TY - JOUR

T1 - A fully stochastic approach to limit theorems for iterates of Bernstein operators

AU - Konstantopoulos, Takis

AU - Yuan, Linglong

AU - Zazanis, Michael A.

N1 - Publisher Copyright:
© 2017 Elsevier GmbH

PY - 2018/6

Y1 - 2018/6

N2 - This paper presents a stochastic approach to theorems concerning the behavior of iterations of the Bernstein operator Bn taking a continuous function f∈C[0,1] to a degree-n polynomial when the number of iterations k tends to infinity and n is kept fixed or when n 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 Bn a number of times k=k(n) to a polynomial f when k(n)∕n tends to a constant.

AB - This paper presents a stochastic approach to theorems concerning the behavior of iterations of the Bernstein operator Bn taking a continuous function f∈C[0,1] to a degree-n polynomial when the number of iterations k tends to infinity and n is kept fixed or when n 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 Bn a number of times k=k(n) to a polynomial f when k(n)∕n tends to a constant.

KW - Bernstein operator

KW - Diffusion approximation

KW - Markov chains

KW - Stochastic calculus

KW - Stochastic compositions

KW - Wright–Fisher model

UR - http://www.scopus.com/inward/record.url?scp=85035139236&partnerID=8YFLogxK

U2 - 10.1016/j.exmath.2017.10.001

DO - 10.1016/j.exmath.2017.10.001

M3 - Article

AN - SCOPUS:85035139236

SN - 0723-0869

VL - 36

SP - 143

EP - 165

JO - Expositiones Mathematicae

JF - Expositiones Mathematicae

IS - 2

ER -