On complete convergence for weighted sums of i.i.d. random variables with application to moving average processes
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Let {Y_(i),−∞ < i < ∞} be a doubly infinite sequence of i.i.d. random variables with E|Y_(1)|<∞, {a_(ni),−∞ < i < ∞, n≥1} an array of real numbers. Under some conditions on {a_(ni)}, we obtain necessary and sufficient conditions for ∑_(n=1)^(∞)1/nP(│∑_(i=-∞)^(∞)a_(ni){Y_(i)-EY_(i)│＞ne)<∞. We examine whether the result of Spitzer [11] holds for the moving average process, and give a partial solution.
Abstract
Let {Y_(i),−∞ < i < ∞} be a doubly infinite sequence of i.i.d. random variables with E|Y_(1)|<∞, {a_(ni),−∞ < i < ∞, n≥1} an array of real numbers. Under some conditions on {a_(ni)}, we obtain necessary and sufficient conditions for ∑_(n=1)^(∞)1/nP(│∑_(i=-∞)^(∞)a_(ni){Y_(i)-EY_(i)│＞ne)<∞. We examine whether the result of Spitzer [11] holds for the moving average process, and give a partial solution.
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