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

Abstract

Let $r\ge 1$ , $1\le p<2$ , and $\alpha ,\beta >0$ with $1/\alpha +1/\beta =1/p$ . Let $\{a_{nk},1\le k\le n,n\ge 1\}$ be an array of constants satisfying ${sup}_{n\ge 1}{n}^{-1}{\sum }_{k=1}^n{\left|a_{nk}\right|}^{\alpha }<\infty$ , and let $\{X_n,n\ge 1\}$ be a sequence of identically distributed ${\rho }^{\ast }$ -mixing random variables. For each of the three cases $\alpha <rp$ , $\alpha =rp$ , and $\alpha >rp$ , we provide moment conditions under which $\sum _{n=1}^{\infty }{n}^{r-2}P\left\{\underset{1\le m\le n}{max}\right|\sum _{k=1}^m{a}_{nk}{X}_k|>\epsilon n^{1/p}\}<\infty ,\forall \epsilon >0.$ We also provide moment conditions under which $\sum _{n=1}^{\infty }{n}^{r-2-q/p}E{\left(\underset{1\le m\le n}{max}\right|\sum _{k=1}^m{a}_{nk}{X}_k|-\epsilon n^{1/p})}_{+}^q<\infty ,\forall \epsilon >0,$ where $q>0$ . 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).