O tober WSPC Pro eedings Trim Size: 75in x 5in suquet revision

O tober WSPC Pro eedings Trim Size: 75in x 5in suquet revision

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Niveau: Supérieur, Master, Bac+5
O tober 19, 2006 18:17 WSPC - Pro eedings Trim Size: 9.75in x 6.5in suquet-revision 1 REPRODUCING KERNEL HILBERT SPACES AND RANDOM MEASURES ChARLES SUQUET Laboratoire P. Painlevé, UMR CNRS 8524, Bât M2, Cité S ientique, Université Lille I F59655 Villeneuve d'As q Cedex, Fran e We show how to use Guilbart's embedding of signed measures into a R.K.H.S. to study some limit theorems for random measures and sto hasti pro esses. Key words: Mathemati s Subje t Classi ation: 1. R.K.H.S. and metri s on signed measures In the late seventies, C. Guilbart [4, 5? introdu ed an embedding into a reprodu - ing kernel Hilbert spa e (R.K.H.S.) H of the spa e M of signed measures on some topologi al spa e X. He hara terized the inner produ ts on M indu ing the weak topology on the subspa e M+ of bounded positive measures and established in this setting a Glivenko-Cantelli theorem with appli ations to estimation and hypothesis testing. In this ontribution we present a onstru tive approa h of Guilbart's em- bedding following [20?. This embedding provides a Hilbertian framework for signed random measures.

  • wspc - pro eedings

  • let µ•1

  • measure

  • positive measure

  • random measures

  • topologi al dual


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Informations

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b
and
Theorem
(14)
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em-
In
al
this
asur

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ase,
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(15)
sample

follo
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ob
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ergence
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r
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sum
w
a
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theorem

limit
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tral


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,
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space
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ert
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ecial
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an
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olo
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ak
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oth
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ariables
Then
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me
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Let

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e
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2 2X s := ESi n nPn
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C([0,1],H)
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1/2ω(W,u) u ln(1/u)
•ξn
1/2 ou ln(1/u) H ([0,1],H)ρ
f : [0,1]→H
kfk :=kf(0)k +ω (f,1)<∞ lim ω (f,u) = 0,ρ ρ ρH u→0
kf(t)−f(s)kHω (f,u) := sup .ρ
ρ(t−s)0<t−s≤u
αρ ρ(u) =u L(1/u)
0 < α ≤ 1/2 L
o • •H ([0,1],H) ξ W μρ n
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β > 1/2
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t→∞
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α < 1/2
A = 1
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p(α)−1 •p(α) := (1/2−α) Ekμ k < ∞K
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1/β•Eexp(dkμ k ) d > 0K
•μ k = 1,...,nk
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alen

uit
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in
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functions
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-value
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d
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er
than
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(24)
in
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satised
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e
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Assume
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.
alternativ
Theorem
2.4.(
• ∗ ∗μ +μ k∈I :={k +1,...,m }c n• k(H ) ν =A k • cμ k∈I :={1,...,n}\Ink n
μ = 0 nc
•ξn
X

S (a,b) = ν , 0≤a<b≤ 1.n k
na<k≤nb
D [0,1] j D ={0,1}j 0
−j j−1 − −jD = (2l−1)2 ; 1≤l≤ 2 j ≥ 1 r∈ D j ≥ 0 r :=r−2j j
+ −jr :=r+2 DI(n,ρ)
1 1 − + DI(n,ρ) := max max S (r ,r)−S (r,r ) .n n−j K1≤j≤logn r∈D2 ρ(2 ) j
jlog 2 log(2 ) =j ln
tln(e ) =t
ρ
• −1/2μ (H ) n DI(n,ρ)0
Z
∞ j−1 Y 2
• (j+1)/2 −jP(Z ≤z) = P(kγ k ≤ 2 ρ(2 )z , z≥ 0,K
j=1
•γ H
•μ [ε,∞) ε> 0
−1/2n DI(n,ρ)
∗ ∗ ∗ρ (H ) l := m − kA
n ∗ ∗ou kμ k l ln c1/2 Klim n =∞, u := min ;1− .n
n→∞ ρ(u ) n nn
pr−1/2n DI(n,ρ)−−−−→∞.
n→∞
μ nc
α ∗ρ(t) = t l = o(n)
β∗ −δ −1 1/2l n →∞ δ = (1−2α)(2−2α) ρ(t) =t ln (c/t) β > 1/2
γ−1u =n ln n γ > 2βn
−γ∗ ∗l = o(n) l ln n → ∞
∗n−l =o(n)
asur
genc
a
e
.
of
h
the
(
pr

o
that

end
(27)
.
is

uniform
stand
on
Consider
any
o
interval
on
Then
to
dene
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the
the
dy
this
adic
dy
,
h


ts
(26)

h
.
r
Theorem
ges
2.5
assume
is
es
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obtained
(28)
from

Theorem
T
2.2
hiev
and
o
from
satises
[16
Assume

use
Th.
the
2
satised
and

Prop.
eha
3.

