Soient des variables aléatoires
(
X
i
)
i
∈
N
∗
{\displaystyle (X_{i})_{i\in \mathbb {N} ^{*}}}
i.i.d. définies sur un espace de probabilité
(
Ω
,
T
,
P
)
{\displaystyle (\Omega ,{\mathcal {T}},\mathbb {P} )}
espace mesurable
(
X
,
A
)
{\displaystyle ({\mathcal {X}},{\mathcal {A}})}
F
{\displaystyle {\mathcal {F}}}
(
X
,
A
)
{\displaystyle ({\mathcal {X}},{\mathcal {A}})}
F
{\displaystyle {\mathcal {F}}}
α
n
⟶
L
n
→
+
∞
G
dans
ℓ
∞
(
F
)
{\displaystyle \alpha _{n}{\underset {n\to +\infty }{\overset {\mathcal {L}}{\longrightarrow }}}\mathbb {G} \qquad {\text{ dans }}\ell ^{\infty }({\mathcal {F}})}
avec
α
n
{\displaystyle \alpha _{n}}
F
{\displaystyle {\mathcal {F}}}
G
{\displaystyle \mathbb {G} }
F
{\displaystyle {\mathcal {F}}}
processus gaussien centré dont la fonction de covariance est donnée par
∀
f
,
g
∈
F
,
Cov
(
G
(
f
)
,
G
(
g
)
)
=
E
[
f
(
X
)
g
(
X
)
]
−
E
[
f
(
X
)
]
E
[
g
(
X
)
]
.
{\displaystyle \forall f,g\in {\mathcal {F}},\quad {\text{Cov}}(\mathbb {G} (f),\mathbb {G} (g))=\mathbb {E} [f(X)g(X)]-\mathbb {E} [f(X)]\mathbb {E} [g(X)].}
Puisqu'une classe de Donsker
F
{\displaystyle {\mathcal {F}}}
P
=
P
X
{\displaystyle P=\mathbb {P} ^{X}}
X
i
{\displaystyle X_{i}}
F
{\displaystyle {\mathcal {F}}}
P
{\displaystyle P}
En particulier, le théorème de Donsker revient à dire que la classe des fonctions indicatrices
F
=
{
x
↦
1
{
x
≤
t
}
:
t
∈
R
}
{\displaystyle {\mathcal {F}}=\{x\mapsto \mathbf {1} _{\{x\leq t\}}:t\in \mathbb {R} \}}
Les conditions suffisantes évoquées dans cette partie[ 1] théorème de Dudley [ 2]
modifier
Soit
F
{\displaystyle {\mathcal {F}}}
∫
0
+
∞
H
[
]
(
F
,
ε
,
L
2
(
P
)
)
d
ε
<
+
∞
,
{\displaystyle \int _{0}^{+\infty }{\sqrt {H_{[\ ]}({\mathcal {F}},\varepsilon ,L_{2}(P))}}\mathrm {d} \varepsilon <+\infty ,}
où
H
[
]
(
F
,
ε
,
d
)
{\displaystyle H_{[\ ]}({\mathcal {F}},\varepsilon ,d)}
N
(
F
,
ε
,
d
)
{\displaystyle N({\mathcal {F}},\varepsilon ,d)}
F
{\displaystyle {\mathcal {F}}}
ε
{\displaystyle \varepsilon }
d
{\displaystyle d}
F
{\displaystyle {\mathcal {F}}}
On note
H
(
F
,
ε
,
d
)
{\displaystyle H({\mathcal {F}},\varepsilon ,d)}
N
(
F
,
ε
,
d
)
{\displaystyle N({\mathcal {F}},\varepsilon ,d)}
F
{\displaystyle {\mathcal {F}}}
ε
{\displaystyle \varepsilon }
d
{\displaystyle d}
F
{\displaystyle {\mathcal {F}}}
∫
0
+
∞
sup
Q
H
(
F
,
ε
|
|
F
|
|
Q
,
2
,
L
2
(
P
)
)
d
ε
{\displaystyle \int _{0}^{+\infty }\sup _{Q}{\sqrt {H({\mathcal {F}},\varepsilon ||F||_{Q,2},L_{2}(P))}}\mathrm {d} \varepsilon }
F
{\displaystyle F}
F
