Contributions à l'étude des processus gaussiens

HAL Id: tel-00287738

https://tel.archives-ouvertes.fr/tel-00287738

Submitted on 12 Jun 2008

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Contributions à l’étude des processus gaussiens

Ivan Nourdin

To cite this version:

Ivan Nourdin. Contributions à l’étude des processus gaussiens. Mathématiques [math]. Université

Pierre et Marie Curie - Paris VI, 2008. �tel-00287738�

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L +   *9
X
* I 9
* 
)*+ + *9
X
*K 

b

X

₁

(n)



X

1

)     + *+= 
L = *9
*
 + 9
= = 

B

, =

n

_{→ ∞}

L9


n

2H

−1

n

−1

X

k=0

(∆B

k/n

)

2 p.s.

−→ 1.

I?FF K 
L9
  E*+>
  I?FF K =

n

2H−1

X

b

1

(n)

− X

1

p.s.

−→ −

X

1

2

6= 0.

 *9
L =
9
*9
  + 9+9

X



X

t

= x

0

+

Z

t

0

σ(X

s

)dB

s

,

t

∈ [0, 1],

I?FF%K )9 
 E9
*9

σ : R

_{→ R}

X
+L 
 E9>+  D) I?FFK
 )+  E*+  9<
F O9E9L ))99
 =

σ

9 >>
 X+  <9
  
9 9
 + 9

φ : R

2

_{→ R}



σ

L  9 + 
= 9+9


φ(x, y) = x +

Z

y

0

σ(φ(x, z))dz,

x, y

∈ R.

I?FF ^K >=9
 +9 = + 9+9

X

 I?FF%K  9

 ) 

X

t

= φ(x

0

, B

t

),

t

∈ [0, 1].

 ) +L + 9

φ

 + )9) L   >X9 ) L 
 ,

∀x, y, z ∈ R, φ(φ(x, y), z) = φ(x, y + z).


+
 * )9)L  ) 9 * =+= +9))>
 +> I9 !" .#   !" H # ) 9 + >G9 KL9
<9  + 
+9X 
  I?FFK ,

b

X

₁

(n)

≈ φ



x

0

, B

1

−

1

2

V

(2)

n

σ

0

◦ φ(x

0

,

·)



.

I?FF&K J
L +    + I K *9
X
* 

b

X

₁

(n)



X

1

 >
  +    *9> ) 9>
 D> )9 = 

V

n

(2)

(f )

) 9

f = σ

0

◦ φ(x

0

,

·)

F 9
 9
 >


 
 9
+X>
 ) + *9> )+= FJ +  *G> +

b

X

(n)

9*  I?FF%KL 
9 9
 + *G>  D) +

e

X

(n)


 ,

e

X

₀

(n)

= x

0



e

X

_(k+1)/n

(n)

= e

X

_k/n

(n)

+ σ( e

X

_k/n

(n)

)∆B

k/n

+

1

2

σσ

0

_{( e}

_X

(n)

k/n

)(∆B

k/n

)

2

_,

_{k = 0, . . . , n}

− 1

F I?FF CK

?^
      
 * *L 9
) >9
 I*EF  !" H#K = + 
+9X  I?FF &K   + E9>

e

X

₁

(n)

_{≈ φ x}

0

, B

1

+ V

n

(3)

(f )

,

) 9 
 *
 E9
*9

f

 ) 

 I  >
 N ) +*K 

φ



σ

(j)

) 9

j = 0, 1, 2

F J
L *  * +    *9>)9>
 D>)9= 

V

n

(3)

(f )

= @9 
+ *+F 
EL)+ X
+>
L9
9<
 
 E9>+  D) I?FF&K*

V

n

(κ)

I

κ

_{≥ 2}

K  + 

V

n

(2)

