In the first part of this article, we discuss and generalize the complete convergence introduced by Hsu and Robbins (1947) to the r-complete convergence introduced by Tartakovsky (1998). We also establish its relation to the r-quick convergence first introduced by Strassen (1967) and extensively studied by Lai (1976). 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.