A strong law of large numbers for nonnegative random variables and applications

For a sequence of nonnegative random variables {Xn,n≥1}{Xn,n≥1} with finite means and partial sums Sn=∑i=1nXi,n≥1, and a sequence of positive numbers {bn,n≥1}{bn,n≥1} with bn↑∞bn↑∞, sufficient conditions are given under which (Sn−ESn)/bn→0(Sn−ESn)/bn→0 almost surely. Our result generalizes the strong law of large numbers obtained by Korchevsky (2015). Some applications for dependent random variables are also provided.