Cyclic independence: Boolean and monotone

TL;DR

This paper introduces cyclic-Boolean and cyclic-monotone independence concepts, explores their convolution formulas, limit theorems, and classifies infinitely divisible distributions, with applications to graph eigenvalues.

Contribution

It develops a new framework for cyclic independence variants, extending free probability tools to graph adjacency matrices and their convolutions.

Findings

Formulas for cyclic-Boolean and cyclic-monotone convolutions

Limit theorems for sums of cyclic-independent variables

Classification of infinitely divisible distributions in this framework

Abstract

The present paper introduces a modified version of cyclic-monotone independence which originally arose in the context of random matrices, and also introduces its natural analogy called cyclic-Boolean independence. We investigate formulas for convolutions, limit theorems for sums of independent random variables, and also classify infinitely divisible distributions with respect to cyclic-Boolean convolution. Finally, we provide applications to the eigenvalues of the adjacency matrices of iterated star products of graphs and also iterated comb products of graphs.