On some properties of a positive linear operator via the Moran model in population genetics
Takahiro Aoyama, Ryuya Namba
TL;DR
This work defines the Moran operator $\mathcal{M}_n$ on $C([0,1])$ from the Moran population-genetics model and proves it uniformly approximates any continuous function. It then establishes a Kelisky–Rivlin type limit for iterates, showing $\mathcal{M}_n^k f$ converges to the linear interpolation $f(1)x+f(0)(1-x)$ as $k\to\infty$. A diffusion-limit result reveals that suitably scaled iterates converge to the Wright–Fisher diffusion semigroup, with a rate $O(n^{-1})$ under extra smoothness, and a functional limit (Donsker-type) showing path convergence in Hölder spaces. The findings connect operator theory with diffusion models in population genetics and suggest pathways to jump-process limits via Cannings-type generalizations, enriching both approximation theory and probabilistic modelling.
Abstract
We introduce a positive linear operator acting on the Banach space of all continuous functions on the unit interval via the Moran model studied in population genetics. We show that this operator, named the Moran operator, uniformly approximates every continuous function on the unit interval. Furthermore, some limit theorems for the iterates of the Moran operator are obtained.