Let {$Y_i$,-$infty$ < i < $infty$} be a doubly infinite sequence of i.i.d. random variables with E|$Y_1$| < $infty$, {$a_{ni}$,-$infty$ < i < $infty$ n $geq$ 1} an array of real numbers. Under some conditions on {$a_{ni}$}, we obtain necessary and sufficient conditions for $sum;_{n=1}^{infty}frac{1}{n}P(|sum;_{i=-infty}^{infty}a_{ni}(Y_i-EY_i)|$>$n{epsilon})$<{infty}$. We examine whether the result of Spitzer [11] holds for the moving average process, and give a partial solution.