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

作者：Konstantopoulos, Takis^*; Yuan, Linglong; Zazanis, Michael A.

来源：Expositiones Mathematicae, 2018, 36(2): 143-165.

DOI：10.1016/j.exmath.2017.10.001

• Bernstein operator
• Markov chains
• Stochastic compositions
• Wright-Fisher model
• Stochastic calculus
• Diffusion approximation

This paper presents a stochastic approach to theorems concerning the behavior of iterations of the Bernstein operator B-n taking a continuous function f is an element of 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 B-n a number of times k = k(n) to a polynomial f when k(n)/n tends to a constant.

• 出版日期2018
• 单位西交利物浦大学