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<p>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 in Bull Lond Math Soc 42:467–477, 2010). 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.</p>

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

  • 期刊论文

  • 发表日期：

    2026年3月19日

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  • Masanori Hino,
  • Ryuya Namba

摘要

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 in Bull Lond Math Soc 42:467–477, 2010). 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.