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# Lectures on Gaussian approximations with Malliavin calculus

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Description

Niveau: Supérieur, Doctorat, Bac+8
Lectures on Gaussian approximations with Malliavin calculus Ivan Nourdin Université de Lorraine, Institut de Mathématiques Élie Cartan B.P. 70239, 54506 Vandoeuvre-lès-Nancy Cedex, France March 20th, 2012 Overview. In a seminal paper of 2005, Nualart and Peccati [37] discovered a surprising central limit theorem (called the Fourth Moment Theorem in the sequel) for sequences of multiple stochastic integrals of a xed order: in this context, convergence in distribution to the standard normal law is equivalent to convergence of just the fourth moment. Shortly afterwards, Peccati and Tudor [44] gave a multidimensional version of this characterization. Since the publication of these two beautiful papers, many improvements and developments on this theme have been considered. Among them is the work by Nualart and Ortiz-Latorre [36], giving a new proof only based on Malliavin calculus and the use of integration by parts on Wiener space. A second step is my joint paper [25] (written in collaboration with Peccati) in which, by bringing together Stein's method with Malliavin calculus, we have been able (among other things) to associate quantitative bounds to the Fourth Moment Theorem. It turns out that Stein's method and Malliavin calculus t together admirably well. Their interaction has led to some remarkable new results involving central and non-central limit theorems for functionals of innite-dimensional Gaussian elds.

• hermite polynomials

• gaussian limit

• hq has

• central limit

• function ? ?

• centered random

• fondation des sciences mathématiques

Subjects

##### Hermite polynomials

Informations

ρ :Z→R E[X X ] =ρ(k−l) k,l> 1 ρ(0) = 1k l
X N(0,1)k
φ :R→R

1 22 2 −x /2√E[φ (X )] = φ (x)e dx<∞.1
2π R
H ,H ,...0 1
2 3H = 1 H =X H =X −1 H =X −3X q H0 1 2 3 q
XH =H +qHq q+1 q−1
22 −x /2φ L (R,e dx)
∞∑
φ(x) = a H (x).q q
q=0
d> 0 q> 0 a = 0 φq
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U,V ∼N(0,1)
{
0 p =q
E[H (U)H (V)] =p q qq!E[UV] p =q.
p = 0 E[H (X )] = 0 q > 1q 1
a = E[φ(X )]0 1
∞∑
2 2E[φ (X )] = q!a .1 q
q=0
{X } φ :R→Rk k>1∑
da =E[φ(X )] = 0 |ρ(k)| <∞ ρ0 1 k∈Z
{X } d φ d> 1 n→∞k k>1
n∑1 law 2V = √ φ(X ) → N(0,σ ),n k
n
k=1

∞∑ ∑
2 2 qσ = q!a ρ(k) ∈ [0,∞).q
q=d k∈Z
2σ ∈ [0,∞)
{Y }k k>1
2σ > 0 FY
−1F FYY
−1F (u) = inf{y∈R : u6F (y)}, u∈ (0,1).YY
law−1U ∼ U F (U) = Y[0,1] 1Y∫ 2X1 1 −t /2√ e dt U[0,1]−∞2π
law
φ(X ) = Y1 1
( )∫ x1 2−1 −t /2φ(x) =F √ e dt , x∈R.
Y 2π −∞
ρ(0) = 1 ρ(k) = 0 k = 0 {X }k k>1
N(0,1)
 
n n ∞∑ ∑ ∑1 1law law 2 √ Y = √ φ(X ) → N 0, q!a ,k k qn n
k=1 k=1 q=d
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−D 1d> 2 ρ(k)∼|k| |k|→∞ D∈ (0, )
d∑ndD/2−1n φ(X )kk=1 ∑
d|ρ(k)| = ∞
k∈Z
d> 2
∑ ∑ [nt]n
k=1 k=1
t> 0
σ
Vn
2N(0,σ )
2 4V σ 3σn

  2
∞ n ∞ n∑ ∑ ∑ ∑1 12   E[V ] = E a H (X ) = a a E[H (X )H (X )]q q p q p qn k k ln n
q=d k=1 p,q=d k,l=1
∞ n ∞∑ ∑ ∑ ∑ ( )1 |r|2 q 2 q= q!a ρ(k−l) = q!a ρ(r) 1− 1 .{|r|<n}q qn n
q=d k,l=1 q=d r∈Z
q>d r∈Z
( )|r|2 q 2 qq!a ρ(r) 1− 1 →q!a ρ(r) n→∞.{|r|<n}q qn

