Error analysis of Aitken's Δ2 process

AbstractA sequence to sequence transformation, called the Δ2 process by its developer Aitken, and recently analyzed by Daniel Shanks, is successful in accelerating convergence in many convergent sequences and inducing convergence in some divergent ones. It is shown here that the Δ2 process applied to sequences whose terms have Cauchy distributions results in sequences whose terms still have the Cauchy distribution and that repeated applications of the Δ2 process to a sequence with terms having uniform distribution (simulating round-off error) and to sequences with terms having a normal distribution (simulating measurement error) yields, in both cases, sequences whose terms approach the Cauchy distribution. The result for the uniform distribution is proven, that for the normal distribution is referenced