Some uniform in bandwidth functional results for the tail uniform empirical and quantile processes

0.25 MB·English

by  Davit Varron

Some uniform in bandwidth functional
results for the tail uniform empirical
2
1
0 and quantile processes
2

n
a Davit Varron
J

6 e-mail: [email protected]
2
Abstract: For fixed t∈[0,1) and h>0, consider the local uniform em-

]
T piricalprocess
.S Dn,h,t(s):=n−1/2hXn 1[t,t+hs](Ui)−hsi, s∈[0,1],
h i=1
t
a wheretheUiareindependentanduniformlydistributedon[0,1].Weinves-
m
tigate the functional limit behaviour of Dn,h,t uniformly in hn ≤ h ≤hn
[
whennhn/loglogn→∞andhn→0.

1
AMS 2000 subject classifications:Primary62G20, 62G30.
v
7 Keywords and phrases: Empirical processes, Strassen laws of the iter-
1
atedlogarithm.
5
5
.
1
0 1. Introduction
2
1
: Let(U ) beanindependent,identicallydistributed(i.i.d.)sequenceofrandom
v i i 1
≥
i
X variables that are uniformly distributed on [0,1]. Define the empirical distribu-
r tion function based on (U ,...,U ) by F (t) := n 1♯ 1 i n, U t , t
a 1 n n − i
{ ≤ ≤ ≤ } ∈
[0,1] and denote by F (t) the left-continuous inverse of F , namely F (t) :=
n← n n←
inf s 0, F (s) t . We also define the empirical (resp. quantile) process by
n
{ ≥ ≥ }
α (t):=√n(F (t) t),t [0,1] (resp. β (t):=√n(F (t) t), t [0,1]). The
n n − ∈ n n← − ∈
frameworkof this paper is the almostsure behaviour of the local empiricaland
quantile processes. Namely, given t [0,1) we focus on studying the following
∈
1
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D. Varron/Uniform inbandwidth results for tailempiricals 2
processes, as n and h 0.
→∞ →
(s):=α (t+hs) α (t), s [0,1], (1.1)
n,h,t n n
D − ∈
(s):=β (t+hs) β (t), s [0,1]. (1.2)
Dn′,h,t n − n ∈
Mason (1988) was the first to establish a functional law of the iterated log-
arithm for the local empirical process (see also Einmahl and Mason (1997)
for a generalization of this result to empirical processes indexed by functions).
To cite this result, we need to introduce some further notations first. Write
log (u) := log(log(u 3)). We say that a sequence (h ) of strictly positive
2 ∨ n n≥1
constants satisfies the local strong invariance conditions when, ultimately as
n ,
→∞
h 0, nh , nh /log n . (1.3)
n ↓ n ↑∞ n 2 →∞
Given a sequence (x ) of elements of a metric space (E,d), we say that
n n 1
≥
x K when K is non void and coincides with the set of all cluster points
n
of (x ) . In our framework, (E,d) is the space B([0,1]) of all real bounded
n n 1
≥
CADLAG trajectories on [0,1], endowed with the usual sup norm, namely
||
g := sup g(s) , s [0,1] . Consider the space AC[0,1] of all absolutely
|| {| | ∈ }
continuousfunctions on[0,1].Foranyg AC[0,1],wedefine the usuallycalled
∈
Hilbertian norm of g as
1
g 2 := g˙2(x)dx, (1.4)
|| ||H
Z
0
where g˙ is any version of the derivative of g with respect to the Lebesgue mea-
sure. The usually called Strassen ball can be defined as follows:
:= g AC([0,1]), g(0)=0, g 1 . (1.5)
H
S ∈ || || ≤
n o
As a corollary of a strong approximation result, Mason (1988) showed that,
given a sequence (h ) fulfilling (1.3) and given t [0,1), we have, almost
n n 1
≥ ∈
surely
Dn,hn,t (1.6)
(2h log n)1/2 S
n 2
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D. Varron/Uniform inbandwidth results for tailempiricals 3
