On Bahadur representation for sample quantiles under α-mixing sequence

In this paper, by relaxing the mixing coefficients to α(n) = O(n −β ), β > 3, we investigate the Bahadur representation of sample quantiles under α-mixing sequence and obtain the rate as $${O(n^{-\frac{1}{2}}(\log\log n\cdot\log n)^{\frac{1}{2}})}$$ . Meanwhile, for any δ > 0, by strengthening the mixing coefficients to α(n) = O(n −β ), $${\beta > \max\{3+\frac{5}{1+\delta},1+\frac{2}{\delta}\}}$$ , we have the rate as $${O(n^{-\frac{3}{4}+\frac{\delta}{4(2+\delta)}}(\log\log n\cdot \log n)^{\frac{1}{2}})}$$ . Specifically, if $${\delta=\frac{\sqrt{41}-5}{4}}$$ and $${\beta > \frac{\sqrt{41}+7}{2}}$$ , then the rate is presented as $${O(n^{-\frac{\sqrt{41}+5}{16}}(\log\log n\cdot \log n)^{\frac{1}{2}})}$$ . Copyright Springer-Verlag 2014