Remark on convergence rate for weighted sums of $\rho ^*$-mixing random variables

Abstract

Let $\{X,X_n,n\ge 1\}$ be a sequence of identically distributed $\rho ^*$-mixing random variables, $\{a_{nk}, 1\le k\le n, n\ge 1\}$ an array of real numbers with $\sup _{n\ge 1}n^{-1}\sum ^n_{k=1}|a_{nk}|^\alpha <\infty$ for some $0<\alpha \le 2$. Under the almost optimal moment conditions, the paper shows that

$$\begin{aligned} \sum ^\infty _{n=1}n^{-1}P\bigg \{\max _{1\le m\le n}\bigg |\sum ^m_{k=1}a_{nk}X_k \bigg |>\varepsilon n^{1/\alpha }(\log n)^{1/\gamma }\bigg \}<\infty ,\quad \ \forall \ \varepsilon >0, \end{aligned}$$

where $0<\gamma <\alpha$. The main result extends that of Chen and Sung (Statist Probab Lett 92:45–52, 2014) from negatively associated random variables to $\rho ^*$-mixing random variables and the method of the proof is different completely.