In the particular case where t = 0, Einmahl and Mason (1988) showed that
also satisfies (1.6). They showed that result by making use of a local
Dn′,hn,t
Bahadur Kiefer representation (see their Theorem 5). The almost sure limit
behavior of when t (0,1) has been investigated by Deheuvels (1997),
Dn′,hn,t ∈
who showed that the above mentioned process may obey functional limit laws
that are different from (1.6). The aim of the present paper is the following:
given two sequences h < h fulfilling (1.3), does (1.6) still hold uniformly in
n n
h h h ? Namely, do we have almost surely
n n
≤ ≤
n,h,t
lim sup inf D g =0, (1.7)
n→∞ hn≤h≤hn g∈S(cid:12)(cid:12)(2hlog2n)1/2 − (cid:12)(cid:12)
(cid:12)(cid:12) (cid:12)(cid:12)
g , liminf sup (cid:12)(cid:12)Dn,h,t g =(cid:12)(cid:12)0? (1.8)
∀ ∈S n→∞ hn≤h≤hn(cid:12)(cid:12)(2hlog2n)1/2 − (cid:12)(cid:12)
(cid:12)(cid:12) (cid:12)(cid:12)
The remainder of this paper is organ(cid:12)i(cid:12)sed as follows. In(cid:12)(cid:12)2, we state our main
§
results on . We then show how this results leadto a localBahadur-Kiefer
n,h,t
D
type representation that holds uniformly in h. The proofs of our main results
follow in 3, 4 and 5.
§
2. Mains results
Our first result is a weaker form of assertion (1.7).
Theorem 1. Let (h ) and (h ) be two sequences satisfying (1.3) as well
n n 1 n n 1
≥ ≥
as h < 1h . Then, given t [0,1), we have, almost surely:
n 2 n ∈
n,h,t
lim sup inf D g =0. (2.1)
n→∞ hn≤h≤hn g∈√2S(cid:12)(cid:12)(2hlog2n)1/2 − (cid:12)(cid:12)
(cid:12)(cid:12) (cid:12)(cid:12)
The proof of Theorem 1 is written(cid:12)i(cid:12)n 3. (cid:12)(cid:12)
§
Remark:Conditionh <h /2is justtechnical,asthis resultisreallyinterest-
n n
ing when (h ) and (h ) are sequences that tend to 0 at different rates
n n 1 n n 1
≥ ≥
(typically n−α1 and n−α2, 0<α1 <α2 <1). Clearly, Theorem 1 seems unsat-
isfactory, as one would expect the limit set to be instead of √2 . As it will
S S
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D. Varron/Uniform inbandwidth results for tailempiricals 4
be pointedout inthe proofofTheorem1 (see 3.2), it is possible to prove(1.7)
§
when
β >0, lim log(h /h )/(logn)β =0. (2.2)
n n
∀ n
→∞
However,(2.2) is a very restrictive condition,imposing (h ) and (h ) to
n n 1 n n 1
≥ ≥
have rates of convergence to zero that are very close one to each other. In 3,
§
we shall try to point out the main difficulty that imposes us to weaken (1.7)
to (2.1). Showing that (1.7) is true or false without imposing (2.2) remains an
open problem.
The second step of our investigation is to determine the validity of (1.8). This
assertion turns out to be false as soon as h /h 0, which is a consequence
n n
→
of our next result. We first need to introduce some further notations. Given an
integerk 2,weendowthespace(B[0,1])k withtheproductsup-norm,namely
≥
g ,...,g :=max g ,..., g , and we define
1 k k 1 k
|| || {|| || || ||}
k
:= (g ,...,g ) (AC[0,1])k, g 2 1 . (2.3)
Sk 1 k ∈ || j ||H≤
n Xj=1 o
Now consider sequences 0 < h < ... < h < 1 satisfying, ultimately as
n,1 n,k
n ,
→∞
h /h 0, l=1,...,k 1, (2.4)
n,l n,l+1
↓ −
h 0, nh . (2.5)
n,k n,1
↓ ↑∞
Our second main result is the following functional limit law, which is provedin
4.
