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

Abstract

We investigate a Belinschi-Nica type semigroup for free and Boolean max-convolutions. We prove that this semigroup at time one connects limit theorems for freely and Boolean max-infinitely divisible distributions. Moreover, we also construct a max-analogue of Booleanclassical Bercovici-Pata bijection, establishing the equivalence of limit theorems for Boolean and classical max-infinitely divisible distributions.

Max-convolution semigroups and extreme values in limit theorems for the free multiplicative convolution

“…In particular, we have B ∨ 1 (B II,α ) = F II,α for all α > 0 (see [29]). In addition, we have the following relation.…”

“…It is known that B t • B s = B t+s and B ∨ t • B ∨ s = B ∨ t+s for all t, s ≥ 0, so that the families {B t } t≥0 and {B ∨ t } t≥0 are semigroups with respect to the composition of operators. The semigroups {B t } t≥0 and {B ∨ t } t≥0 are called Belinschi-Nica semigroup (see [9]) and max-Belinschi-Nica semigroup (see [29]), respectively. Considering these semigroups is important to understand relations between free and Boolean type limit theorems or freemax and Boolean-max type limit theorems.…”

“…Proof. By [9, 10], we have B 1 (b + α ) = f + α .Moreover, by[29], we haveB ∨ 1 (B II, α 1−α ) = F II, α1−α . By Theorem 1.3 and Example 4.1, we have…”