. Hermitian matrix identities
  9. Hypergeometric identities and Gegenbauer polynomials
  10. Free probability
  11. Leading order calculation
  12. Proofs of Theorems \ref{['thm2']} and \ref{['thm1']}
  13. Step 1
  14. Step 2
  15. Step 3
  16. ...and 6 more sections

Key Result

Theorem 1.3

Let $\mu$ and $\nu$ be compactly-supported probability measures on $\mathbb{R}$. Then, for any $f \in C^{4}(\mathbb{R})$ with $f^{(4)} \geq 0$, we have where $*$ and $\boxplus$ denote the classical and free convolutions respectively. Conversely, if $f \in C^{4}(\mathbb{R})$ is such that (main_ineq) holds for any $\mu$ and $\nu$, then the fourth derivative of $f$ is non-negative.

Theorems & Definitions (32)

  • Theorem 1.3
  • Theorem 1.4
  • Corollary 1.5
  • proof
  • Theorem 2.3
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • Lemma 2.7
  • ...and 22 more