§
Theorem 2. Under assumptions (2.4) and (2.5) we have almost surely
Dn,hn,1,t ,..., Dn,hn,k,t . (2.6)
(2h log n)1/2 (2h log n)1/2 Sk
n,1 2 n,k 2
(cid:16) (cid:17)
Here refers to the Banach space B([0,1])k, .
k
||·||
(cid:0) (cid:1)
Note that is the unit ball of the reproducing kernel Hilbert space of
k
S
(W ,...,W ), where W ,...,W are independent Wiener processes on [0,1].
1 k 1 k
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D. Varron/Uniform inbandwidth results for tailempiricals 5
Theorem 2 describes an asymptotic independence phenomenon which has been
earlierinvestigatedby Deheuvels (2000)andDeheuvels et al. (1999).The proof
of Theorem 2 is provided in 4. Now, to see that (1.8) is false, choose g as the
§
identity function so as (g,g) / , which entails that inf g g ,g g
2 1 2 2
∈ S {|| − − ||
, (g ,g ) >ǫ for some ǫ >0. By Theorem 2 we have, almost surely,
1 2 2 0 0
∈S }
n,h,t
liminf sup D g
n→∞ hn≤h≤hn(cid:12)(cid:12)(2hlog2n)1/2 − (cid:12)(cid:12)
(cid:12)(cid:12) (cid:12)(cid:12)
liminf Dn,(cid:12)h(cid:12)n,t g, Dn(cid:12)(cid:12),hn,t g
≥ n→∞ (cid:12)(cid:12)(2hnlog2n)1/2 − (2hnlog2n)1/2 − (cid:12)(cid:12)2
(cid:12)(cid:12) (cid:12)(cid:12)
ǫ0, (cid:12)(cid:12) (cid:12)(cid:12)
≥
which invalidates (1.8).
A local Bahadur-Kiefer representation
A consequence of Theorem 1 is the following local Bahadur-Kiefer represen-
tation, which is very largely inspired from Einmahl and Mason (1988, Theo-
rem 5). For 0 < h < 1 and n 1 we set a (h) := (hlog n/n)1/2, b (h) :=
≥ n 2 n
log(nh), d (h):=2log n+b (h), r (h):=(a (h)d (h))1/2 and
n 2 n n n n
R (h):= + .
n Dn,h,0 Dn′,h,0
(cid:12)(cid:12) (cid:12)(cid:12)
Theorem 3. Under the conditions(cid:12)(cid:12)(cid:12)(cid:12)of Theorem 1, w(cid:12)(cid:12)(cid:12)(cid:12)ith t = 0, we have, almost
surely
limsup sup r (h) 1R (h) 21/2. (2.7)
n − n
≤
n→∞ hn≤h≤hn
The proof of Theorem 3 is provided in 5.
§
Remark: In view of Theorem 5 of Einmahl and Mason (1988), Theorem 3
seems to be non optimal since a factor 21/4 can be drop when h =h . This is
n n
a consequence of the fact that we were only able to prove(2.1) insteadof (1.7).
3. Proof of Theorem 1
Our proof is divided into two subsections. In 3.1, we establish a large devia-
§
tion result which holds uniformly in h h h . Then we make use of that
n n
≤ ≤
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D. Varron/Uniform inbandwidth results for tailempiricals 6
(uniform) large deviation principle to prove Theorem 1 in 3.2.
§
3.1. A uniform large deviation principle
3.1.1. Definitions
Large deviation results are commonly used when proving functional laws of
the iterated logarithm such as (1.6). As a uniformity in h h h ap-
n n
≤ ≤
pears in Theorem 1, we shall make use of a large deviation principle that
holds uniformly in h. This tool was first used by Mason (2004). From now
on, (ǫ ) will denote a triangular array of strictly positive numbers
n,i n≥1,i≤pn
satisfying max ǫ 0 as n . We call a rate function in a metric
1≤i≤pn n,i → → ∞
space(E,d) anypositive realfunctionJ onE suchthat,for eacha 0,the set
≥
g E, J(g) a is a compact set of (E,d).
