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

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.