Homomorphisms relative to additive convolutions and max-convolutions: free, boolean and classical cases

Noise as a Boolean algebra of $σ$-fields

A noise is a kind of homomorphism from a Boolean algebra of domains to the lattice of $σ$-fields. Leaving aside the homomorphism we examine its image, a Boolean algebra of $σ$-fields. The largest extension of such Boolean algebra of $σ$-fields, being well-defined always, is a complete Boolean algebra if and only if the noise is classical, which answers an old question of J. Feldman.

Free Subordination and Belinschi-Nica Semigroup

We realize the Belinschi-Nica semigroup of homomorphisms as a free multiplicative subordination. This realization allows to define more general semigroups of homomorphisms with respect to free multiplicative convolution. For these semigroups we show that a differential equation holds, generalizing the complex Burgers equation. We give examples of free multiplicative subordination and find a relation to the Markov-Krein transform, Boolean stable laws and monotone stable laws. A similar idea works for additive subordination, and in particular we study the free additive subordination associated t…

On a remarkable semigroup of homomorphisms with respect to free multiplicative convolution

Let M denote the space of Borel probability measures on the real line. For every nonnegative t we consider the transformation $\mathbb B_t : M \to M$ defined for any given element in M by taking succesively the the (1+t) power with respect to free additive convolution and then the 1/(1+t) power with respect to Boolean convolution of the given element. We show that the family of maps {\mathbb B_t|t\geq 0} is a semigroup with respect to the operation of composition and that, quite surprisingly, every $\mathbb B_t$ is a homomorphism for the operation of free multiplicative convolution. We prove t…

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

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 Boolean-classical Bercovici-Pata bijection, establishing the equivalence of limit theorems for Boolean and classical max-infinitely divisible distributions.

Boolean Extremes and Dagum Distributions

We study the max-convolution and max-stable laws for Boolean independence and prove that these are Dagum distributions (also known as log-logistical distributions).

自由独立性、单调独立性与布尔独立性混合情形的一般极限定理

General limit theorems for mixtures of free, monotone, and boolean independence

本文在$\u27b1$-无卷积框架下研究三类独立性混合的有向图卷积运算$\boxplus_G$，给出有向图序列下极限存在的充分条件，涵盖多种经典极限定理，并验证多类示例，得到多正则有向图新极限定理。