=
9
*9
   *G>  ++

κ

_{− 1}

I
 

 = + *G> +   ++ ?L= + *G>  +
I?FFCK  ++ AL*FKF89 
9D9
   !" H # ) 9   ) *F J 
  
 
9 )9<+> 
+L*   +    *9>)9>
 D>) 9= 

V

n

(κ)

(f )

)9 9 


κ

≥ 2

 9 E9
*9

f : R

→ R

>>
 X+L X
+9
 = +   *9
X
*  *G> + ) 9 + =9
QU\ a_P[U

X

t

= x

0

+

Z

t

0

σ(X

s

)dB

s

+

Z

t

0

b(X

s

)ds,

t

∈ [0, 1],

 9

 I*EF !!" =#KL =

H >

1

_/

₂

L )

n

2H

−1

X

b

₁

(n)

_{− X}

1

p.s.

−→ −

1

2

Z

1

0

D

s

X

1

σ

0

(X

s

)ds,

I?FF KK 9

D

X
 +   ++
)   )) 9 

B

F  +  
)   ) 

 =
9
+ *9>)  9

+9X <9

I

H =

1

_/

₂

KL9<
 ) $Y  9 C 
?KK?F 
 
L 
 * *L

√

n b

X

₁

(n)

_{− X}

1

Loi

−→ −

√

1

2

Y

1

Z

1

0

σσ

0

(X

s

)Y

s

−1

dW

s

,

I?FF?BK 9

Y

 9+9

 =9

++ 9 
 +9  9

W

X
 
>9>
 <9


 L 
)

 

B

F  :#  $ 
% #%#%  
9
 >


  +   

V

n

(κ)

(f )

I *

κ

_{≥ 2}



f : R

_{→ R}

>>
 X+K =

Z = B

 
>9>
 <9

E*9

 
*

H

=+*9
= 
 + 
++

(0, 1)

F 9>>
9
 ) 9

 
 < *)9
 + @ *9

F O9 <9 L
9

9  * 9

κ

 )F
 +9L

f

 *9

 ,

n

κH−1

V

n

(κ)

(f )

p.s.

−→ µ

κ

Z

1

0

f (B

s

)ds,

9

µ

κ

X
 + >9>
 9 

κ


 <+ +9  +9

N

_{(0, 1)}

F  *9
 X
*
 )  *+  >9
  )) 
  
+  D)  +9 X

9>< I9 G9> X9 =KF 

κ

 >)L9 =

κ

 ) > = 9


     ?& *


V

n

(κ)

(f )

L + *9>)9>
 D>)9=  )+ *+  9<
F 9>>
9
 @ ) + * 9 + E9
*9

f

  
=>
 X+  ?F +  +9 <
*9

L ) + N  Z  @9 &L 9<G
 @9 ?KL   ( X+ A& 9 
*9 O== ^AL ^ L = 9
L
 + ) 

κ

 + + 

H

,

κ

) 

H <

3

_/

₄

₌

_⇒

√

_{n n}

κH

−1

_V

(κ)

n

(1)

− µ

κ

Loi

−→ N (0, σ

2

H,κ

)

I?F F??K

κ

) 

H =

3

_/

₄

₌

_⇒

r

n

ln n

n

κH

−1

_V

(κ)

n

(1)

− µ

κ

Loi

−→ N (0, σ

3/4,κ

2

)

κ

) 

H >

3

_/

₄

₌

_{⇒ n}

H

_n

κH

−1

_V

(κ)

n

(1)

− µ

κ

Loi

−→ Z

1

I?F F?AK

κ

> )  

H

≤

1

_/

₂

₌

_{⇒ n}

κH−

1

2

V

(κ)

n

(1)

Loi

−→ N (0, σ

2

H,κ

)

I?F F?K

κ

> )  

H >

1

_/

₂

₌

_{⇒ n}

(κ−1)H

_V

(κ)

n

(1)

Loi

−→ N (0, σ

2

H,κ

).

