A fully stochastic approach to limit theorems for iterates of Bernstein operators



Konstantopoulos, Takis, Yuan, Linglong ORCID: 0000-0002-7851-1631 and Zazanis, Michael A
(2018) A fully stochastic approach to limit theorems for iterates of Bernstein operators. Expositiones Mathematicae, 36 (2). pp. 143-165.

[thumbnail of BernsIter_Final.pdf] Text
BernsIter_Final.pdf - Author Accepted Manuscript

Download (324kB) | Preview

Abstract

This paper presents a stochastic approach to theorems concerning the behavior of iterations of the Bernstein operator taking a continuous function to a degree- polynomial when the number of iterations tends to infinity and is kept fixed or when tends to infinity as well. In the first instance, the underlying stochastic process is the so-called Wright–Fisher model, whereas, in the second instance, the underlying stochastic process is the Wright–Fisher diffusion. Both processes are probably the most basic ones in mathematical genetics. By using Markov chain theory and stochastic compositions, we explain probabilistically a theorem due to Kelisky and Rivlin, and by using stochastic calculus we compute a formula for the application of a number of times to a polynomial when tends to a constant.

Item Type: Article
Uncontrolled Keywords: Bernstein operator, Markov chains, Stochastic compositions, Wright-Fisher model, Stochastic calculus, Diffusion approximation
Depositing User: Symplectic Admin
Date Deposited: 01 Apr 2020 11:02
Last Modified: 18 Jan 2023 23:56
DOI: 10.1016/j.exmath.2017.10.001
Related URLs:
URI: https://livrepository.liverpool.ac.uk/id/eprint/3081328