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

Let ({X,X_n,nge 1}) be a sequence of identically distributed (rho ^*)-mixing random variables, ({a_{nk}, 1le kle n, nge 1}) an array of real numbers with (sup _{nge 1}n^{-1}sum ^n_{k=1}|a_{nk}|^alpha u003cinfty ) for some (0u003calpha le 2). Under the almost optimal moment conditions, the paper shows that $$begin{aligned} sum ^infty _{n=1}n^{-1}Pbigg {max _{1le mle n}bigg |sum ^m_{k=1}a_{nk}X_k bigg |u003evarepsilon n^{1/alpha }(log n)^{1/gamma }bigg } 0, end{aligned}$$where (0u003cgamma u003calpha ). 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.