Asymptotic behaviors of fractional binomial distributions derived from the generalized binomial theorem

TL;DR

This paper explores the asymptotic properties of fractional binomial distributions, including deviation principles and convergence rates, extending classical binomial results to a fractional context.

Contribution

It derives large and moderate deviation principles and a Berry--Esseen estimate for fractional binomial distributions, advancing understanding of their asymptotic behavior.

Findings

Established large deviation principles

Derived moderate deviation principles

Provided Berry--Esseen type estimates

Abstract

A fractional binomial distribution, introduced by Hino and Namba (2024) via the generalized binomial theorem, is a fractional variant of the classical binomial distribution. Building upon previous work that established limit theorems, such as the weak law of large numbers and the central limit theorem, for fractional binomial distributions, this paper further investigates their asymptotic behaviors. Specifically, we derive the large and moderate deviation principles and a Berry--Esseen type estimate for these distributions.