An Lp-view of the Bahadur-Kiefer Theorem

Abstract

Let ]] > < /equationsource > < /equationsource > < /equationsource > < /equationsource > < /equationsource > < /equationsource > < /equationsource > < /equationsource > < /equationsource > < /equationsource > < /equationsource > < /equationsource > < /inlineequation > ]] > ]] > ]] > ]] > ]] > ]] > ]] > ]] > ]] > ]] > ]] > an]]>]]>]]>]]>]]>]]>]]>]]>]]>]]>]]>]]>\alpha_n and bn\beta_n be respectively the uniform empirical and quantile processes, and define Rn = an + bnR_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: $\|R_n\| /\sqrt{\| \alpha_n \| }\, \sim \, n^{-1/4} (\log n)^{1/2}$\|R_n\| /\sqrt{\| \alpha_n \| }\, \sim \, n^{-1/4} (\log n)^{1/2} almost surely, as nn goes to infinity, where || f|| = sup0 £ t £ 1 |f(t)|\| f\| =\sup_{0\le t\le 1} |f(t)| is the L¥L^\infty-norm. We prove that ||Rn||2 /Ö{|| an ||1} ~ n-1/4\|R_n\|_2 /\sqrt{\| \alpha_n \|_1}\, \sim \, n^{-1/4} almost surely, where || ||p\| \, \cdot \, \|_p is the LpL^p-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 /Ö{|| an ||(p/2)}n^{1/4} \|R_n\|_p /\sqrt{\| \alpha_n \|_{(p/2)}} converges almost surely to a finite positive constant whose value is explicitly known.

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An Lp-view of the Bahadur–Kiefer theorem
Mikl´ os CS¨ORG˝O∗
School of Mathematics and Statistics, Carleton University, Ottawa, Ont K1S 5B6, Canada.
E-mail: mcsorgo@math.carleton.ca
Zhan SHI∗∗
Laboratoire de Probabilit´ es, Universit´ e Paris VI, 4 Place Jussieu, F-75252 Paris Cedex 05,
France. E-mail: zhan@proba.jussieu.fr
Dedicated to Endre Cs´ aki and P´ al R´ ev´ esz on the occasion of their 70th birthdays
Summary.
cesses, 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 equiv-
alence: ?Rn?/??αn? ∼ n−1/4(logn)1/2almost surely, as n goes to infinity, where ?f? =
where ? · ?pis 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)
Let αn and βn be respectively the uniform empirical and quantile pro-
sup0≤t≤1|f(t)| is the L∞-norm. We prove that ?Rn?2/??αn?1 ∼ n−1/4almost surely,
converges almost surely to a finite positive constant whose value is explicitly known.
Keywords.
modulus of continuity for Brownian motion, Brownian bridge, Kiefer process.
Empirical process, quantile process, Bahadur–Kiefer representation, Lp-
2000 Mathematics Subject Classification. Primary 60F25; Secondary 62G30.
∗
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.
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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
70-th birthdays.
1. Introduction
Let {Ui}i≥1be a sequence of independent and identically distributed random variables,
whose common law is the uniform distribution in (0,1). Define the uniform empirical
process
αn(t)def
= n1/2?Fn(t) − t?,
0 ≤ t ≤ 1,
where Fn(·) is the empirical distribution function based on the first n observations, i.e.,
Fn(t)def
=1
n
n
?
i=1
1 l{Ui≤t},
0 ≤ t ≤ 1.
Likewise, we can define the uniform empirical quantile process
βn(t)def
= n1/2?F−1
n(t) − t?,
0 ≤ t ≤ 1,
where F−1
inverse function (quantile function) of Fn. The process
n(t)def
= inf{s > 0 : Fn(s) ≥ t} (for 0 < t ≤ 1) and F−1
n(0)def
= F−1
n(0+) is the
Rn(t)def
= 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 representation theorem.
Theorem A (Kiefer [23], Shorack [28], Deheuvels and Mason [17]). We have,
(1.1)lim
n→∞n1/4(logn)−1/2?Rn?
??αn?
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= 1,
a.s.,

