A DeMoivre-Laplace theorem of all orders of regularity

Abstract

The DeMoivre-Laplace Theorem states that the binomial probability distribution B(N; 1/2) tends for N to infinity to the Gaussian distribution. We extend this theorem to the difference quotients of the family of the binomial distributions with varying N, showing that they converge to the corresponding differential quotients of the time-dependent Gaussian distribution. The convergence holds for difference quotients of all order.