[Paper Review] Limit theorems for classical, freely and Boolean max-infinitely divisible distributions

Yuki Ueda|arXiv (Cornell University)|Jul 28, 2019

Random Matrices and Applications14 references4 citations

TL;DR

This paper establishes a max-analogue of the Boolean-classical Bercovici-Pata bijection using a Belinschi-Nica-type semigroup for max-convolutions, proving that the time-one map of this semigroup connects limit theorems for Boolean and classical max-infinitely divisible distributions. It further shows that this semigroup links free and Boolean max-infinitely divisible laws, unifying extreme value limit theorems across classical, free, and Boolean probability theories via a novel max-Belinschi-Nica semigroup.

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

Motivation & Objective

• To extend the Boolean-classical Bercovici-Pata bijection to the max-infinitely divisible setting using a max-analogue of the Belinschi-Nica semigroup.
• To unify limit theorems for classical, free, and Boolean max-infinitely divisible distributions through a semigroup structure on max-convolutions.
• To establish a correspondence between limit theorems in Boolean and classical extreme value theory via a novel max-Bercovici-Pata bijection.
• To clarify the role of the time-one map of the max-Belinschi-Nica semigroup in connecting free and Boolean max-infinitely divisible laws.

Proposed method

• Introduces a max-analogue of the Belinschi-Nica semigroup via the map $\mathbf{B}_t^M(\mu) := (\mu^{\boxplus(1+t)})^{\uplus\frac{1}{1+t}}$ for $t \geq 0$ and $\mu \in \mathcal{P}^+$, defining a semigroup under composition.
• Defines max-convolution operations $\lor$ (classical), $\cup$ (Boolean), and $\Box$ (free) on distribution functions, with $F \lor G = FG$, $F \cup G = \frac{F}{1 - (1 - F)(1 - G)}$, and $F \Box G = F \boxplus G$.
• Constructs a map $\mathcal{X}: \Delta_+ \to \Delta_+$ via $\mathcal{X}(F)(x) = \frac{1}{2 - F(x)}$ for $x \geq 0$, which acts as the Boolean-classical max-Bercovici-Pata bijection.
• Proves that the time-one map $\mathbf{B}_1^M$ coincides with the max-analogue of the Boolean-free Bercovici-Pata bijection, linking Boolean and free max-infinitely divisible laws.
• Uses weak convergence arguments and asymptotic analysis of $F_n^{\lor k_n}$ and $F_n^{\Box \lor k_n}$ to establish equivalence between limit theorems under different convolution types.
• Applies the semigroup structure to derive convergence results: $F_n^{\lor k_n} \xrightarrow{w} F$ if and only if $F_n^{\cup \lor k_n} \xrightarrow{w} \mathcal{X}^{-1}(F)$, and $F_n^{\Box \lor k_n} \xrightarrow{w} \Lambda^\lor(F)$.

Experimental results

Research questions

• RQ1 How can the Boolean-classical Bercovici-Pata bijection be extended to the max-infinitely divisible setting using a semigroup framework?
• RQ2 What is the role of the time-one map of the max-Belinschi-Nica semigroup in connecting free and Boolean max-infinitely divisible distributions?
• RQ3 Can a max-analogue of the Bercovici-Pata bijection be constructed to unify limit theorems across classical, Boolean, and free extreme value theories?
• RQ4 Under what conditions does convergence in free max-convolution imply convergence in classical max-convolution?
• RQ5 How do the asymptotic behaviors of $F_n^{\lor k_n}$, $F_n^{\cup \lor k_n}$, and $F_n^{\Box \lor k_n}$ relate under weak convergence?

Key findings

• The map $\mathcal{X}(F)(x) = \frac{1}{2 - F(x)}$ for $x \geq 0$ defines a Boolean-classical max-Bercovici-Pata bijection, with $\mathcal{X}^{-1}(F)(x) = \frac{F(x)}{1 + F(x)}$.
• The time-one map $\mathbf{B}_1^M$ of the max-Belinschi-Nica semigroup coincides with the Boolean-free Bercovici-Pata bijection, establishing a direct link between Boolean and free max-infinitely divisible laws.
• For sequences $\{F_n\}$ and $\{k_n\}$, $F_n^{\lor k_n} \xrightarrow{w} F$ if and only if $F_n^{\cup \lor k_n} \xrightarrow{w} \mathcal{X}^{-1}(F)$, proving the equivalence of limit theorems in classical and Boolean max-infinitely divisible settings.
• If $F_n^{\Box \lor k_n} \xrightarrow{w} F$ with $F > 0$ on $[0,\infty)$, then $F_n^{\lor k_n} \xrightarrow{w} \Pi_{1,F}^\lor = \mathcal{X}\left(\frac{1}{2 - F}\right)$, showing that free max-limit laws imply classical max-limit laws.
• The semigroup $\{\mathbf{B}_t^M\}_{t \geq 0}$ satisfies $\mathbf{B}_t^M \circ \mathbf{B}_s^M = \mathbf{B}_{t+s}^M$, confirming it is a well-defined semigroup under composition.
• The convergence $F_n^{\Box \lor k_n} \xrightarrow{w} \Lambda^\lor(F)$ holds if and only if $F_n^{\lor k_n} \xrightarrow{w} \Pi_{1,F}^\lor$, establishing a complete correspondence between free and classical max-limit theorems.