On complete convergence and complete moment convergence for weighted sums of ... formula ...-mixing random variables

摘要

Let r ≥ 1, 1 ≤ p  2, and α, β  0 with 1/α + 1/β = 1/p. Let {ank, 1 ≤ k ≤ n, n ≥ 1} be an array of constants satisfying supn≥1n−1∑k=1n|ank|α<∞, and let {Xn, n ≥ 1} be a sequence of identically distributed ρ^∗-mixing random variables. For each of the three cases α  rp, α = rp, and α  rp, we provide moment conditions under which ∑n=1∞nr−2P{max1≤m≤n|∑k=1mankXk|>εn1/p}<∞,∀ε>0. We also provide moment conditions under which ∑n=1∞nr−2−q/pE(max1≤m≤n|∑k=1mankXk|−εn1/p)+q<∞,∀ε>0, where q  0. Our results improve and generalize those of Sung (Discrete Dyn. Nat. Soc. 2010:630608, ) and Wu et al. (Stat. Probab. Lett. 127:55–66, ).