{\displaystyle {\mathcal {F}}}
(
X
,
A
)
{\displaystyle ({\mathcal {X}},{\mathcal {A}})}
|
|
F
|
|
Q
,
2
2
=
∫
F
2
d
Q
>
0
{\displaystyle ||F||_{Q,2}^{2}=\int F^{2}\mathrm {d} Q>0}
Les classes
F
ε
:=
{
f
−
g
:
f
,
g
∈
F
,
|
|
f
−
g
|
|
P
,
2
<
ε
}
{\displaystyle {\mathcal {F}}_{\varepsilon }:=\{f-g:f,g\in {\mathcal {F}},||f-g||_{P,2}<\varepsilon \}}
F
∞
2
{\displaystyle {\mathcal {F}}_{\infty }^{2}}
P
{\displaystyle P}
ε
>
0
{\displaystyle \varepsilon >0}
P
F
2
<
+
∞
{\displaystyle PF^{2}<+\infty }
F
{\displaystyle F}
F
{\displaystyle {\mathcal {F}}}
Alors
F
{\displaystyle {\mathcal {F}}}
P
{\displaystyle P}
La première condition est généralement appelée « condition d'entropie uniforme ».
Condition avec les deux entropies
modifier
Soit
F
{\displaystyle {\mathcal {F}}}
∫
0
+
∞
log
N
[
]
(
F
,
ε
,
L
2
,
∞
(
P
)
)
d
ε
+
∫
0
+
∞
log
N
(
F
,
ε
,
L
2
(
P
)
)
d
ε
<
+
∞
.
{\displaystyle \int _{0}^{+\infty }{\sqrt {\log N_{[\ ]}({\mathcal {F}},\varepsilon ,L_{2,\infty }(P))}}\mathrm {d} \varepsilon +\int _{0}^{+\infty }{\sqrt {\log N({\mathcal {F}},\varepsilon ,L_{2}(P))}}\mathrm {d} \varepsilon <+\infty .}
De plus, on suppose que
F
{\displaystyle {\mathcal {F}}}
F
{\displaystyle F}
P
(
F
>
x
)
=
o
(
x
−
2
)
,
x
→
∞
{\displaystyle P(F>x)=o(x^{-2}),x\to \infty }
F
{\displaystyle {\mathcal {F}}}
P
{\displaystyle P}
↑ (en) Aad W. Van Der Vaart et Jon. A. Wellner, Weak convergence and empirical processes with applications to statistics , Springer, p. 127↑ (en) Michel Ledoux et Michel Talagrand, Probability in Banach spaces , Springer, p. 321 
![{\displaystyle \forall f,g\in {\mathcal {F}},\quad {\text{Cov}}(\mathbb {G} (f),\mathbb {G} (g))=\mathbb {E} [f(X)g(X)]-\mathbb {E} [f(X)]\mathbb {E} [g(X)].}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7648c7821e0a9c31a8e93072daa019b7f7253280)
![{\displaystyle \int _{0}^{+\infty }{\sqrt {H_{[\ ]}({\mathcal {F}},\varepsilon ,L_{2}(P))}}\mathrm {d} \varepsilon <+\infty ,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/306e350da94aa09b36bf2cd8f6dd80ea841e8c32)
![{\displaystyle H_{[\ ]}({\mathcal {F}},\varepsilon ,d)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3feaaf94f2fda071c83690ef2c41d4afdfd6020a)
![{\displaystyle \int _{0}^{+\infty }{\sqrt {\log N_{[\ ]}({\mathcal {F}},\varepsilon ,L_{2,\infty }(P))}}\mathrm {d} \varepsilon +\int _{0}^{+\infty }{\sqrt {\log N({\mathcal {F}},\varepsilon ,L_{2}(P))}}\mathrm {d} \varepsilon <+\infty .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b7fbebcf515eb1215240ad2991c3a1fbde95c962)
