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

We study mixtures of free, monotone, and Boolean independence described by a directed graph G=(V,E) in the context of T-free convolutions from [18]. We prove general limit theorems for the associated additive convolution operations ⊞G. For a sequence of digraphs Gn=(Vn,En), we give sufficient conditions for the limit μˆ=limn→∞⊞Gn(μn) to exist whenever the Boolean convolution powers μn⊎|Vn| converge to some μ. 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 Gn is that the normalized counts of digraph homomorphisms from rooted trees into Gn converge as n→∞, and we verify this for several families of examples where the Gn’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.