*L

Z

1

X
 + +  > )  ?  RPS\U``V` aU S`Ub<*QTTF 
9
>


 *X
+*9
*



f

F

H =

1

_/

₂

I*  =

B

 
>9>
 <9


 KL +    *9> ) 9>
 D> )9 = 

V

n

(κ)

(f )

I)9

f

*9

KE)) L + +>L

9 >9>
<9




W

[ba_RUbaQbT 

B

F+ )*>
L 
 !" ' # I9   !" # )9 
 N> >9
9
+
 + G9>  $
XG D> ) 9=L  *9 L 9 
+ +<+ 
 
*  )+ X
+  
9
*KL +  )9 = 9
L +9=

n

_{→ ∞}

,

κ

) 

=

_⇒

_√

1

n

n

−1

X

k=0

f (B

k/n

) (

√

n∆B

k/n

)

κ

− µ

κ

Loi

−→

p

µ

2κ

− µ

2

κ

Z

1

0

f (B

s

)dW

s

.

I?F F? K

κ

>)

=

_⇒

√

1

n

n

−1

X

k=0

f (B

k/n

)(

√

n∆B

k/n

)

κ

Loi

−→

Z

1

0

f (B

s

)

q

µ

2κ

− µ

2

κ+1

dW

s

+ µ

κ+1

dB

s

.

I?F F?%K 9
9
 >


  * E*9

 I*    * 9

H

6=

1

_/

₂

KF 
 + *
 +  M
 M   ? I9  ? )9  +  >> D)L *9
*

 +

p

9
 
X+ 9*G= ) ))9  
>9>
 <9

E*9

KL +  >9
L =

f

   
 
 GD) 9G  D)  *9
*  )+ N)9

++L = 9
 ,

κ

)  

H

_{∈ (}

1

_/

₂

_,

3

_/

₄

₎



⇒

√

1

n

n−1

X

k=0

f (B

k/n

) (n

H

∆B

k/n

)

κ

− µ

κ

Loi

−→ σ

H,κ

Z

1

0

f (B

s

)dW

s

,

I?F F?^K 9L 
*9 
 E9L

W

X
 
>9>
 <9


 
)

 

B

F J >
 L I?F F?^K >9
 
*9>)9>
 >+  I?F F? KF M *9

H <

3

_/

₄


 I?F F?^K   N) += )  I?F F??K  I?F F?AKF  *9
L + *9

H >

1

_/

₂

 
 )>
 *G
=L *9>>
9 + N)+=9
 >


F

?C
      


[NNT

_{− 22]}



[NR

_{− 27]}

L
9 9+9
 *9>)+>
 + )9<+>  *9> )9>
 D>)9=  9
  )9   >9>
 <9

E*9

F +  9 = + <9
*  )9 >
  <
*    + + )9+D
> H>F
L *  8+  )
O 9 
 )  J
G9
D ++*  )L
9 9
 >9
 + + 
 , @ 01 26 ? ?? 7S[UbT

q

Vb UbT[UP

≥ 2

L UT

f : R

→ R

VbU ( Sb\T[Sb _P['QbT

(H

q

)

f

∈ C

2q

UTL RSVP TSV`

p

_{∈ (0, ∞)}

UT

0

_{≤ i ≤ 2q}

>

sup

_t∈[0,1]

E



_|f

(i)

_(B

t

)

|

p

<

_∞

-Fb RS`U

S

_n

(q)

(f ) =

n−1

X

k=0

f (B

k/n

) H

q

n

H

∆B

k/n

S

H

q

U`T *U

q

[
&U

RS*db3&U a46UP&[TU a_'b[ RQP >

H

q

(x) = (

−1)

q

e

−

x2

2

d

q

dx

q



e

−

x2

2



.

- Fb `VRRS`U

0 < H <

1

_/

_2q

-$*SP`L 9VQba

n

→ ∞

L

n

qH

−1

V

_n

(q)

(f )

−→

L

2

(

−1)

q

2

q

Z

1

0

f

(q)

(B

s

)ds.

