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

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.

💡 Research Summary

The paper introduces a fractional extension of the classical binomial distribution and its associated Bernstein operator by exploiting the generalized binomial theorem of Hara and Hino. For a real order α > 0 and integer n ≥ 1, the authors define fractional binomial coefficients
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