{ ∈ ≤ }
Definition 3.1. Let (E,d) be a metric space and let be a σ-algebra included
0
T
in the Borel σ-algebra of (E,d). Let (X ) be a triangular array of
n,i n≥1,i≤pn
random variables that are measurable for (E, ). We say that (X )
T0 n,i n≥1,i≤pn
satisfies the uniform large deviation principle (ULDP) for (ǫ ) , a rate
n,i n≥1,i≤pn
function J and whenever
0
T
1. For each closed set F we have
0
∈T
limsupmaxǫ log P X F J(F), (3.1)
n,i n,i
n→∞ i≤pn (cid:16) (cid:16) ∈ (cid:17)(cid:17)≤−
2. For each open set O we have
0
∈T
liminf minǫ log P X O J(O). (3.2)
n,i n,i
n→∞ i≤pn (cid:16) (cid:16) ∈ (cid:17)(cid:17)≥−
Remark: In this definition, weintroduce a subσ-algebra because we will
0
T
consider repeatedly (E,d) as the metric space (B([0,1], ). As the
n,h,t
|| · || D
are not Borel measurable in that space, we shall consider as the σ-algebra
0
T
spawned by the open balls of (B([0,1], ). We will sometimes take (E,d)
|| · ||
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D. Varron/Uniform inbandwidth results for tailempiricals 7
as a finite dimensional vector space, in which case will denote the Borel σ-
0
T
algebra. Another way to avoid measurability problems is to consider inner and
outer probabilities (see, e.g.,Van der Vaart and Wellner (1996), Chapter 1).
The next result is a consequence of the work of Arcones (2003).
Proposition3.1. Let(X ) beatriangular arrayof random variables
n,i n≥1,i≤pn
takingvalues in B([0,1])andmeasurable for .Let (ǫ ) beatriangu-
T0 n,i n≥1,i≤pn
lar array of strictly positive real numbers. Assume that the following conditions
hold:
1. For each p 1 and (s ,...,s ) (0,1)p satisfying s =s for each i=j,
1 p i j
≥ ∈ 6 6
the triangular array X (s ),...,X (s ) satisfies the ULDP
n,i 1 n,i p n≥1,i≤pn
in Rp for (ǫ ) (cid:0) and a rate function(cid:1)I .
n,i n≥1,i≤pn s1,...,sp
2. For any τ >0 we have
limlimsupmaxlog P sup X (s) X (s) >τ = .
n,i ′ n,i
δ↓0 n→∞ i≤pn (cid:16) (cid:16)|s−s′|<δ| − | (cid:17)(cid:17) −∞
Then (X ) satisfies the ULDP in (B([0,1], ) for (ǫ ) ,
n,i n≥1,i≤pn || · || n,i n≥1,i≤pn
and the following rate function:
0
T
I(g):= sup I g(s ),...,g(s ) , g B([0,1]).
s1,...,sp 1 p ∈
p≥1,(s1,...,sp)∈(0,1)p (cid:0) (cid:1)
Now consider the following rate function on B([0,1]) that is known to rule
the large deviation properties of a Wiener process:
g 2 , when g AC[0,1];
J(g):= || ||H ∈ (3.3)

, when g / AC[0,1].
 ∞ ∈
Notice that = g B([0,1]), g(0) = 0, J(g) 1 . The main tool that will
S { ∈ ≤ }
be used to achieve our proof of Theorem 1 is the following ULDP.
Proposition 3.2. Let (h ) and (h ) be two sequences satisfying con-
n n 1 n n 1
≥ ≥
ditions of Theorem 1 and let (h ) be a triangular array satisfying
n,i n≥1,i≤pn
h h h for each n 1, i p . Then the triangular array
n n,i n n
≤ ≤ ≥ ≤
(2h log n) 1/2
(cid:16) n,i 2 − Dn,hn,i,t(cid:17)n≥1,i≤pn
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D. Varron/Uniform inbandwidth results for tailempiricals 8
satisfies the ULDP in (B([0,1]), ) for , the rate function J given in (3.3)
0
||·|| T
and the (constant in i p ) triangular array (1/log n) .
