Complete q th moment convergence for arrays of random variables

Let {Xni,1≤i≤n,n≥1} be an array of random variables with EXni=0 and E|Xni|q<∞ for some q≥1. For any sequences {an,n≥1} and {bn,n≥1} of positive real numbers, sets of sufficient conditions are given for complete q th moment convergence of the form ∑n=1∞bnan−qE(max1≤k≤n|∑i=1kXni|−ϵan)+q<∞, ∀ϵ>0, where x+=max{x,0}. From these results, we can easily obtain some known results on complete q th moment convergence.