F
with
or
e
general
.
estimates

on
H?lder
the
variable

ne
v
law
ergence

rate
Condition
in

(27),
is
see
dep
Prop.
.
4
measure
in
y
[16
ws

short
The
that

on
of
and
the
satises

,
of
andom
test
an

and
onver
function
b
the
y
Theorem

w
The
w
.
natural
.

and
(28)
as
vided
e
)
ovarianc


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follo
(
ws
leads
from
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the
that
next
the
result
and
whic

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uous
is
b
an

same
epidemics
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of
of
ology
Th.
where
5
andom
in
gative
[16
non

to
Theorem
in
2.6.
onver
L
T
et
discuss
the
(28),
satisfying
for
(24).
y
Under6
with
do
,
not
in
end
element
some
,
When
write
signed
andom
whic
r
ma
Gaussian
,
o
allo
zer
us
an

me
epidemics
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h
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for
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length
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(25).
and
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assume
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that
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me
for
r
e
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wher
me
(27)
that
rite
(24)
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e
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weight
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and
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2.5.
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).
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goal,
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e
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the
Here
9
tin
if
functionals
(28)
In
Then
oth
y
one
b

denote
long
us

Let
that
sums
in
if
top
function
.
distribution
partial
with
easy
adaptation2L (0,1)
ϕ
2L (0,1)
2L (0,1)
Suquet,
ac
Y.V.
es
1.
autor
16.
epr
mesures
o
rim
duisants
as
et
o
mesur
de
e
(2)
empirique.
a
M?tho
An
des
theory
splines
theorem,
en
dimensional
es-
es
timation
Stat.
fonctionnel
Inst.
le
,
,
Oliv
Thesis,

Univ
Oliv
ersit

y
a
of
pro
Lille
,
1,

F
ob
rance
Suquet,
(1980).
al
2.
ac
A.
of
Berlinet,
des
V
Ch.
ariables
sid?r?es
al?atoires
(1993).
?
Berry-Ess?en,
v
in
aleurs
(1995).
dans
Suquet,
les
6.5in

v
?
(1995).
no
Suquet,
y
v
au

repro
Opp
duisan

t,
v
C.R.A.S.
theorems
290
Pr
,
A.
s?rie

A,
functional
973975
or
(1980).
No
3.
ausk
A.

Berlinet
app
and
for
Ch.
17.
Thomas-Agnan,
o
R
Thesis,
epr
rance
o
pr?-hilb

b
kernel
,
Hilb
ergences
ert
al?atoires
sp
al?atoires
ac
37
es

in
?
pr
Math.
ob
Ch.
ability

and
,

P
,
and
Klu
Berlinet,
w
10
er
v
A
LPQD

Portugaliae
Publishers,
,
Boston,
P

and
h
9.75in
t,

London
L
(2004).
Notes
4.
,
C.
and
Guilbart,
(Eds),
?tude
elets
des
(1995).
pr
v,
o
of
duits
and
sc
probabilit
alair
The
es
Appl.
sur
(1956).
l'esp
k
ac
Ch.
e
sucien
des
the
mesur
tral
es.
of
Estimation
al
p
17
ar
221243
pr
Ra?
oje
and

esting
T
of
ests
rameters,
?
in
noyaux.

Th?se

d'Etat,
esses
Lille
Suquet,
1,
autor
F
et
rance
atoir
(1978).
ersit
5.
1,
C.
18.
Guilbart,
top
Pro
sur
duits
?
scalaires
Pub.
sur
Paris

(1990).
des
Con
mesures,

A
de
nnales
signe
de
v
l'Institut
ertiennes,
Henri
Univ.

1-2,
ar
Ch.
?
sur
,
et

th?or?mes
B,
l.
15

,
(1995).
333354
Tigh
(1979).
Math.
6.
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