The study of the

Let n and n be respectively the uniform empirical and quantile processes , and deene R n = n + n , which usually is referred to as the Bahadur{Kiefer process. The well-known Bahadur{Kiefer theorem connrms the following remarkable equivalence: kR n k= p k n k n ?1=4 (log n) 1=2 almost surely, as n goes to innnity, where kfk = sup 0t1 jf(t)j is the L 1-norm. We prove that kR n k 2 = p k n k 1 n ?1=4 almost surely, where k k 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 kR n k p = p k n k (p=2) converges almost surely to a nite positive explicit constant. We also extend our result to a more general Bahadur{Kiefer process.