[Paper Review] Multiplicative Monotone Convolution

Hari Bercovici|ArXiv.org|Jun 23, 2004

Mathematical Analysis and Transform Methods3 citations

TL;DR

This paper introduces and characterizes multiplicative monotonic convolution, a new operation for probability measures under monotonic independence, using a modified Cauchy transform and generating functions. It establishes that the convolution preserves support on the unit circle and positive real line, and fully characterizes infinite divisibility via one-parameter semigroups, proving uniqueness up to rotation for the generating function of the semigroup.

ABSTRACT

We show that the monotonic independence introduced by Muraki can also be used to define a multiplicative convolution. We also find a method for the calculation of this convolution based on an appropriate form of the Cauchy transform. We discuss infinite divisibility in the multiplicative monotonic context as well.

Motivation & Objective

• To define and study a multiplicative convolution operation based on monotonic independence in noncommutative probability.
• To develop a computational method for multiplicative monotonic convolution using a transformed Cauchy integral and moment generating series.
• To characterize infinite divisibility in the multiplicative monotonic setting via one-parameter convolution semigroups.
• To prove that the multiplicative monotonic convolution of measures on the unit circle or R+ remains within the same class.
• To establish uniqueness of the semigroup generator up to rotation, resolving non-uniqueness in the infinite divisibility framework.

Proposed method

• The paper uses a modified Cauchy transform and analytic function theory to compute the multiplicative monotonic convolution.
• It defines two operations: $\circlearrowright$ on $\mathfrak{M} \times \mathbb{C}$ and $\circlearrowright_0$ on $\mathfrak{M}$, with the latter preserving the first moment under convolution.
• The convolution is computed via the $\eta$-transform, which satisfies a differential equation $\frac{d}{d\tau}\eta_{\mu_\tau}(z) = \eta_{\mu_\tau}(z) B(\eta_{\mu_\tau}(z))$.
• The generating function $A(z) = zB(z)$ is analytic in the unit disk with $\Re B(z) \leq 0$, ensuring the existence of a unique semigroup.
• The method extends to measures with unbounded support by using the $\circlearrowright$ operation, while $\circlearrowright_0$ is restricted to compactly supported measures.
• The paper proves that the limit of discrete approximations $\mu_{m/2^n}$ converges weakly to a unique measure $\mu_\tau$, establishing the existence of the semigroup.

Experimental results

Research questions

• RQ1 Can a multiplicative convolution be defined under monotonic independence, and how does it differ from standard free or Boolean convolutions?
• RQ2 How can the multiplicative monotonic convolution be computed efficiently using analytic tools like the Cauchy transform?
• RQ3 Does the multiplicative monotonic convolution preserve the support of probability measures on the unit circle $\mathbb{T}$ or on $\mathbb{R}_+$?
• RQ4 What is the structure of one-parameter semigroups under multiplicative monotonic convolution, and how are they generated?
• RQ5 Is infinite divisibility under multiplicative monotonic convolution uniquely characterized, and what is the role of the generating function?

Key findings

• The multiplicative monotonic convolution $\circlearrowright_0$ preserves the class of compactly supported probability measures on $\mathbb{R}_+$ and on the unit circle $\mathbb{T}$.
• For any probability measure $\mu$ on $\mathbb{T}$ with $\int_{\mathbb{T}} \zeta \, d\mu(\zeta) \neq 0$, there exists a unique $\circlearrowright_0$-infinitely divisible measure $\mu_\tau$ such that $\mu_1 = \mu$, up to rotation.
• The semigroup $\{\mu_\tau\}$ is generated by a function $A(z) = zB(z)$ with $\Re B(z) \leq 0$, and the $\eta$-transform satisfies a differential equation $\frac{d}{d\tau}\eta_{\mu_\tau}(z) = \eta_{\mu_\tau}(z) B(\eta_{\mu_\tau}(z))$.
• The limit of discrete approximations $\mu_{m/2^n}$ converges weakly to a unique measure $\mu_\tau$, proving existence of the semigroup for all $\tau \geq 0$.
• The semigroup is not unique in general, but uniqueness holds up to rotation: if two measures have the same first moment and generate the same convolution square, they are equal.
• The operation $\circlearrowright$ on $\mathfrak{M} \times \mathbb{C}$ allows extension to measures with unbounded support, while $\circlearrowright_0$ is restricted to compactly supported measures.