- Fb `VRRS`U

H =

1

_/

_2q

- $*SP`L 9VQba

n

_{→ ∞}

L

1

√

n

V

(q)

n

(f )

Loi

−→

(

−1)

q

2

q

Z

1

0

f

(q)

(B

s

)ds + c

1

_/

2q

,q

Z

1

0

f (B

s

)dW

s

,

S

W

U`T Vb &SVU&UbT <PS,b[Ub `TQbaQPa[ba_RUbaQbT aU

B

L UT

c

1

_/

2q

,q

VbU \Sb`TQbTU  U%R*[\[TU 

a_RUbaQbT `UV*U&UbT aU

q

-- Fb `VRRS`U

1

_/

_2q

_{< H < 1}

₋

1

_/

_2q

- $*SP`L 9VQba

n

_{→ ∞}

L

1

√

n

V

(q)

n

(f )

Loi

−→ c

H,q

Z

1

0

f (B

s

)dW

s

,

S

W

U`T Vb &SVU&UbT <PS,b[Ub `TQbaQPa [ba_RUbaQbT aU

B

L UT

c

H,q

VbU \Sb`TQbTU U%R*[\[TU

a_RUbaQbT `UV*U&UbT aU

q

UT

H

-- Fb `VRRS`U

H = 1

−

1

_/

_2q

- $*SP`L 9VQba

n

→ ∞

L

1

√

n log n

V

(q)

n

(f )

Loi

−→ c

1

−

1

/

2q

,q

Z

1

0

f (B

s

)dW

s

,

S

W

U`TVb &SVU&UbT <PS,b[Ub `TQbaQPa[ba_RUbaQbTaU

B

L UT

c

1−

1

_/

2q

,q

VbU \Sb`TQbTU U%R*[\[TU

a_RUbaQbT `UV*U&UbT aU


     ?K - Fb `VRRS`U

H > 1

₋

1

_/

_2q

- $*SP`L 9VQba

n

_{→ ∞}

L

n

q(1−H)−1

V

(q)

n

(f )

L

2

−→

Z

1

0

f (B

s

)dZ

s

(q)

,

S

Z

(q)

U`T *URPS\U``V` a46UP&[TU a4SPaPU

q

\Sb`TPV[T 8 RQPT[P aU

B

-
+>
L 
*9>)9
 + >9
>

x

κ


 + <  )9+D
> H>L9
9<
 + + 
L =   *9>) * I?F F? KL I?F F?%K  I?F F?^K ,
 .3 6 ? ?'? 7S[T

κ

_{≥ 1}

Vb UbT[UPL UT

f : R

_{→ R}

VbU(Sb\T[Sb _P['QbT

(H

κ

)

-$*SP`L RSVP

n

→ ∞

> - VQba

H <

1

_/

₂

UT

κ

U`T [&RQ[PL

n

H

−1

n

−1

X

k=0

f (B

k/n

)(n

H

∆B

k/n

)

κ

L

2

−→ −

µ

κ+1

₂

Z

1

0

f

0

(B

s

)ds.

I?F F?&K - VQba

H >

1

_/

₂

UT

κ

U`T [&RQ[PL

n

−H

n−1

X

k=0

f (B

k/n

)(n

H

∆B

k/n

)

κ

L

2

−→ κµ

κ−1

Z

1

0

f (B

s

)dB

s

= κµ

κ−1

Z

B

1

0

f (x)dx.

- VQba

H <

1

_/

₄

UT

κ

U`T RQ[PL

n

2H−1

n

−1

X

k=0

f (B

k/n

)



(n

H

∆B

k/n

)

κ

− µ

κ



L

2

−→

1

4



κ

2



µ

κ

−2

Z

1

0

f

00

(B

s

)ds.

