Let \((\xi_ k)\) be a sequence of functions on a finite measure space (X,A,\(\mu)\) such that \(\xi_ k\in L_ 2\) and \(M\xi_ k=0\). If the sequence \((\xi_ k)\) is (C,\(\alpha)\)-summable to zero almost everywhere for some \(\alpha >0\), then one says that the \(\alpha\)-strong law of large numbers (\(\alpha\)-s.l.l.n.) holds. The estimate \(S_ n(x)=\sum^{n}_{k=0}\xi_ k(x)=o(n^{\alpha})\) a.e. \((0<\alpha <1)\) is sufficient for the \(\alpha\)-s.l.l.n. The author derives several other conditions sufficient for the \(\alpha\)-s.l.l.n.