Let ${X_{nk}}, u_n; leq ;k leq ;u_n,; n; geq; 1}$ be an array of rowwise independent, but not necessarily identically distributed, random variables with $EX_{nk}$=0 for all k and n. In this paper, we povide a domination condition under which ${sum^{u_n}}_=u_n; S_{nk}/n^{1/p},; 1; leq; p;<2$ converges completely to zero.