On complete convergence and complete moment convergence for weighted sums of ρ∗ $\rho^{*}$-mixing random variables

Abstract

Abstract Let r≥1 $r\geq1$, 1≤p0 $\alpha, \beta>0$ with 1/α+1/β=1/p $1/\alpha+1/\beta=1/p$. Let {ank,1≤k≤n,n≥1} $\{a_{nk}, 1\leq k\leq n,n\geq1\}$ be an array of constants satisfying supn≥1n−1∑k=1n|ank|αrp $\alpha>rp$, we provide moment conditions under which ∑n=1∞nr−2P{max1≤m≤n|∑k=1mankXk|>εn1/p}0. $$\sum^{\infty}_{n=1}n^{r-2}P \Biggl\{ \max_{1\leq m\leq n} \Biggl\vert \sum^{m}_{k=1}a_{nk}X_{k} \Biggr\vert >\varepsilon n^{1/p} \Biggr\} 0. $$ We also provide moment conditions under which ∑n=1∞nr−2−q/pE(max1≤m≤n|∑k=1mankXk|−εn1/p)+q0, $$\sum^{\infty}_{n=1}n^{r-2-q/p} E \Biggl( \max_{1\leq m\leq n} \Biggl\vert \sum^{m}_{k=1}a_{nk}X_{k} \Biggr\vert -\varepsilon n^{1/p} \Biggr)_{+}^{q}0, $$ where q>0 $q>0$. Our results improve and generalize those of Sung (Discrete Dyn. Nat. Soc. 2010:630608, 2010) and Wu et al. (Stat. Probab. Lett. 127:55–66, 2017).

Keywords

• ρ ∗ $\rho^{*}$ -mixing random variables
• Complete convergence
• Complete moment convergence
• Weighted sum