A general approach rate to the strong law of large numbers

We obtain the strong growth rate for the sum Sn of random variables by using a Hajek-Renyi type maximal inequality. Under the same conditions as that in Fazekas and Klesov [2000. A general approach to the strong law of large numbers. Theory Probab. Appl. 45, 436-449], we get sharper results for some dependent sums.