Let ${X, X_n, n geq 1}$ be a sequence of independent and identically distributed(i.i.d) random variables with EX = 0 and $E$mid$X$mid$^p < infty$</TEX> for some $p geq 1$. Let ${a_{ni}, 1 leq i leq n, n geq 1}$ be a triangular arrary of constants. The almost sure(a.s) convergence of weighted sums $sum_{i=1}^{n} a_{ni}X_i$ can be founded in Choi and Sung[1], Chow[2], Chow and Lai[3], Li et al. [4], Stout[6], Sung[8], Teicher[9], and Thrum[10].