≤ n 2 n≥1,i≤pn
Proof of Proposition 3.2: We shall make use of Proposition 3.1, and we
hencehavetoshowthatconditions1and2ofthispropositionaresatisfied.This
verification will be a consequence of two separate lemmas. The next proposi-
tion, which shall be useful to prove our first lemma, follows directly from the
argumentsofEllis(1984).Here < , > denotes the usualscalarproductonRp.
· ·
Proposition 3.3. Let (X ) be a triangular array of random vectors
n,i n≥1,i≤pn
taking values in Rp, and let (ǫ ) be a triangular array of strictly posi-
n,i n≥1,i≤pn
tivereal numbers.Assumethat thereexists apositive realfunction ℓ (which may
take infinite values) on Rp such that the following conditions are satisfied.
1. ℓ is convex and lower semi continuous on Rp.
2. The definition set D(ℓ) := λ Rp, ℓ(λ) < has an interior that
{ ∈ ∞}
contains the null vector.
3. ℓ is differentiable on the interior of D(ℓ) and, for each sequence (λ )
n n 1
≥
converging to a boundary point of D(ℓ) we have ℓ(λn) Rp . Here
||∇ || →∞
Rp denotes the usual Euclidian norm.
||·||
4. For each λ D(ℓ), we have
∈
nl→im∞ mi≤apnx(cid:12)(cid:12)ǫn,ilog(cid:16)E(cid:16)exp(cid:0)ǫ−n,1i <λ,Xn,i >(cid:1)(cid:17)(cid:17)−ℓ(λ)(cid:12)(cid:12)=0.
5. For each λ / D(ℓ(cid:12)), we have (cid:12)
∈
nl→im∞ im≤ipnnǫn,ilog(cid:16)E(cid:16)exp(cid:0)ǫ−n,1i <λ,Xn,i >(cid:1)(cid:17)(cid:17)=∞.
Then (X ) satisfies the ULDP in Rp for (ǫ ) with the fol-
n,i n≥1,i≤pn n,i n≥1,i≤pn
lowing rate function:
J(s):= sup <λ,s> ℓ(λ), s Rp.
λ Rp − ∈
∈
e
We now state our first lemma.
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D. Varron/Uniform inbandwidth results for tailempiricals 9
Lemma 3.1. Let p 1 and (s ,...,s ) [0,1]p be arbitrary, with s < s <
1 p 1 2
≥ ∈
... < s . Under the assumptions of Proposition 3.2, the triangular array of
p
Rp-valued random vectors
(2h log n) 1/2 (s ),..., (s )
(cid:16) n,i 2 − (cid:0)Dn,hn,i,t 1 Dn,hn,i,t p (cid:1)(cid:17)n≥1,i≤pn
satisfiestheULDPfor(ǫ ) withthefollowingratefunction(withs :=
n,i n≥1,i≤pn 0
0).
p x x 2
J (x ,...,x ):= (s s ) i+1− i , (x ,...,x ) Rp.
s1,...,sp 1 p i+1− i s s 1 p ∈
i+1 i
Xi=0 (cid:16) − (cid:17)
Proof of Lemma 3.1.
We shall make use of Proposition 3.3. Fix λ = (λ ,...,λ ) Rp and and
1 p
∈
write the as sums of i.i.d. random variables, namely
Dn,hn,i,t
p n
(2h log n) 1/2 λ (s )=(2nh log n) 1/2 Zk , (3.4)
n,i 2 − jDn,hn,i,t j n,i 2 − n,hn,i,t
j=1 k=1
X X
where
p
Zk := λ 1 (U ) h s , k =1,...,n.
