Precise rates in complete moment convergence for ρ-mixing sequences ✩

Let X1,X2, . . . be a strictly stationary sequence of ρ-mixing random variables with mean zeros and positive, finite variances, set Sn = X1 + ··· + Xn. Suppose that limn→∞ES2 n/n = σ2 > 0, ∞n=1 ρ2/q (2n) < ∞, where q > 2δ + 2. We prove that, if EX2 1(log+ |X1|)δ <∞for any 0 < δ 1, then lim ↓0 2δ ∞ n=2 (log n)δ−1 n2 ES2 nI |Sn| σ n log n = E|N|2δ+2 δ , where N is the standard normal random variable. © 2007 Elsevier Inc. All rights reserved.