An L_p-view of the Bahadur-Kiefer Theorem

Summary

Let <InlineEquation ID=IE”1”><EquationSource Format=”TEX”><![CDATA[<InlineEquation ID=IE”2”><EquationSource Format=”TEX”><![CDATA[<InlineEquation ID=IE”3”><EquationSource Format=”TEX”><![CDATA[<InlineEquation ID=IE”4”><EquationSource Format=”TEX”><![CDATA[<InlineEquation ID=IE”5”><EquationSource Format=”TEX”><![CDATA[<InlineEquation ID=IE”6”><EquationSource Format=”TEX”><![CDATA[<InlineEquation ID=IE”7”><EquationSource Format=”TEX”><![CDATA[<InlineEquation ID=IE”8”><EquationSource Format=”TEX”><![CDATA[<InlineEquation ID=IE”9”><EquationSource Format=”TEX”><![CDATA[<InlineEquation ID=IE”10”><EquationSource Format=”TEX”><![CDATA[<InlineEquation ID=IE”11”><EquationSource Format=”TEX”><![CDATA[<InlineEquation ID=IE”12”><EquationSource Format=”TEX”><![CDATA[$]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>\alpha_n$ and $\beta_n$ be respectively the uniform empirical and quantile processes, and define $R_n = \alpha_n + \beta_n$, which usually is referred to as the Bahadur--Kiefer process. The well-known Bahadur-Kiefer theorem confirms the following remarkable equivalence: $\|R_n\| /\sqrt{\| \alpha_n \|  }\, \sim \, n^{-1/4} (\log n)^{1/2}$ almost surely, as $n$ goes to infinity, where $\| f\| =\sup_{0\le t\le 1} |f(t)|$ is the $L^\infty$-norm. We prove that $\|R_n\|_2 /\sqrt{\| \alpha_n \|_1}\, \sim \, n^{-1/4}$ almost surely, where $\| \, \cdot \, \|_p$ is the $L^p$-norm. It is interesting to note that there is no longer any logarithmic term in the normalizing function. More generally, we show that $n^{1/4} \|R_n\|_p /\sqrt{\| \alpha_n \|_{(p/2)}}$ converges almost surely to a finite positive constant whose value is explicitly known.