Remark on convergence rate for weighted sums of $$\rho ^*$$ ρ ∗ -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 $$\begin{aligned} \sum ^\infty _{n=1}n^{-1}P\bigg \{\max _{1\le m\le n}\bigg |\sum ^m_{k=1}a_{nk}X_k \bigg |>\varepsilon n^{1/\alpha }(\log n)^{1/\gamma }\bigg \} 0, \end{aligned}$$ 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.