- VQba

H =

1

_/

₄

UT

κ

U`T RQ[PL

1

√

n

n

−1

X

k=0

f (B

k/n

)



(n

H

∆B

k/n

)

κ

−µ

κ



Loi

−→

1

₄



κ

2



µ

κ

−2

Z

1

0

f

00

(B

s

)ds+σ

1

/

4

,κ

Z

1

0

f (B

s

)dW

s

,

I?F F?CK RSVP

σ

1

_/

4

,κ

VbU \Sb`TQbTU U%R*[\[TU UT

W

Vb &SVU&UbT <PS,b[Ub `TQbaQPa [ba_RUb ) aQbT aU

B

-- VQba

1

_/

₄

_{< H <}

3

_/

₄

UT

κ

U`T RQ[PL

1

√

n

n−1

X

k=0

f (B

k/n

)



(n

H

∆B

k/n

)

κ

− µ

κ



Loi

−→ σ

H,κ

Z

1

0

f (B

s

)dW

s

,

I?F F?KK RSVP

σ

H,κ

VbU\Sb`TQbTU U%R*[\[TU UT

W

Vb &SVU&UbT <PS,b[Ub `TQbaQPa[ba_RUbaQbT aU

-AB
      - VQba

H =

3

_/

₄

UT

κ

U`T RQ[PL

1

√

n log n

n

−1

X

k=0

f (B

k/n

)



(n

H

∆B

k/n

)

κ

− µ

κ



Loi

−→ σ

3

4

,κ

Z

1

0

f (B

s

)dW

s

,

RSVP

σ

3

_/

4

,κ

VbU \Sb`TQbTU U%R*[\[TU UT

W

Vb &SVU&UbT <PS,b[Ub `TQbaQPa [ba_RUb ) aQbT aU

B

-
- VQba

H >

3

_/

₄

UT

κ

U`T RQ[PL

n

1−2H

n−1

X

k=0

f (B

k/n

)



(n

H

∆B

k/n

)

κ

− µ

κ



L

2

−→ µ

κ−2



κ

2

 Z

1

0

f (B

s

)dZ

s

(2)

,

RSVP

Z

(2)

*U RPS\U``V` aU S`Ub<*QTT \Sb`TPV[T 8 RQPT[P aU

B

- : #  $ 
% #%#%   "% %

1

_/

₄

   "#     # % 
 + G9> ?F F?L + *

q = 2



H =

1

_/

₄

  E9>
 +  
 *9
@*  *
 ) I Z YD K (
9
^? *9
*

 + *9>)9>
 D>)9=  9
  ) 9  ) 9 + 9+9
 + =9
 + *G+ 9*G= X )  
< <+
* )*>)F + )*>
L
99


u = u(t, x)

+ 9+9
 + =9

u

t

=

1

2

u

xx

+ ˙

W (t, x),

t

≥ 0, x ∈ R,

I?F%F ABK * *9
9

+

u(0, x) = 0

F N9


x

∈ R

 *9
 9
 + )9*

F = (F

t

)

t

≥0


 ) 

F

t

= u(t, x)

) 9

t

≥ 0

F J+9

F

 X
L  ) 9 E9
*9
 *9
*

E(F

s

F

t

) =

1

√

2π

(t + s)

1

2

− |t − s|

1

2

.

J
L

F

 
) 9*X
9>+ 
*

1

_/

₄

* 
 E9
*9
 *9
*  )9*G  *++  <9

E*9

 
*

1

_/

₄

F 
 9
*+ ^?L (
9
*9
@* =

F

 + X+  *G
X>
  <+ 
F 9

g : R

→ R

X+ 

t

≥ 0

L I9
@*K

g(F

t

) = g(0) +

Z

t

0

g

0

(F

s

)dF

s

+ c

Z

t

0

g

00

(F

s

)dW

s

,

I?F%F A?K 9

c


 *
 *9

 
++ I 9

 N)+*>
KL9

W

X
 
>9 >
 <9


 [ba_RUbaQbT 

F

F 9>> + N)+= (
9

 9
*+ ^?L + >

++  >9
 I?F%F A?K  
+D + *9> ) 9>
 D> )9  =  *
9
 )9  IX
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Z

