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Fractional binomial distributions induced by the generalized binomial theorem and their applications

Masanori Hino, Ryuya Namba

TL;DR

The paper develops a cohesive fractional framework for binomial-type structures using the generalized binomial theorem, introducing the α-fractional binomial distribution and the α-fractional Bernstein operator. It derives explicit moment formulas, uniform bounds, and characteristic functions, and proves fundamental limit theorems including LLN, CLT, and a fractional Poisson limit, along with a multivariate fractional multinomial extension. The operator-theoretic part shows uniform approximation by B_{α,n}, a Kelisky–Rivlin-type result for iterates, and convergence of iterates to a Wright–Fisher diffusion semigroup, with extensions to spatially varying α(x) yielding more general diffusion limits. Collectively, the work provides a unified fractional probability-approximation theory that blends generalized binomial expansions, Whitney-number structures, and diffusion limits, with potential applications to flexible count models and stochastic-process modeling.

Abstract

We develop a fractional extension of the classical binomial distribution and the associated Bernstein operator, formulated within the framework of the generalized binomial theorem (Hara and Hino [Bull.\ London Math.\ Soc. \textbf{42} (2010), 467--477]). This provides a new probabilistic structure not representable as the law of the sum of independent and identically distributed random variables. Despite this nonstandard nature, we establish several of its fundamental analytic and probabilistic properties, including limit theorems,through a unified framework based on the generalized binomial theorem.We further analyze the properties of the fractional Bernstein operator associated with the fractional binomial distribution. In particular, we prove that the iterates of the operator converge to a generalized Wright--Fisher diffusion semigroup after a proper diffusive rescaling.

Fractional binomial distributions induced by the generalized binomial theorem and their applications

TL;DR

The paper develops a cohesive fractional framework for binomial-type structures using the generalized binomial theorem, introducing the α-fractional binomial distribution and the α-fractional Bernstein operator. It derives explicit moment formulas, uniform bounds, and characteristic functions, and proves fundamental limit theorems including LLN, CLT, and a fractional Poisson limit, along with a multivariate fractional multinomial extension. The operator-theoretic part shows uniform approximation by B_{α,n}, a Kelisky–Rivlin-type result for iterates, and convergence of iterates to a Wright–Fisher diffusion semigroup, with extensions to spatially varying α(x) yielding more general diffusion limits. Collectively, the work provides a unified fractional probability-approximation theory that blends generalized binomial expansions, Whitney-number structures, and diffusion limits, with potential applications to flexible count models and stochastic-process modeling.

Abstract

We develop a fractional extension of the classical binomial distribution and the associated Bernstein operator, formulated within the framework of the generalized binomial theorem (Hara and Hino [Bull.\ London Math.\ Soc. \textbf{42} (2010), 467--477]). This provides a new probabilistic structure not representable as the law of the sum of independent and identically distributed random variables. Despite this nonstandard nature, we establish several of its fundamental analytic and probabilistic properties, including limit theorems,through a unified framework based on the generalized binomial theorem.We further analyze the properties of the fractional Bernstein operator associated with the fractional binomial distribution. In particular, we prove that the iterates of the operator converge to a generalized Wright--Fisher diffusion semigroup after a proper diffusive rescaling.
Paper Structure (23 sections, 31 theorems, 208 equations, 1 figure, 1 table)

This paper contains 23 sections, 31 theorems, 208 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Fractional binomial distribution and fractional Bernstein operator
  3. Outline of the paper
  4. Future perspectives
  5. Remarks
  6. Notation
  7. The fractional binomial distribution and its properties
  8. A fractional-order Taylor series
  9. Moments of the fractional binomial distribution
  10. Uniform bound of $Z_{\alpha, x}^{(n)}$
  11. Uniform estimates for moments of the fractional binomial distribution
  12. Characteristic function of the fractional binomial distribution
  13. The fractional multinomial distribution
  14. Limit theorems for the fractional binomial distribution
  15. The weak law of large numbers
  16. ...and 8 more sections

Key Result

Proposition 1.1

For $\alpha>0$, $n \in \mathbb{N}$, and $\lambda \in (0, 1]$, we have Here, the second term on the right-hand side is regarded as zero if $\alpha/2 \in \mathbb{N}$.

Figures (1)

  • Figure 1: Variation of the fractional binomial distribution $\mu_{\alpha, x}^{(n)}$ with respect to $\alpha$. Shown are the probability mass functions for $n=20$ and $x=0.35$, where the parameter $\alpha$ takes the values $0.2$, $0.6$, $1$, $1.4$, $1.8$, and $2.2$.

Theorems & Definitions (61)

  • Proposition 1.1: HH
  • Definition 1.2: Fractional binomial distribution
  • Definition 1.3: Fractional Bernstein operator
  • Proposition 2.1: cf. HH
  • Proposition 2.2: cf. HH
  • Lemma 2.3
  • proof
  • Remark 2.4
  • Theorem 2.5
  • proof
  • ...and 51 more