ON SOME SERIES APPROXIMATIONS ARISING FROM THE PROBABILISTIC PROOF OF WEIERSTRASS APPROXIMATION

In this paper, we first present Bernstein’s proof of the Weierstrass approximation theorem using probabilistic arguments. We then apply the method to develop approximations for a Dirichlet series on the critical strip. We prove two limit theorems, the first one by considering sums of independent Bernoulli random variables, and the second one by looking at sums of independent geometric random variables.