Counting distributions from the perspective of combinants

TL;DR

This paper explores counting distributions through combinants, especially oscillatory ones, revealing that specific compound distributions based on the Binomial Distribution are necessary to describe their behavior, aiding in understanding underlying processes.

Contribution

It introduces analytical forms of compound distributions based on the Binomial Distribution to describe oscillatory combinants in counting distributions.

Findings

Oscillatory combinants indicate specific underlying distribution structures.

Compound distributions based on the Binomial Distribution can accurately model these behaviors.

Analytical forms provided facilitate further analysis of counting distributions.

Abstract

We present a comprehensive insight into counting distributions from the perspective of the combinants extracted from them. In particular, we focus on cases where these combinants exhibit oscillatory behavior that can provide an invaluable new source of information about the dynamics of the process under study. We show that such behavior can be described only by specific combinations of compound distributions based on the Binomial Distribution and provide their analytical forms which can be used in further investigations and which can be helpful in the analysis of all other types of counting distributions.