Central limit measure for V-monotone independence

TL;DR

This paper investigates the central limit distribution for V-monotone independence, proving its absolute continuity and providing an implicit form of its density, supported by computational visualization.

Contribution

It introduces the explicit analysis of the central limit distribution for V-monotone independence, including its density and properties, expanding non-commutative probability theory.

Findings

The distribution is absolutely continuous with respect to Lebesgue measure.

An implicit form of the density function is derived.

A computer-generated graph visualizes the density.

Abstract

We study the central limit distribution $\mu$ for V-monotone independence. Using its Cauchy--Stieltjes transform, we prove that $\mu$ is absolutely continuous with respect to the Lebesgue measure on $\mathbb{R}$ and we give its density $\rho$ in an implicit form. We present a computer generated graph of $\rho$.