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

TL;DR

This paper introduces a fractional extension of the binomial distribution based on the generalized binomial theorem, exploring its properties, limit theorems, and applications to fractional Bernstein operators and diffusion processes.

Contribution

It develops a novel fractional binomial distribution framework and analyzes its fundamental properties and applications, extending classical probabilistic and operator theories.

Findings

Established limit theorems for the fractional binomial distribution

Proved convergence of fractional Bernstein operator iterates to a Wright--Fisher diffusion

Provided a unified framework based on the generalized binomial theorem

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.