For $N = 1,2,\cdots$ let $S_N = \sum_k a_{N,k}X_k$ where $a_{N,k}$ is a real number for $N,k = 1,2, \cdots$ and $\{Xk\}$ is a sequence of not necessarily independent random variables. For the case $0 < t < 1$, with assumptions closely related to $E|X_k|^t < \infty$ it is shown that the rate of convergence of $P(|S_N| > \varepsilon)$ to zero is related to $\sum_k |a_{N,k}|^t$. The theorems presented here extend some of the results in the literature to not necessarily independent sequences $\{X_k\}$.