  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.
