Boolean convolutions and regular variation

TL;DR

This paper explores how regularly varying probability measures affect Boolean convolutions, introducing Boolean subexponentiality, and establishing new results analogous to Breiman's theorem for multiplicative Boolean convolution.

Contribution

It introduces Boolean subexponentiality, extends classical and free subexponentiality concepts, and derives an analog of Breiman's theorem for Boolean convolutions involving regularly varying measures.

Findings

Regularly varying tails belong to Boolean subexponential distributions.

Established an analog of Breiman's theorem for multiplicative Boolean convolution.

Analyzed the behavior of the Belinschi-Nica map in this context.

Abstract

In this article we study the influence of regularly varying probability measures on additive and multiplicative Boolean convolutions. We introduce the notion of Boolean subexponentiality (for additive Boolean convolution), which extends the notion of classical and free subexponentiality. We show that the distributions with regularly varying tails belong to the class of Boolean subexponential distributions. As an application we also study the behaviour of the Belinschi-Nica map. Breiman's theorem study the classical product convolution between regularly varying measures. We derive an analogous result to Breiman's theorem in case of multiplicative Boolean convolution. In proving these results we exploit the relationship of regular variation with different transforms and their Taylor series expansion.