Quick and Complete Convergence in the Law of Large Numbers with Applications to Statistics

Abstract

In the first part of this article, we discuss and generalize the complete convergence introduced by Hsu and Robbins in 1947 to the r-complete convergence introduced by Tartakovsky in 1998. We also establish its relation to the r-quick convergence first introduced by Strassen in 1967 and extensively studied by Lai. Our work is motivated by various statistical problems, mostly in sequential analysis. As we show in the second part, generalizing and studying these convergence modes is important not only in probability theory but also to solve challenging statistical problems in hypothesis testing and changepoint detection for general stochastic non-i.i.d. models.