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1

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B

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√

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t

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t

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E



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2

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0

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;   

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W*G  F 89 
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# F 89 
L F **  JF ++* IABBCK , V*T[QP[QTU bSP&Q* QRRPS %[&QT[Sb V`[b
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F

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L
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A^


   


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9
 


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9>>
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X
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κ = 2

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1

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√

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3

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)

 +
99
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

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n

−1

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k=0

n

H

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n

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n

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i

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∗)

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n

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2

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2

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K , 

(U, V )

 
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9>+
 

p, q

∈ N

∗

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p

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p



p = q

.

I AFAF?K  +    * *+*+  
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9
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1

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_n

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9<
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σ

H,2

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9

F

n

=

1

√

n

n−1

X

k=0



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H

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2

− 1



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n

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X
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N

_{(0, σ}

2

H,2

)

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(G

1

, . . . , G

p

)

 
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9>+   9

9

r

ij

= E(G

i

G

j

)

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L )9 9

k

1

, . . . , k

p

∈ N

∗

L

E[H

k

1

(G

1

) . . . H

k

p

(G

p

)] =



k

1

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p

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2

q

_q!

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1

r

i

1

j

1

. . . r

i

q

j

q



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p

i=1

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0

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0


9
9

P

1

 
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i

1

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1

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i

1

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1

, . . . , i

q

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q

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E

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k

1


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1

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L

k

p


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p

F 
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p = κ

* +

k

i

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κ

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κ



F

n

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9
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q

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L
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H

F 
 + L
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X =

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q

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r



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q

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n

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n

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q

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n

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}

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n→∞

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q

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n

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4

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r = 1, . . . , q

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L

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n

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r

f

n

k

H

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q

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n

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k

2

H

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L

2

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n


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d

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1

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n

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q

1

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1

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q

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d

n

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i

n

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H

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q

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(f

j

n

)I

q

i

(f

i

n

)



= δ

ij

,

S

δ

ij

a_`[
bU *U `d&<S*U aU
PSbU\CUP- $*SP` *U` aUV% Q``UPT[Sb` `V[QbTU` `SbT _ 9V[Q ) *UbTU` >

IK 9VQba

n

_{→ ∞}

L *Q `V[TU

{F

n

}

n≥1

\SbUP
U Ub *S[ UP` Vb U\TUVP Q*_ QTS[PU
QV``[Ub `TQbaQPa

N

_d

_{(0, I}

_d

₎



I

d

U`T *Q &QTP[\U [aUbT[T_

d

_{× d}

 L IK RSVP \Q9VU

j = 1, . . . , d

L 9VQba

n

_{→ ∞}

L *Q `V[TU

{I

q

j

(f

j

n

)

}

n

≥1

\SbUP
U Ub *S[ UP` *Q QP[Q<*U Q*_ QTS[PU
QV``[UbbU `TQbaQPa

N

_{(0, 1)}

-
 *9
=
* >>  I> 9
  >)9
 )9 + ))+*9
K  G9> AFF A = 9
L
EL9>=>
 *9
X
* <+ 
 + G9> AFF? ,  73/3 '? ? ? 7S[T

q

Vb UbT[UP

≥ 2

L UT `S[T

{I

q

(f

n

)

}

n

≥1

VbU `V[TU a4[bT_
PQ*U` &V* ) T[R*U` _P['QbT

--

a4VbU RQPT UT VbU aU` 9VQTPU \Sba[T[Sb` [ ) [ aV T_SP
&U -- a4QVTPU RQPT- $*SP`L RSVP TSVT

h

∈ H

L 9VQba

n

→ ∞

L *U \SVR*U

I

q

(f

n

), X(h)

