Boolean convolutions and regular variation

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 studies 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.2010 Mathematics Subject Classification: 60G70; 46L53; 46L54.

Tail probabilities of random linear functions of regularly varying random vectors

“…Recently, Tillier and Wintenberger [26] have extended Breiman's multivariate result to vectors of random length, determined for instance by a Poisson random variable. In a more general setting, Chakraborty and Hazra [5], extend Breiman's result for multiplicative Boolean convolution of regularly varying measures. Finally, the monograph Buraczewski, Damek and Mikosch [4] provides many applications of Breiman's result and its generalizations in the area of stochastic modeling with power-law tail.…”

Limit theorems for classical, freely and Boolean max-infinitely divisible distributions

“…More generally, we can conclude that three type max-convolution preserve tails of distribution functions as follows. However, the following proposition has already been obtained in [10] and [11] to study behavior at tails of free and Boolean subexponential distributions, but for readers convenience we include the proof. Proposition 4.12.…”