Precise asymptotics in complete moment convergence of moving-average processes

In this paper, we discuss moving-average process X-k = Sigma(infinity)(i=-infinity) a(i+k)epsilon(i), where {epsilon(i); -infinity < i < infinity) is a doubly infinite sequence of i.i.d random variables with mean zeros and finite variances, {a(i);-infinity<i<infinity) is an absolutely summable sequence of real numbers. Set S-n = Sigma(n)(k=1) ,X-k, n >= 1. Suppose E\epsilon(1)\(3)<infinity, we prove that, if E\epsilon(1)\(r)<infinity, for 1<p<2 and r>1 +p/2, then lim(epsilon SE arrow 0) epsilon(2(e-p)/(2-p)-1) Sigma(infinity)(n=1) n(r/p-2-1/p) E{\S-n\ - epsilon n(1/p)}(+) = p(2-p)/(r-p)(2r- p-2) E\Z](2(r-p)/(2-p)), where Z has a normal distribution with mean 0 and variance tau(2) = sigma(2)(Sigma(infinity)(i=-infinity)a(i))(2). (C) 2006 Elsevier B.V. All rights reserved.