On the $q$-generalised multinomial/divergence correspondence

TL;DR

This paper establishes an asymptotic link between the $q$-deformed multinomial distribution and a family of divergence measures, extending Tsallis entropy through higher-order terms in nonextensive statistics.

Contribution

It introduces a novel asymptotic correspondence between $q$-multinomial PMFs and generalized divergence measures, expanding Tsallis entropy with new divergence forms.

Findings

Asymptotic correspondence between $q$-multinomial and divergence measures

Emergence of a family of divergence measures extending Tsallis entropy

Fundamental properties of the new divergence measures

Abstract

The asymptotic correspondence between the probability mass function of the $q$-deformed multinomial distribution and the $q$-generalised Kullback-Leibler divergence, also known as Tsallis relative entropy, is established. The probability mass function is generalised using the $q$-deformed algebra developed within the framework of nonextensive statistics, leading to the emergence of a family of divergence measures in the asymptotic limit as the system size increases. The coefficients in the asymptotic expansion yield Tsallis relative entropy as the leading-order term when $q$ is interpreted as an entropic parameter. Furthermore, higher-order expansion coefficients naturally introduce new divergence measures, extending Tsallis relative entropy through a one-parameter generalisation. Some fundamental properties of these extended divergences are also explored.