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Double $q$-Wigner Chaos and the Fourth Moment

Todd Kemp, Akihiro Miyagawa

TL;DR

This work extends the Fourth Moment Theorem to finite chaos expansions consisting of two stochastic integrals of opposite parity, in both free Wigner chaos and $q$-Wigner chaos. The authors establish that convergence to the appropriate Gaussian-type limit is governed entirely by the first four mixed moments, equivalently by the contraction behavior of the kernels, via a polarization identity for the fourth cumulant in the free setting and a $q$-analogue for the $q$-Wigner case. The free-case result yields convergence to a centered semicircular law, while the $q$-Wigner case yields convergence to a mixed $Q$-Gaussian distribution with a specified correlation structure. The key technical innovation is a precise cumulant decomposition for sums of opposite-parity chaos, together with explicit contraction-based criteria, mirroring recent Wiener-Itô fourth-mMoment theorems in the classical setting and extending them to noncommutative, free, and $q$-deformed frameworks.

Abstract

In this paper, we prove the Fourth Moment Theorem for sequences of (noncommutative) random variables given as sums of two stochastic integrals in two different parity orders of chaos, both in the free Wigner chaos setting and a $q$-Gaussian generalization. Specifically, we prove that convergence to the appropriate central limit distribution is mediated entirely by the behavior of the first four (mixed) moments of the two stochastic integrals, which in turn controls the $L^2$ norms of partial integral contractions of those kernels. The key step in both the free and $q$-Gaussian settings is a polarization identity for fourth cumulants of sums which holds only when the two terms have differing parities. These results are analogous to the recent preprint Fourth-Moment Theorems for Sums of Multiple Integrals by Basse-O'Connor, Kramer-Bang, and Svedsen in the classical Wiener-Itô chaos setting.

Double $q$-Wigner Chaos and the Fourth Moment

TL;DR

This work extends the Fourth Moment Theorem to finite chaos expansions consisting of two stochastic integrals of opposite parity, in both free Wigner chaos and -Wigner chaos. The authors establish that convergence to the appropriate Gaussian-type limit is governed entirely by the first four mixed moments, equivalently by the contraction behavior of the kernels, via a polarization identity for the fourth cumulant in the free setting and a -analogue for the -Wigner case. The free-case result yields convergence to a centered semicircular law, while the -Wigner case yields convergence to a mixed -Gaussian distribution with a specified correlation structure. The key technical innovation is a precise cumulant decomposition for sums of opposite-parity chaos, together with explicit contraction-based criteria, mirroring recent Wiener-Itô fourth-mMoment theorems in the classical setting and extending them to noncommutative, free, and -deformed frameworks.

Abstract

In this paper, we prove the Fourth Moment Theorem for sequences of (noncommutative) random variables given as sums of two stochastic integrals in two different parity orders of chaos, both in the free Wigner chaos setting and a -Gaussian generalization. Specifically, we prove that convergence to the appropriate central limit distribution is mediated entirely by the behavior of the first four (mixed) moments of the two stochastic integrals, which in turn controls the norms of partial integral contractions of those kernels. The key step in both the free and -Gaussian settings is a polarization identity for fourth cumulants of sums which holds only when the two terms have differing parities. These results are analogous to the recent preprint Fourth-Moment Theorems for Sums of Multiple Integrals by Basse-O'Connor, Kramer-Bang, and Svedsen in the classical Wiener-Itô chaos setting.

Paper Structure

This paper contains 8 sections, 10 theorems, 81 equations.

Table of Contents

  1. Introduction
  2. Main Results
  3. Preliminaries
  4. Semicircle distribution and free cumulants
  5. Wigner chaos
  6. $q$-Wigner chaos
  7. Proof for Wigner chaos
  8. Proof for $q$-Wigner chaos

Key Result

Theorem 1.1

Let $f_k$ be a sequence of real-valued symmetric functions in $L^2(\mathbb{R}^n_+)$ normalized so that $n!\|f_k\|_2\to 1$ as $k\to\infty$. Then the following three conditions are equivalent.

Theorems & Definitions (17)

  • Theorem 1.1: Nourdin, Peccati, NuPec2005
  • Theorem 1.2: K, Nourdin, Peccati, Speicher, 1529a405-f4b1-33eb-aeac-05c9f76e60d3
  • Theorem 1.3: Deya, Norredine, Nourdin, MR3089666
  • Theorem 1.4: Basse-O'Connor, Kramer-Bang, Svedsen, basseoconnor2025fourthmoment
  • Theorem 1.5
  • Theorem 1.6
  • Remark 1.7
  • Lemma 3.1
  • proof
  • Theorem 3.2
  • ...and 7 more