# The Bahadur representation for sample quantiles under dependent sequence

## TL;DR

This paper establishes the Bahadur representation for sample quantiles in dependent sequences, providing specific convergence rates under different mixing conditions, which enhances understanding of quantile behavior in dependent data.

## Contribution

It extends the Bahadur representation to dependent sequences with new convergence rates under weaker mixing conditions.

## Key findings

- Rate of $O(n^{-3/4}\log n)$ under $ ext{O}(n^{-3})$ mixing
- Rate of $O(n^{-1/2}(\log n)^{1/2})$ under summable mixing coefficients
- Results applicable to dependent data analysis and statistical inference

## Abstract

On the one hand, we investigate the Bahadur representation for sample quantiles under $\varphi$-mixing sequence with $\varphi(n)=O(n^{-3})$ 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 $\sum\nolimits_{n=1}^\infty\varphi^{1/2}(n)<\infty$, a rate $O(n^{-1/2}(\log n)^{1/2})$, $a.s.$, is also obtained.