Steins method and Malliavin calculus for independent random - - PowerPoint PPT Presentation

▶
stein s method and malliavin calculus for independent
SMART_READER_LITE
LIVE PREVIEW

Steins method and Malliavin calculus for independent random - - PowerPoint PPT Presentation

Jul 17, 2023 190 likes •580 views

Steins method Motivation Discrete Malliavin calculus Convergence results Steins method and Malliavin calculus for independent random variables H el` ene Halconruy under the supervision of Laurent Decreusefond T el ecom

slide-1
SLIDE 1

1/ 14 Stein’s method Motivation Discrete Malliavin calculus Convergence results

Stein’s method and Malliavin calculus for independent random variables

H´ el` ene Halconruy under the supervision of Laurent Decreusefond T´ el´ ecom ParisTech - Universit´ e de Paris Saclay bla bla

slide-2
SLIDE 2

2/ 14 Stein’s method Motivation Discrete Malliavin calculus Convergence results

Introduction

Theorem (Central Limit Theorem)

Let (Xn, n ≥ 1) be a sequence of i.i.d. twice integrable random variables defined on (Ω, A, P) √n 1 n

n

  • k=1

Xk

  • − E [X1]
  • D

− − − − →

n→∞ N(0, var[X1]).

slide-3
SLIDE 3

2/ 14 Stein’s method Motivation Discrete Malliavin calculus Convergence results

Introduction

Theorem (Central Limit Theorem)

Let (Xn, n ≥ 1) be a sequence of i.i.d. twice integrable random variables defined on (Ω, A, P) √n 1 n

n

  • k=1

Xk

  • − E [X1]
  • D

− − − − →

n→∞ N(0, var[X1]).

→ Rate of convergence for the law of numbers.

slide-4
SLIDE 4

2/ 14 Stein’s method Motivation Discrete Malliavin calculus Convergence results

Introduction

Theorem (Central Limit Theorem)

Let (Xn, n ≥ 1) be a sequence of i.i.d. twice integrable random variables defined on (Ω, A, P) √n 1 n

n

  • k=1

Xk

  • − E [X1]
  • F (X1,··· ,Xn)

D

− − − − →

n→∞ N(0, var[X1]).

→ Rate of convergence for the law of numbers. Question : how to estimate dist(F ∗Pn, P) where Pn := ⊗n

k=1PXk

and for P, Q measures on a Polish space F distW (P, Q) = sup

g∈T

  • g dP −
  • g dQ
  • and T = {g ∈ C1(R, R) : g′∞ ≤ 1}?
slide-5
SLIDE 5

2/ 14 Stein’s method Motivation Discrete Malliavin calculus Convergence results

Introduction

Theorem (Central Limit Theorem)

Let (Xn, n ≥ 1) be a sequence of i.i.d. twice integrable random variables defined on (Ω, A, P) √n 1 n

n

  • k=1

Xk

  • − E [X1]
  • F (X1,··· ,Xn)

D

− − − − →

n→∞ N(0, var[X1]).

→ Rate of convergence for the law of numbers. Question : how to estimate dist(F ∗Pn, P) where Pn := ⊗n

k=1PXk

and for P, Q measures on a Polish space F distW (P, Q) = sup

g∈T

  • g dP −
  • g dQ
  • and T = {g ∈ C1(R, R) : g′∞ ≤ 1}?
slide-6
SLIDE 6

3/ 14 Stein’s method Motivation Discrete Malliavin calculus Convergence results

Stein’s method (1)

  • Caracterization of the target measure P.

For all g ∈ T ,

  • Lg dQ = 0 ⇐

⇒ Q = P. L is the Stein operator associated to P.

slide-7
SLIDE 7

3/ 14 Stein’s method Motivation Discrete Malliavin calculus Convergence results

Stein’s method (1)

  • Caracterization of the target measure P.

For all g ∈ T ,

  • Lg dQ = 0 ⇐

⇒ Q = P. L is the Stein operator associated to P.

  • Resolution of the Stein equation

Lϕg = g −

  • g dP,

(1) for any g ∈ T , then g ∈ T ⇐ ⇒ ϕg ∈ F.

slide-8
SLIDE 8

3/ 14 Stein’s method Motivation Discrete Malliavin calculus Convergence results

Stein’s method (1)

  • Caracterization of the target measure P.

For all g ∈ T ,

  • Lg dQ = 0 ⇐

⇒ Q = P. L is the Stein operator associated to P.

