Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A Fixed-Point Approach to Non-Commutative Central Limit Theorems

Jad Hamdan

Abstract

We show how the renormalization group approach can be used to prove quantitative central limit theorems (CLTs) in the setting of free, Boolean, bi--free and bi--Boolean independence under finite third moment assumptions. The proofs rely on the construction of a contractive metric over the space of probability measures over $\mathbb{R}$ or $\mathbb{R}^2$, which has the appropriate analogue of a Gaussian distribution as a fixed point (for instance, the semi--circle law in the case of free independence). In all cases, this yields a convergence rate of $1/\sqrt{n}$, and we show that this can be improved to $1/n$ in some instances under stronger assumptions.

A Fixed-Point Approach to Non-Commutative Central Limit Theorems

Abstract

We show how the renormalization group approach can be used to prove quantitative central limit theorems (CLTs) in the setting of free, Boolean, bi--free and bi--Boolean independence under finite third moment assumptions. The proofs rely on the construction of a contractive metric over the space of probability measures over or , which has the appropriate analogue of a Gaussian distribution as a fixed point (for instance, the semi--circle law in the case of free independence). In all cases, this yields a convergence rate of , and we show that this can be improved to in some instances under stronger assumptions.
Paper Structure (9 sections, 18 theorems, 50 equations)

This paper contains 9 sections, 18 theorems, 50 equations.

Table of Contents

  1. Introduction
  2. Preliminary definitions and notation.
  3. Analytic theory of free and Boolean convolution.
  4. Fixed-point proofs of the free and Boolean CLTs
  5. Proof of the Boolean and free CLT
  6. Bi--free harmonic analysis
  7. Bi-free $R$ and Cauchy transforms
  8. Extension to bi--free and bi--Boolean
  9. Other Potential Generalizations

Key Result

Theorem 2.2

Let $p$ be a positive integer and $\mu$ a probability measure on the real line. If $\mu$ admits a $p$--th moment, then $R_\mu$ admits the Taylor expansion where $(\kappa_n(\mu))_{n\in\mathbb{N}}$ are the free cumulants of $\mu$ and the limit is as $z\to 0$ non--tangentially, meaning $|z|\to 0$ and $|\Re(z)|\leq -\alpha \Im{z}$ for some $\alpha>0$.

Theorems & Definitions (30)

  • Remark 2.1
  • Theorem 2.2: Benaych-Georges
  • Proposition 2.3: Arizmendi--Salazar
  • Theorem 3.1: Free Berry--Esseen
  • Theorem 3.2: Boolean Berry--Esseen
  • Remark 3.3
  • Remark 3.4
  • Proposition 3.5
  • proof
  • Remark 3.6
  • ...and 20 more