n,hn,i,t j [t,t+hn,isj] k − n,i j
j=1
X (cid:0) (cid:1)
These n randomvariables are i.i.d with mean 0 and variance-covariancematrix
given by hn,iλ′Σn,iλ, with Σn,i(l,l′) := min(sl,sl′) hn,islsl′. Now define the
−
matrix Σ(l,l′) := min(sl,sl′). Clearly, as hn,i hn 0 we have Σn,i Σ
≤ → →
uniformly in i as n . By standard computations we have, for each n 1
→ ∞ ≥
and i p :
n
≤
p
(log2n)−1log E(cid:18)exp(cid:16)log2n(2hn,ilog2n)−1/2Xj=1λjDn,hn,i,t(sj)(cid:17)(cid:19)!
n
= log Eexp r Z1 , (3.5)
log n n,i n,hn,i,t
2
(cid:16) (cid:0) (cid:1)(cid:17)
where r :=(log n/2nh )1/2. Recall that max r 0 as n , since
n,i 2 n,i i≤pn n,i → →∞
h satisfies (1.3), and notice that the Zk are centered and almost surely
n n,hn,i,t
boundedbypmax λ .ThisensuresthatthefollowingTaylorexpansion
j=1,...,p j
| |
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D. Varron/Uniform inbandwidth results for tailempiricals 10
isvalid,foreachn 1, i p (hereεdenotesarealfunctionsatisfyingε(u) 0
n
≥ ≤ →
as u 0):
→
r2 h
E exp rn,iWn1,hn,i,t =1+ n,i2n,iλ′Σn,iλ(1+ε(rn,i)). (3.6)
(cid:16) (cid:0) (cid:1)(cid:17)
Combining (3.5) and (3.6), we get
p
nl→im∞ mi≤apnx(cid:12)(cid:12)log(cid:18)E(cid:18)exp(cid:16)(2hn,ilolgo2g2nlno)g12/2njP=1λjDn,hn,i,t(sj)(cid:17)(cid:19)(cid:19) − 14λ′Σλ(cid:12)(cid:12)=0.
(cid:12) (cid:12)
(cid:12) (cid:12)
As the func(cid:12)tion ℓ(λ) := λ′(Σ/4)λ obviously satisfies conditions of Propo(cid:12)sition
3.3, the proof of Lemma 3.1 is concluded by noticing that
p x x 2
sup <t,x> ℓ(t)=x′Σ−1x= (si+1 si) i+1− i .(cid:3)
t Rp − − si+1 si
∈ Xi=0 (cid:16) − (cid:17)
Our next lemma shows that condition 2 of Proposition 3.1 is fulfilled.
Lemma3.2. UndertheassumptionsofProposition3.2,wehave,foreachτ >0
(s) (s)
limlimsupmaxlog P sup Dn,hn,i,t −Dn,hn,i,t ′ τ = .
δ↓0 n→∞ i≤pn (cid:16) (cid:16)|s−s′|<δ(cid:12) (2hn,ilog2n)1/2 (cid:12)≥ (cid:17)(cid:17) −∞
(cid:12) (cid:12)
Proof of Lemma 3.2. (cid:12) (cid:12)
Fix τ >0 and introduce a parameter δ >0 that will be chosensmall enough
inthesequel.Theproofofthislemmareliesonanexponentialinequalityforthe
oscillations of the local empirical process, which is due to Einmahl and Mason
(1988) (see their Inequality 1). For positive numbers a,b with a+b 1, write
≤
ω (a,b):= sup α (s+s) α (s) . (3.7)
n n ′ n
| − |
0 s b,
0≤s′≤a
≤ ≤
Fact 1 (Einmahl,Mason,1988). Fix 0<ε 1/2. There exists K(ε)< such
≤ ∞
that, for any n 1, λ>0, a>0, b>0 fulfilling a+b 1 and 0<a<1/4,
≥ ≤
(1 ε)λ2 λ
P ωn(a,b) λ K(ε)ba−1exp − Ψ . (3.8)
≥ ≤ − 2a √na
(cid:16) (cid:17) (cid:16) (cid:16) (cid:17)(cid:17)
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