  This paper presents a stochastic approach to theorems concerning the behavior
of iterations of the Bernstein operator $B_n$ taking a continuous function $f
\in C[0,1]$ 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.

Probability