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

We study mixtures of free, monotone, and boolean independence described by directed graphs (digraphs). For a sequence of digraphs Gₙ = (Vₙ,Eₙ), we give sufficient conditions for the limit μ = 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. In particular, we obtain a new limit theorems for multiregular digraphs, as well as recovering several limit theorems in prior work.