An L p -view of the Bahadur-Kiefer Theorem
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AN Lp-VIEW OF THE BAHADUR–KIEFER THEOREM ´ s Cso ¨ rgo ˝ ∗ (Ottawa) and Zhan Shi∗∗ (Paris) Miklo Dedicated to Endre Cs´ aki and P´ al R´ev´esz on the occasion of their 70th birthdays
Abstract Let αn and βn be respectively the uniform empirical and quantile processes, and define Rn = αn +βn , which usually is referred to as the Bahadur–Kiefer process. The well-known Bahadur–Kiefer theorem confirms the following remarkable equivalence: Rn / αn ∼ n−1/4 (log n)1/2 almost surely, as n goes to infinity, where f = sup0≤t≤1 |f (t)| is the L∞ -norm. We prove that Rn 2 / αn 1 ∼ n−1/4 almost surely, where · p is the Lp -norm. It is interesting to note that there is no longer any logarithmic term in the normalizing function. More generally, we show that n1/4 Rn p / αn (p/2) converges almost surely to a finite positive constant whose value is explicitly known.
Prologue This paper is an updated version of our 1998 technical report [10]. An Lp view of a more general version [11] that is based on these results had already been published by us in 2001. However our original results together with their proofs are being published here first at the same time. In addition to being the bases on which [11] is built, our 1998 technical report [10] has also played a seminal role in the papers [7], [14], [15] and [16]. We are pleased to have the honour of publishing this updated version in celebration of the work of our distinguished friends, Endre Cs´aki and P´ al R´ev´esz, on the occasion of their 70th birthdays.
Mathematics subject classification number: Primary 60F25; Secondary 62G30. Key words and phrases: Empirical process, quantile process, Bahadur–Kiefer representation, Lp -modulus of continuity for Brownian motion, Brownian bridge, Kiefer process. ∗ Research supported by an NSERC Canada Grant at Carleton University, Ottawa, and by a Paul Erd˝ os Visiting Professorship of the Paul Erd˝ os Summer Research Center of Mathematics, Budapest. ∗∗ Research supported by a Fellowship of the Paul Erd˝ os Summer Research Center of Mathematics, Budapest.
0031-5303/2005/$20.00 c Akad´ emiai Kiad´ o, Budapest
Akad´ emiai Kiad´ o, Budapest Springer, Dordrecht
¨ rgo ˝ and z. shi m. cso
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1. Introduction Let {Ui }i≥1 be a sequence of independent and identically distributed random variables, whose common law is the uniform distribution in (0, 1). Define the uniform empirical process def 0 ≤ t ≤ 1, αn (t) = n1/2 Fn (t) − t , where Fn (·) is the empirical distribution function based on the first n observations, i.e., n def 1 Fn (t) = ½{Ui ≤t} , 0 ≤ t ≤ 1. n i=1 Likewise, we can define the uniform empirical quantile process def βn (t) = n1/2 Fn−1 (t) − t , 0 ≤ t ≤ 1, def
def
where Fn−1 (t) = inf{s > 0 : Fn (s) ≥ t} (for 0 < t ≤ 1) and Fn−1 (0) = Fn−1 (0+) is the inverse function (quantile function) of Fn . The process def
Rn (t) = αn (t) + βn (t),
0 ≤ t ≤ 1,
which is often referred to as the (0, 1)–uniform Bahadur–Kiefer process, enjoys some remarkable properties. Let us recall the following Bahadur–Kiefer rep
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