\SbUP
U Ub *S[ UP`

N, X(h)

L S

N

_{∼ N (0, 1)}

U`T [ba_RUbaQbTU aU

X(h)

-9>> N>)+ *
 ))+*9
  G9> AFF?  AFF AL 9
) *L 
*L +    ,

p

9
 
X+ 9*G= E*9

 I9* UT Q*- ?KLE9
*9

++ = =  ) 9* X
< I G+ UT Q*- ?^KL >) +9*+ 9
*9
 >9>
 <9

E*9

 I H 8+ AKL *G> ))9N>9
)9+=9
 
++9*G=*+ !!" =#L G9> +> GE=
* ) 9 + *G> )  +9    ) * G9>9X
 I
**  **   L L ** %?KL 
+D ) 9
+  + )G I Z+ 

   AK UT Q*-  AKL >9
 + 
* 9>+ IO 9  
 ^%KL 9
  )9   >9>
 <9

E*9

  !!@"' '#L9
 )9   >9>
 <9

 !"' #L 9
 )
* )9  )9* X
  **9>
 9

 I Z
98+
UT Q*- KL *F  ; 
%  #!!  #!!%#%       J * 9

 ** I9  !" ' A #   !" '.KL
9
9 9>>  *>>
 ) = + *9
9
IK 
 + OG9> AFF?  *9><
  >++ * + >G9   (
I9 )+ <K  )>L ) N>)+L 9<
 I 
 >
 9

>
 > ) +K  
9
*  D )  
 , @ 01 26 '??? Sb`[a_PSb` VbU`V[TU

(Z

n

)

n

≥1

aU *Q ( SP&U

Z

n

= I

2

(f

n

)

L QU\

f

n

∈ H

2

L UT bSTSb`

κ

(n)

r

L

r

_{≥ 1}

L *U` \V&V*QbT` aU

Z

n

- 7VRRS`Sb` 9VU

κ

(n)

₂

_{−→ 1}

9VQba

n

_{→ ∞}

-$*SP`L 9VQba

n

_{→ ∞}

L Sb Q 9VU

Z

n

Loi

−→ Z ∼ N (0, 1)

`[ UT `UV*U&UbT `[

κ

(n)

₄

_{−→ 0}

- IQb` \U \ Q`L Sb Q &&U QVTS&QT[9VU&UbT &[UV%L 8 `QS[P >

sup

z

∈R

|P (Z

n

≤ z) − P (Z ≤ z)| ≤

s

κ

(n)

4

6

+ (κ

(n)

2

− 1)

2

.

7[L aUR*V`L Sb QL 9VQba

n

_{→ ∞}

L

κ

(n)

₂

_{− 1}

κ

(n)

₄

6

+ (κ

(n)

2

− 1)

2

−→ 0,

κ

(n)

3

q

κ

(n)

₄

6

+ (κ

(n)

2

− 1)

2

−→ α

UT

κ

(n)

8



κ

(n)

₄

6

+ (κ

(n)

2

− 1)

2



2

−→ 0

Q*SP`

P (Z

n

≤ z) − P (Z ≤ z)

q

κ

(n)

₄

6

+ (κ

(n)

2

− 1)

2

−→

α

6

√

2π

(1

− z

2

_)e

−

z2

2

,

9VQba

n

_{→ ∞.}

I AF F K Jb RQPT[\V*[UPL `[

α

6= 0

L [* U%[`TU

c > 0

UT

n

0

≥ 1

TU* 9VUL RSVP TSVT

n

≥ n

0

>

sup

z∈R

|P (Z

n

≤ z) − P (Z ≤ z)| ≥ c

s

κ

(n)

4

6

+ (κ

(n)

2

− 1)

2

.

9
*

 + *+= G9>  Z  @9 I?F F??KL 9
) +9 
9
* + + 
 =L 
9 *9


*L *9
 + ) > + 9*
 
 <9
  D) ZD
 I?F F??K ,