The Bahadur representation for sample quantiles under dependent sequence

Abstract

On the one hand, we investigate the Bahadur representation for sample quantiles under $\phi$-mixing sequence with $\phi (n)=O(n^{-3})$ and obtain a rate as $O(n^{-\frac{3}{4}}\log n)$, $a.s.$. On the other hand, by relaxing the condition of mixing coefficients to ${\sum }_{n=1}^{\infty }{\phi }^{1/2}(n)<\infty$, a rate $O(n^{-1/2}(\log n{)}^{1/2})$, $a.s.$, is also obtained.