A note on the Berry–Esseen bounds for $\rho$-mixing random variables and their application

Recently, Wang and Hu [Theory Probab. Appl., 63 (2019), pp. 479–499] established the Berry–Esseen bounds for $\rho$-mixing random variables (r.v.'s) with the rate of normal approximation $O(n^{-1/6}\log n)$ by using the martingale method. In this paper, we establish some general results on the rates of normal approximation, which include the corresponding ones of Wang and Hu. The rate can be as high as $O(n^{-1/5})$ or $O(n^{-1/4}\log^{1/2} n)$ under some suitable conditions. As applications, we obtain the Berry–Esseen bounds of sample quantiles based on $\rho$-mixing random samples. Finally, we also present some numerical simulations to demonstrate finite sample performances of the theoretical result.