Classical Scale Mixtures of Boolean Stable Laws

TL;DR

This paper investigates Boolean stable laws, demonstrating their scale mixtures coincide with free and monotone mixtures, and establishes their infinite divisibility properties across classical, free, and monotone convolutions.

Contribution

It introduces the concept of multiplicative monotone convolution and proves the infinite divisibility of Boolean stable laws under various convolutions, including a conjecture on free Bessel laws.

Findings

Scale mixtures of Boolean stable laws are both classically and freely infinitely divisible for certain parameters.

The paper proves the multiplicative infinite divisibility of Boolean stable laws with specific parameters.

It confirms the existence of free Bessel laws as probability measures, resolving a conjecture.

Abstract

We study Boolean stable laws, $\mathbf{b}_{\alpha,\rho}$, with stability index $\alpha$ and asymmetry parameter $\rho$. We show that the classical scale mixture of $\mathbf{b}_{\alpha,\rho}$ coincides with a free mixture and also a monotone mixture of $\mathbf{b}_{\alpha,\rho}$. For this purpose we define the multiplicative monotone convolution of probability measures, one is supported on the positive real line and the other is arbitrary. We prove that any scale mixture of $\mathbf{b}_{\alpha,\rho}$ is both classically and freely infinitely divisible for $\alpha\leq1/2$ and also for some $\alpha>1/2$. Furthermore, we show the multiplicative infinite divisibility of $\mathbf{b}_{\alpha,1}$ with respect classical, free and monotone convolutions. Scale mixtures of Boolean stable laws include some generalized beta distributions of second kind, which turn out to be both classically and freely infinitely divisible. One of them appears as a limit distribution in multiplicative free laws of large numbers studied by Tucci, Haagerup and M\"oller. We use a representation of $\mathbf{b}_{\alpha,1}$ as the free multiplicative convolution of a free Bessel law and a free stable law to prove a conjecture of Hinz and M{\l}otkowski regarding the existence of the free Bessel laws as probability measures. The proof depends on the fact that $\mathbf{b}_{\alpha,1}$ has free divisibility indicator 0 for $1/2<\alpha$.