On Bahadur representation for sample quantiles under α-mixing sequence

In this paper, by relaxing the mixing coefficients to α(n) = O(n <Superscript>−β</Superscript>), β > 3, we investigate the Bahadur representation of sample quantiles under α-mixing sequence and obtain the rate as <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$${O(n^{-\frac{1}{2}}(\log\log n\cdot\log n)^{\frac{1}{2}})}$$</EquationSource> </InlineEquation>. Meanwhile, for any δ > 0, by strengthening the mixing coefficients to α(n) = O(n <Superscript> −β </Superscript>), <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$${\beta > \max\{3+\frac{5}{1+\delta},1+\frac{2}{\delta}\}}$$</EquationSource> </InlineEquation>, we have the rate as <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$${O(n^{-\frac{3}{4}+\frac{\delta}{4(2+\delta)}}(\log\log n\cdot \log n)^{\frac{1}{2}})}$$</EquationSource> </InlineEquation>. Specifically, if <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$${\delta=\frac{\sqrt{41}-5}{4}}$$</EquationSource> </InlineEquation> and <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$${\beta > \frac{\sqrt{41}+7}{2}}$$</EquationSource> </InlineEquation>, then the rate is presented as <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$${O(n^{-\frac{\sqrt{41}+5}{16}}(\log\log n\cdot \log n)^{\frac{1}{2}})}$$</EquationSource> </InlineEquation>. Copyright Springer-Verlag 2014