Conditionally monotone independence I: Independence, additive convolutions and related convolutions

TL;DR

This paper introduces a new type of independence called conditionally monotone independence in non-commutative probability, unifying several existing products and exploring their properties, limit theorems, and deformations.

Contribution

It defines the conditionally monotone product, unifies multiple independence concepts, and develops associated cumulants, limit theorems, and deformation analysis.

Findings

Unified monotone, Boolean, and orthogonal products.

Derived limit distributions for central limit theorem and Poisson law.

Established combinatorial moment-cumulant formulas using monotone partitions.

Abstract

We define a product of algebraic probability spaces equipped with two states. This product is called a conditionally monotone product. This product is a new example of independence in non-commutative probability theory and unifies the monotone and Boolean products, and moreover, the orthogonal product. Then we define the associated cumulants and calculate the limit distributions in central limit theorem and Poisson's law of small numbers. We also prove a combinatorial moment-cumulant formula using monotone partitions. We investigate some other topics such as infinite divisibility for the additive convolution and deformations of the monotone convolution. We define cumulants for a general convolution to analyze the deformed convolutions.