The Bernstein operator is a positive linear operator on the Banach space of continuous functions on $[0, 1]$, which is used to show the celebrated Weierstrass approximation theorem from a probabilistic perspective. In this talk, we introduce an extension of the Bernstein operator to the
$d$-dimensional cases and discuss some limit theorems for the iterates of the operator. As the limit, we capture the $d$-dimensional Wright-Fisher diffusion with mutation which is well-studied in population genetics. Some further possible directions of these limit theorems are discusses as well.
Based on a joint work with Takatoshi Hirano.