Letbe random variables anda triangular array of constants. In this paper we find the various conditions on {ani} and {Xn} under whichconverges to zero almost surely. It is shown that for the identically distributed random variables the conditionsa.s. A generalization of Marcinkiewicz's law of large numbers is given. It is shown that for the i.i.d. case the conditionsandEX= 0 imply thata.s