Some results on complete convergence, complete moment convergence, and complete integral convergence

In this correspondence, we study the relationship between the concepts of complete convergence, complete moment convergence, and complete integral convergence. More specifically, complete moment convergence of order q>0 is shown to be equivalent to complete integral convergence of the same order. A triumvirate of conditions are provided for complete moment convergence of the row sums from an array of rowwise independent random variables. Several illustrative examples are presented.