Un poly de référence : Concentration-of-measure inequalities , Gabor Lugosi.
modifier
P
(
|
|
x
−
m
|
|
≥
t
σ
)
≤
1
/
t
2
{\displaystyle \mathbf {P} \left(||x-m||\geq t\sigma \right)\leq 1/t^{2}}
Une démonstration de 1937 .
Soit
X
{\displaystyle X}
t
{\displaystyle t}
P
(
X
≥
t
)
≤
1
t
E
(
X
)
{\displaystyle \mathbf {P} (X\geq t)\leq {1 \over t}\mathbf {E} (X)}
On peut en déduire que pour toute fonction
ϕ
{\displaystyle \phi }
P
(
X
≥
t
)
≤
E
(
ϕ
(
X
)
)
ϕ
(
t
)
{\displaystyle \mathbf {P} (X\geq t)\leq {\mathbf {E} (\phi (X)) \over \phi (t)}}
ce qui permet d'obtenir une généralisation de Chebychev (citée dans Concentration-of-measure inequalities , Lecture notes by Gabor Lugosi , March 7, 2005):
P
(
|
X
−
E
(
X
)
|
≥
t
)
≤
E
(
|
X
−
E
(
X
)
|
q
t
q
{\displaystyle \mathbf {P} (|X-\mathbf {E} (X)|\geq t)\leq {\mathbf {E} (|X-\mathbf {E} (X)|^{q} \over t^{q}}}
voir mathworld.wolfram .
P
(
∪
i
E
i
)
≤
∑
i
P
(
E
i
)
{\displaystyle P(\cup _{i}E_{i})\leq \sum _{i}P(E_{i})}
Soit
(
X
n
)
1
≤
n
≤
N
{\displaystyle (X_{n})_{1\leq n\leq N}}
σ
2
{\displaystyle \sigma ^{2}}
S
N
{\displaystyle S_{N}}
N
{\displaystyle N}
S
N
=
∑
n
=
1
N
X
n
{\displaystyle S_{N}=\sum _{n=1}^{N}X_{n}}
Alors, pour tout
0
≤
k
≤
2
σ
{\displaystyle 0\leq k\leq 2\sigma }
P
(
|
S
N
|
≥
k
σ
)
≤
2
e
−
k
2
/
4
N
2
{\displaystyle P(|S_{N}|\geq k\sigma )\leq 2e^{-k^{2}/4N^{2}}}
Soit
(
X
n
)
1
≤
n
≤
N
,
(
X
n
′
)
1
≤
n
≤
N
{\displaystyle (X_{n})_{1\leq n\leq N},(X'_{n})_{1\leq n\leq N}}
f
:
I
R
N
↦
I
R
{\displaystyle f:{I\!\!R}^{N}\mapsto {I\!\!R}}
Z
=
f
(
X
1
,
…
,
X
N
)
,
Z
(
i
)
=
f
(
X
1
,
…
,
X
i
−
1
,
X
i
′
,
X
i
+
1
,
…
,
X
N
)
{\displaystyle Z=f(X_{1},\ldots ,X_{N}),Z^{(i)}=f(X_{1},\ldots ,X_{i-1},X'_{i},X_{i+1},\ldots ,X_{N})}
Alors:
V
(
Z
)
≤
1
2
E
[
∑
i
=
1
N
(
Z
−
Z
(
i
)
)
2
]
{\displaystyle \mathbf {V} (Z)\leq {1 \over 2}\,\mathbf {E} \left[\sum _{i=1}^{N}(Z-Z^{(i)})^{2}\right]}
Cité par : Concentration inequalities using the entropy method , BOUCHERON, S., LUGOSI, G., MASSART, P. (2003), Ann. of Probability 31 n°3, 1583-1614.
et en notant
E
i
(
Z
)
=
E
(
Z
|
X
i
,
…
,
X
i
−
1
,
X
i
+
1
,
…
,
X
n
)
{\displaystyle \mathbf {E} _{i}(Z)=\mathbf {E} (Z|X_{i},\ldots ,X_{i-1},X_{i+1},\ldots ,X_{n})}
V
(
Z
(
i
)
)
≤
E
[
∑
i
=
1
N
(
Z
(
i
)
−
E
i
(
Z
)
)
2
]
{\displaystyle \mathbf {V} (Z^{(i)})\leq \mathbf {E} \left[\sum _{i=1}^{N}(Z^{(i)}-\mathbf {E} _{i}(Z))^{2}\right]}
An application of the Efron-Stein inequality in density estimation , Luc Devroye , Annals of Statistics, vol. 15, pp. 1317-1320, 1987.
Elle est utilisée par exemple pour obtenir les résultats du Bootstrap et Jackknife The Jackknife and Bootstrap , by Jun Shao, p28.
Soit
(
X
n
)
1
≤
n
≤
∞
{\displaystyle (X_{n})_{1\leq n\leq \infty }}
S
N
{\displaystyle S_{N}}
N
{\displaystyle N}
S
N
=
∑
n
=
1
N
X
n
{\displaystyle S_{N}=\sum _{n=1}^{N}X_{n}}
Alors, pour tout
λ
>
0
{\displaystyle \lambda >0}
P
(
max
1
≤
k
≤
n
|
S
k
|
≥
λ
)
≤
1
λ
2
V
(
S
n
)
{\displaystyle \mathbf {P} \left(\max _{1\leq k\leq n}|S_{k}|\geq \lambda \right)\leq {1 \over \lambda ^{2}}\mathbf {V} (S_{n})}
cf PlanetMath.org
modifier
modifier
Il y a dans Hull (Options, Futures and Other Derivatives, Fifth Edition ; John C. Hull), page 248, une approximation de la fonction de répartition d'une distribution gaussienne assez pratique :
∀
x
≥
,
N
(
x
)
=
1
−
M
(
x
)
∑
i
=
1
5
a
i
k
(
x
)
i
{\displaystyle \forall x\geq ,\,N(x)=1-M(x)\sum _{i=1}^{5}a_{i}k(x)^{i}}
∀
x
<
0
,
N
(
x
)
=
1
−
N
(
−
x
)
{\displaystyle \forall x<0,\,N(x)=1-N(-x)}
k
(
x
)
=
1
1
+
γ
x
,
γ
=
0.2316419
{\displaystyle k(x)={\frac {1}{1+\gamma x}},\,\gamma =0.2316419}
a
1
=
0.319381530
,
a
2
=
−
0.356563782
,
a
3
=
1.781477937
{\displaystyle a_{1}=0.319381530,\,a_{2}=-0.356563782,\,a_{3}=1.781477937}
a
4
=
−
1.821255978
,
a
5
=
1.330274429
{\displaystyle a_{4}=-1.821255978,\,a_{5}=1.330274429}
M
(
x
)
=
1
2
π
e
−
x
2
/
2
{\displaystyle M(x)={\frac {1}{\sqrt {2\pi }}}\,e^{-x^{2}/2}}
![{\displaystyle \mathbf {V} (Z)\leq {1 \over 2}\,\mathbf {E} \left[\sum _{i=1}^{N}(Z-Z^{(i)})^{2}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b49a1dea525670a1cbaa2f0ee94c4792681971dd)
![{\displaystyle \mathbf {V} (Z^{(i)})\leq \mathbf {E} \left[\sum _{i=1}^{N}(Z^{(i)}-\mathbf {E} _{i}(Z))^{2}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cab936ea209b6c6d8111eae27638aa5757e00191)
