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On the $q$-generalised multinomial/divergence correspondence

Keisuke Okamura

TL;DR

This work extends Wallis-type combinatorial asymptotics to a $q$-deformed multinomial PMF within Tsallis nonextensive statistics, showing that the leading term in the large-$n$ expansion is the Tsallis relative entropy $D_q(P||_0 Q)$. Higher-order corrections naturally generate a one-parameter family of lambda-extended divergences $D_q(P||_λ Q)$, with explicit recursivity and sign properties explored. The analysis combines $q$-algebra, analytic continuation of the Riemann zeta function, and a detailed examination of characteristic values and limits, unifying the combinatorial and information-theoretic perspectives across all real $q$. The results provide a new mathematical bridge between $q$-deformed combinatorics and nonextensive information geometry, with potential applications in statistical physics and information theory where Tsallis-type divergences are relevant.

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.

On the $q$-generalised multinomial/divergence correspondence

TL;DR

This work extends Wallis-type combinatorial asymptotics to a -deformed multinomial PMF within Tsallis nonextensive statistics, showing that the leading term in the large- expansion is the Tsallis relative entropy . Higher-order corrections naturally generate a one-parameter family of lambda-extended divergences , with explicit recursivity and sign properties explored. The analysis combines -algebra, analytic continuation of the Riemann zeta function, and a detailed examination of characteristic values and limits, unifying the combinatorial and information-theoretic perspectives across all real . The results provide a new mathematical bridge between -deformed combinatorics and nonextensive information geometry, with potential applications in statistical physics and information theory where Tsallis-type divergences are relevant.

Abstract

The asymptotic correspondence between the probability mass function of the -deformed multinomial distribution and the -generalised Kullback-Leibler divergence, also known as Tsallis relative entropy, is established. The probability mass function is generalised using the -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 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.
Paper Structure (13 sections, 34 equations)

This paper contains 13 sections, 34 equations.

Table of Contents

  1. Introduction
  2. The generalised multinomial/divergence correspondence
  3. Review and preliminaries
  4. The $\bm{M}_{q}$ formula and the emergent divergence family
  5. Characteristic values and limits
  6. The $\bm{P}\equiv\bm{Q}$ case
  7. Properties of the $\lambda$-extended Tsallis relative entropy
  8. The $\bm{\lambda=0}$ case
  9. The $\bm{\lambda\neq 0}$ case
  10. Relationship to Tsallis entropy
  11. Summary and Conclusion
  12. Asymptotic behaviours of $\bm{L}_{q}$
  13. Note on the $\bm{q}$-deformation of the multinomial distribution PMF