Let ${X, X_n, n geq 1}$ be a sequence of independent and indentically distributed(i, i, d) random variables. Let ${a_{ni}, 1 leq i leq n, geq 1}$ be a triangular array of constants. The purpose of this paper is to find sufficient conditions on $a_{ni}$ and X such that $$ (1) sum_{i = 1}^{n} a_{ni}(X_i - EX_i) o 0 a.s.$$