2 2|ρ(k)| =|E[X X ]|6 E[X ]E[X ] = 11 k+1 1 1+k
( )|r|2 q 2 q 2 dq!a |ρ(r)| 1− 1 6q!a |ρ(r)| 6q!a |ρ(r)| ,{|r|<n}q q qn
∑ ∑ ∑∞ 2 d 2 dq!a |ρ(r)| =E[φ (X )]× |ρ(r)| <∞1q=d r∈Z q r∈Z
2 2 2E[V ]→σ n→∞ σ ∈ [0,∞)n
φ =H q> 1q
φ =P ∈R[X]
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22 −x /2φ φ ∈ L (R,e dx)
N> 1
N n ∞ n∑ ∑ ∑ ∑1 1
V = √ a H (X )+√ a H (X ) =:V +R .n q q k q q k n,N n,Nn n
q=d k=1 q=N+1 k=1
∞∑ ∑
2 2 dsupE[R ]6 q!a × |ρ(r)| → 0 N →∞.n,N q
n>1
q=N+1 r∈Z
∑∞2 2E[φ (X )] = q!a <∞1 q=d q
N n→∞
 
N∑ ∑
law 2 q V → N 0, q!a ρ(k) .n,N q
q=d k∈Z
law 2V =V +R → N(0,σ ) n→∞n n,N n,N
{X }k k>1
2L (Ω)
H := span{X ,X ,...}1 2
NR N > 1
2 2 NL (R ) H ≃ L (R ) H ≃R+ +
2Φ :H→L (R ) e = Φ(X ) k> 1+ k k

ρ(k−l) =E[X X ] = e (x)e (x)dx, k,l> 1k l k l
0
B = (B )t t>0
{∫ }∞
law
{X } = e (t)dB ,k k>1 k t
0 k>1
ek
2e ∈ L (R )+
∥e∥ 2 = 1L (R )+
(∫ ) ∫ ∫ ∫∞ ∞ t t1 q−1
H e(t)dB =q! dB e(t ) dB e(t )... dB e(t ).q t t 1 t 2 t q1 2 q
0 0 0 0
(∫ ) ∫ ∫ ∫2∞ ∞ t ∞1
2e(t)dB = 2 dB e(t ) dB e(t )+ e(t) dtt t 1 t 21 2
0 0 0 0
∫ ∫∞ t1
= 2 dB e(t ) dB e(t )+1,t 1 t 21 2
0 0
φ ψ r s φ⊗ψ
r +s φ⊗ψ(x ,...,x ) = φ(x ,...,x )ψ(x ,...,x )1 r+s 1 r r+1 r+s
⊗qq> 1 e e
e⊗...⊗e e q
q2f ∈ L (R ) f(x ,...,x ) = f(x ,...,x )1 q σ(1) σ(q)+
σ∈S x ,...,x ∈Rq 1 q +
∫ ∫ ∫ ∫∞ t t1 q−1
I (f) = f(t ,...,t )dB ...dB :=q! dB dB ... dB f(t ,...,t ).q 1 q t t t t t 1 q1 q 1 2 q
q
R 0 0 0+
(∫ )∞
⊗qH e(t)dB =I (e ).q t q
0
q > 2 {f }n n>1
q2 2 2 2L (R ) E[I (f ) ] = q!∥f ∥ → σqq n n 2+ L (R )
+
n→∞ σ> 0 n→∞
law 2I (f ) → N(0,σ )q n
law4 4E[I (f ) ] → 3σq n
2q−2r2∥f ⊗ f ∥ 2q−2r → 0 r = 1,...,q−1 f ⊗ f L (R )n r n 2 n r n +L (R )+
f ⊗ f (x ,...,x )n r n 1 2q−2r∫
= f (x ,...,x ,y ,...,y )f (x ,...,x ,y ,...,y )dy ...dy .n 1 q−r 1 r n q−r+1 2q−2r 1 r 1 r
rR+
2I (f ) N(0,σ )q n
43σ
q> 2 f
q2 2 2L (R ) E[I (f) ] =q!∥f∥ = 1qq 2+ L (R )+
√ ∫ √