The Bahadur Representation for Sample Quantiles Under Dependent Sequence

On the one hand, we investigate the Bahadur representation for sample quantiles under φ-mixing sequence with φ(n) = O(n−3) and obtain a rate as O(n−34logn)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O\left(n^{-\frac{3}{4}} \log n\right)$$\end{document}, a.s. On the other hand, by relaxing the condition of mixing coefficients to ∑n=1∞φ1/2(n)<∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sum_{n=1}^{\infty} \varphi^{1 / 2}(n)<\infty$$\end{document}, a rate O(n−1/2(log n)1/2), a.s., is also obtained.