On a fractional binomial process

TL;DR

This paper introduces a fractional generalization of the classical binomial process, incorporating non-Markovian dynamics, which better models real physical processes and retains the binomial limit at large times.

Contribution

The paper develops a fractional binomial process using fractional calculus, extending the classical model to include non-Markovian fluctuations at small times.

Findings

Preserves the binomial limit at large times.

Includes non-binomial fluctuations at small times.

More suitable for modeling real physical processes.

Abstract

The classical binomial process has been studied by \citet{jakeman} (and the references therein) and has been used to characterize a series of radiation states in quantum optics. In particular, he studied a classical birth-death process where the chance of birth is proportional to the difference between a larger fixed number and the number of individuals present. It is shown that at large times, an equilibrium is reached which follows a binomial process. In this paper, the classical binomial process is generalized using the techniques of fractional calculus and is called the fractional binomial process. The fractional binomial process is shown to preserve the binomial limit at large times while expanding the class of models that include non-binomial fluctuations (non-Markovian) at regular and small times. As a direct consequence, the generality of the fractional binomial model makes the proposed model more desirable than its classical counterpart in describing real physical processes. More statistical properties are also derived.