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

• Published in: Journal of inequalities and applications Vol. 2018; no. 1; pp. 1 - 16
• Main Authors: Chen, Pingyan, Sung, Soo Hak
• Format: Journal Article
• Language: English
• Published: SpringerOpen 01.06.2018
• Subjects: Complete convergence; Complete moment convergence; Weighted sum; ρ ∗ $\rho^{}$ -mixing random variables
• ISSN: 1029-242X; 1029-242X
• DOI: 10.1186/s13660-018-1710-2

Abstract Let r≥1 $r\geq1$, 1≤p<2 $1\leq p<2$, and α,β>0 $\alpha, \beta>0$ 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|α<∞ $\sup_{n\geq1}n^{-1}\sum^{n}_{k=1}|a_{nk}|^{\alpha}<\infty$, and let {Xn,n≥1} $\{ X_{n},n\geq1\}$ be a sequence of identically distributed ρ∗ $\rho^{*}$-mixing random variables. For each of the three cases α<rp $\alpha< rp$, α=rp $\alpha=rp$, and α>rp $\alpha>rp$, we provide moment conditions under which ∑n=1∞nr−2P{max1≤m≤n|∑k=1mankXk|>εn1/p}<∞,∀ε>0. $$\sum^{\infty}_{n=1}n^{r-2}P \Biggl\{ \max_{1\leq m\leq n} \Biggl\vert \sum^{m}_{k=1}a_{nk}X_{k} \Biggr\vert >\varepsilon n^{1/p} \Biggr\} < \infty,\quad \forall \varepsilon>0. $$ We also provide moment conditions under which ∑n=1∞nr−2−q/pE(max1≤m≤n|∑k=1mankXk|−εn1/p)+q<∞,∀ε>0, $$\sum^{\infty}_{n=1}n^{r-2-q/p} E \Biggl( \max_{1\leq m\leq n} \Biggl\vert \sum^{m}_{k=1}a_{nk}X_{k} \Biggr\vert -\varepsilon n^{1/p} \Biggr)_{+}^{q}< \infty,\quad \forall\varepsilon>0, $$ 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).