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

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.