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

TL;DR

This paper establishes general limit theorems for mixtures of free, monotone, and Boolean independence using graph-based convolutions, including new results for multiregular digraphs and classical domains.

Contribution

It introduces a unified framework for limit theorems in non-commutative probability involving graph-structured convolutions, extending previous results and including new cases.

Findings

Proves limit theorems for $oxplus_G$ convolutions under graph homomorphism convergence.

Includes new limit theorem for multiregular digraphs.

Recovers several classical limit theorems within this framework.

Abstract

We study mixtures of free, monotone, and Boolean independence described by a directed graph $G = (V,E)$ in the context of $\mathcal{T}$-free convolutions of Jekel and Liu. We prove general limit theorems for the associated additive convolution operations $\boxplus_G$. For a sequence of digraphs $G_n = (V_n,E_n)$, we give sufficient conditions for the limit $\widehat{\mu} = \lim_{n \to \infty} \boxplus_{G_n}(\mu_n)$ to exist whenever the Boolean convolution powers $\mu_n^{\uplus |V_n|}$ converge to some $\mu$. This in particular includes central limit and Poisson limit theorems, as well as limit theorems for each classical domain of attraction. The hypothesis on the sequence of $G_n$ is that the normalized counts of digraph homomorphisms from rooted trees into $G_n$ converge as $n \to \infty$, and we verify this for several families of examples where the $G_n$'s converge in some sense to a continuum limit, or digraphon. In particular, we obtain a new limit theorem for multiregular digraphs, as well as recovering several limit theorems in prior work.