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where ?f?def
= sup0≤t≤1|f(t)| denotes the uniform sup-norm of f.
Together with some well-known laws of the iterated logarithm (LIL’s) for αn(cf. Fact
3.2 in Section 3 for the exact statement), (1.1) immediately implies the following:
limsup
n→∞
liminf
n→∞
n1/4(logn)−1/2(log2n)−1/4?Rn? = 2−1/4,
n1/4(logn)−1/2(log2n)1/4?Rn? =π1/2
a.s.,
(1.2)
81/4,
a.s.,
(1.3)
where log2n
The study of the Bahadur–Kiefer representation was initiated by Bahadur [2], who
proved a “pointwise” version of (1.1). Kiefer [23] pointed out Theorem A, even though he
only proved the convergence in probability, and omitted the proof of the theorem due to
its extreme length. The upper bound in (1.1) was proved by Shorack [28], and the lower
bound by Deheuvels and Mason [17]. We mention that a simplified proof of the lower
bound was since discovered by Einmahl [20]. For a detailed discussion of various aspects
of the Bahadur–Kiefer theorem, as well as extensions to sequential empirical processes, we
refer to Cs¨ org˝ o and Szyszkowicz [13].
Looking at Theorem A, it is remarkable that the ratio between ?Rn? and
suitably normalized, should almost surely converge to a constant. A natural question
would be whether it remains true if the uniform sup-norm ? · ? is replaced by, say, the
L2-norm ? · ?2. For example, one might wonder if either ?Rn?2/??αn? or ?Rn?2/??αn?2
term of some power of log2n.
Somewhat surprisingly, the answer is no: we should use a different normalizing
function. Moreover, under the new normalization, the ratio between ?Rn?2and the square
root of αnunder the L1-norm, converges again to a constant limit with probability one.
More precisely, we have the following Lpversion of the Bahadur–Kiefer representation.
Throughout the paper, we write ?f?p
def
= log(logn).
??αn?,
would still be of order of magnitude which is around n−1/4(logn)1/2, perhaps with an extra
def
= (?1
= p/2. Then
0|f(t)|pdt)1/p.
Theorem 1.1. Let 2 ≤ p < ∞ and qdef
(1.4)lim
n→∞n1/4?Rn?p
??αn?q
= c0(p),
a.s.,
where
(1.5)
c0(p)def
= (E|N|p)1/p=
√2
?Γ((p + 1)/2)
√π
?1/p
,
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and N denotes a Gaussian N(0,1) variable. In particular,
n→∞n1/4?Rn?2
lim
??αn?1
= 1,
a.s.
Remark 1.2. The condition p ≥ 2 is to ensure that ? · ?qis a true metric norm. When
0 < p < 2, it is no longer a metric. Our proof shows that in this case, we still have the
following weaker version of Theorem 1.1: for 0 < p < 2, there exist two finite constants
c1(p) > 0 and c2(p) > 0, depending on p, such that
c1(p) ≤ liminf
n→∞
n1/4?Rn?p
??αn?q
≤ limsup
n→∞
n1/4?Rn?p
??αn?q
≤ c2(p),
a.s.
Remark 1.3. The reason for which the normalizing function in Theorem 1.1 differs from
the one in Theorem A will become clear in Section 3.
From (1.4), it is possible to deduce the almost sure asymptotics of Rn under the
Lp-norm.
Corollary 1.4. For 2 ≤ p < ∞,
limsup
n→∞
n1/4(log2n)−1/4?Rn?p= 21/4c0(p)
liminf
n→∞
where c0(p) is as in (1.5), q = p/2, and c3(q) ∈ (0,∞) and c4(q) ∈ (0,∞) are defined by
2−(q−1)/qq−1/2(q + 2)(q−2)/(2q)
?1
=2−(q−1)/qq1/2(q + 2)(q−2)/(2q)
B(1/2,1/q)
??∞
Here, B(·, ·) is the usual beta function, and C is the set of probability densities f such
that
1
8
−∞
?
c3(q),
a.s.,
n1/4(log2n)1/4?Rn?p= c0(p)
?
c4(q),
a.s.,
c3(q)
def
=
0(1 − xq)−1/2dx
,
(1.6)
c4(q)
def
= inf
f∈C
−∞
|x|qf(x)dx
?1/q
.
(1.7)
?∞
(f?(y))2
f(y)
dy ≤ 1.
Remark 1.5. Comparing this corollary with (1.2)–(1.3), it is immediately noted that Rn
has rather different asymptotics under Lp- and L∞- norms. This is in complete constrast
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to the situation for the empirical process αn. Indeed, ?αn? and ?αn?psatisfy almost the
same LIL’s (from both limsup and liminf points of view), except for the constants, cf.
Lemmas 3.3 and 3.8 and Fact 3.2 in Section 3 (they are stated for the Kiefer process,
but in view of the KMT strong invariance in Fact 3.1, one can immediately deduce the
corresponding LIL’s for αn).
Remark 1.6. The value of the constant c4(q) in (1.7) is in general implicit, except for
q = 1 or 2. Indeed, c4(2) = 1/√8, and it follows from the proof of Lemma 3.3 (cf. Section
3) and Tak´ acs [34, Theorem 1] that
?
where a?
Ai(·).
c4(1) =
2|a?
27
1|3
,
1< 0 denotes the largest real root of Ai?(·), the derivative of the Airy function
As a consequence of (1.1), the continuity of ? · ? on C[0,1] and the fact that we have
(cf. Doob [19], or Theorem 1.5.1 in [9])
(1.8)
P(?B? ≤ x) = 1 + 2
∞
?
k=1
(−1)kexp(−2k2x2), x > 0,
where {B(t); 0 ≤ t ≤ 1} is a Brownian bridge, we also have the following corollary.
Corollary 1.7 (Kiefer [23]). For x > 0,
n→∞P?n1/4(logn)−1/2?Rn? ≤ x?= P??
lim
?B? ≤ x?
?
(1.9)
= 1 + 2
∞
k=1
(−1)kexp(−2k2x4).
In a similar vein, from Theorem 1.1 we conclude the following Lpanalogue of (1.9).
Corollary 1.8. With 2 ≤ p < ∞ and q = p/2 we have
(1.10)lim
n→∞P?n1/4?Rn?p≤ x?= P
?
c0(p)
?
?B?q≤ x
?
,x > 0,
where {B(t); 0 ≤ t ≤ 1} is a Brownian bridge.
For a Brownian bridge {B(t); 0 ≤ t ≤ 1}, Smirnov [30], Anderson and Darling [1]
established the following result (cf., e.g., [9, Theorem 1.5.2]):
?
(1.11)
P
(?B?2)2≤ x
?
= 1 −2
π
∞
?
k=1
(−1)k+1
?2kπ
(2k−1)π
exp(−t2x/2)
√−tsint
dt,x > 0.
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