[Paper Review] Convergence of the Fourth Moment and Infinite Divisibility

Octavio Arizmendi|arXiv (Cornell University)|Jun 11, 2013

Random Matrices and Applications21 references3 citations

TL;DR

This paper establishes that for infinitely divisible distributions under classical, free, Boolean, and monotone convolutions, convergence of the fourth moment to the Gaussian (3), semicircle (2), or arcsine (1.5) value is sufficient for convergence in distribution to the respective limiting law. The proof leverages Jacobi parameter convergence and Bercovici-Pata bijections, extending Nualart-Peccati-type criteria to infinitely divisible and compound Poisson laws.

ABSTRACT

In this note we prove that, for infinitely divisible laws, convergence of the fourth moment to 3 is sufficient to ensure convergence in law to the Gaussian distribution. Our results include infinitely divisible measures with respect to classical, free, Boolean and monotone convolution. A similar criterion is proved for compound Poissons with jump distribution supported on a finite number of atoms. In particular, this generalizes recent results of Nourdin and Poly.

Motivation & Objective

• To extend the Nualart-Peccati moment convergence criterion from multiple Wiener-Itô integrals to infinitely divisible distributions.
• To unify moment convergence criteria across classical, free, Boolean, and monotone convolution frameworks.
• To prove that fourth moment convergence to 3, 2, or 1.5 implies convergence to the standard Gaussian, semicircle, or arcsine law, respectively, for infinitely divisible measures.
• To generalize recent results on compound Poisson limits in free probability to finite-support jump distributions.
• To demonstrate the utility of Boolean independence in analyzing convergence in non-commutative probability.

Proposed method

• Utilizes the Bercovici-Pata bijection to relate classical, free, Boolean, and monotone infinitely divisible laws.
• Applies the Jacobi parameter representation of measures via continued fractions to analyze moment convergence.
• Employs a key lemma showing that convergence of the first $2k+2$ moments implies weak convergence for measures with $k$-atom support.
• Uses the Cauchy transform and its convergence on the upper half-plane to establish weak convergence of measures.
• Applies the continuity of Bercovici-Pata maps to transfer convergence results from Boolean to other convolutional frameworks.
• Establishes that for compound Poisson laws with finite support, convergence of moments up to order $2k+2$ implies convergence to the limiting compound Poisson distribution.

Experimental results

Research questions

• RQ1 Can the Nualart-Peccati fourth moment criterion be extended from multiple Wiener-Itô integrals to infinitely divisible distributions?
• RQ2 Does convergence of the fourth moment to 3, 2, or 1.5 imply convergence to the standard Gaussian, semicircle, or arcsine law in the classical, free, and monotone settings?
• RQ3 Can the moment convergence criterion be generalized to compound Poisson distributions with finite support in free probability?
• RQ4 What role does Boolean independence play in establishing convergence results for other convolutional frameworks?
• RQ5 Is the fourth moment criterion sufficient for convergence in law for multiple integrals in monotone probability?

Key findings

• For classical infinitely divisible laws with mean 0 and variance 1, convergence of the fourth moment to 3 implies convergence in distribution to the standard normal law.
• For free infinitely divisible laws with mean 0 and variance 1, convergence of the fourth moment to 2 implies convergence in distribution to the standard semicircle law.
• For monotone infinitely divisible laws with mean 0 and variance 1, convergence of the fourth moment to 1.5 implies convergence in distribution to the standard arcsine law.
• For free compound Poisson measures with finite-support jump distribution, convergence of the first $2k+2$ moments implies weak convergence to the limiting compound Poisson distribution.
• The result for compound Poisson laws recovers Theorem 4.3 of Nourdin and Poly (2019) as a special case.
• The proof technique using Jacobi parameters and the Cauchy transform provides a unified framework across classical, free, Boolean, and monotone convolution.