Asymptotic normality through factorial cumulants and partitions identities

TL;DR

This paper introduces a novel approach to establishing asymptotic normality using factorial cumulants and partition identities, providing a unified framework for various classical and generalized probabilistic models.

Contribution

It develops a new method based on factorial cumulants and partition identities to prove asymptotic normality across multiple models.

Findings

Asymptotic normality is established for classical discrete distributions.

The approach applies to occupancy problems and generalized allocation schemes.

Results include models related to negative multinomial distribution.

Abstract

In the paper we develop an approach to asymptotic normality through factorial cumulants. Factorial cumulants arise in the same manner from factorial moments, as do (ordinary) cumulants from (ordinary) moments. Another tool we exploit is a new identity for "moments" of partitions of numbers. The general limiting result is then used to (re-)derive asymptotic normality for several models including classical discrete distributions, occupancy problems in some generalized allocation schemes and two models related to negative multinomial distribution.