  • Resolution of the Stein equation

Lϕg = g −

  • g dP,

(1) for any g ∈ T , then g ∈ T ⇐ ⇒ ϕg ∈ F.

  • Equivalent problem

sup

g∈T

  • g dQ −
  • g dP
  • = sup

ϕ∈F

|E [Lϕ(X)] | where X ∼ Q.

slide-9
SLIDE 9

3/ 14 Stein’s method Motivation Discrete Malliavin calculus Convergence results

Stein’s method (1)

  • Caracterization of the target measure P.

For all g ∈ T ,

  • Lg dQ = 0 ⇐

⇒ Q = P. L is the Stein operator associated to P.

  • Resolution of the Stein equation

Lϕg = g −

  • g dP,

(1) for any g ∈ T , then g ∈ T ⇐ ⇒ ϕg ∈ F.

  • Equivalent problem

sup

g∈T

  • g dQ −
  • g dP
  • = sup

ϕ∈F

|E [Lϕ(X)] | where X ∼ Q.

slide-10
SLIDE 10

4/ 14 Stein’s method Motivation Discrete Malliavin calculus Convergence results

Stein’s method (2)

  • Principle :

distT (P, Q) = sup

ϕ∈F

|E [Lϕ(X)] |

slide-11
SLIDE 11

4/ 14 Stein’s method Motivation Discrete Malliavin calculus Convergence results

Stein’s method (2)

  • Principle :

distT (P, Q) = sup

ϕ∈F

|E [Lϕ(X)] | = sup

ϕ∈F

|E [L1ϕ(X)] + E [L2ϕ(X)]|

slide-12
SLIDE 12

4/ 14 Stein’s method Motivation Discrete Malliavin calculus Convergence results

Stein’s method (2)

  • Principle :

distT (P, Q) = sup

ϕ∈F

|E [Lϕ(X)] | = sup

ϕ∈F

|E [L1ϕ(X)] + E [L2ϕ(X)]|

  • Idea : transform L1ϕ(X) into −L2ϕ(X)+ remainder

→ rate of convergence. 3 methods :

  • 1. Exchangeable pairs.
  • 2. Size-biased.
  • 3. Malliavin integration by parts.
slide-13
SLIDE 13

4/ 14 Stein’s method Motivation Discrete Malliavin calculus Convergence results

Stein’s method (2)

  • Principle :

distT (P, Q) = sup

ϕ∈F

|E [Lϕ(X)] | = sup

ϕ∈F

|E [L1ϕ(X)] + E [L2ϕ(X)]|

  • Idea : transform L1ϕ(X) into −L2ϕ(X)+ remainder

→ rate of convergence. 3 methods :

  • 1. Exchangeable pairs.
  • 2. Size-biased.
  • 3. Malliavin integration by parts.
slide-14
SLIDE 14

5/ 14 Stein’s method Motivation Discrete Malliavin calculus Convergence results

Finite dimension differential calculus and Malliavin calculus Analogous terminology Differential calculus on Malliavin calculus on Classical Euclidian spaces Wiener space Vectors Functions Gradient

slide-15
SLIDE 15

5/ 14 Stein’s method Motivation Discrete Malliavin calculus Convergence results

Finite dimension differential calculus and Malliavin calculus Analogous terminology Differential calculus on Malliavin calculus on Classical Euclidian spaces Wiener space Vectors Paths of Brownian motion Functions Gradient

slide-16
SLIDE 16

5/ 14 Stein’s method Motivation Discrete Malliavin calculus Convergence results

Finite dimension differential calculus and Malliavin calculus Analogous terminology Differential calculus on Malliavin calculus on Classical Euclidian spaces Wiener space Vectors Paths of Brownian motion Random variables Functions = Functionals of the paths Gradient

slide-17
SLIDE 17

5/ 14 Stein’s method Motivation Discrete Malliavin calculus Convergence results

Finite dimension differential calculus and Malliavin calculus Analogous terminology Differential calculus on Malliavin calculus on Classical Euclidian spaces Wiener space Vectors Paths of Brownian motion Random variables Functions = Functionals of the paths Gradient Malliavin derivative

slide-18
SLIDE 18

6/ 14 Stein’s method Motivation Discrete Malliavin calculus Convergence results

Stein-Malliavin criterion on the Gaussian space

  • H = L2(T, B, µ) real separable Hilbert space.
  • X = {X(h), h ∈ H} centered I.G.P. and ˜

Q = X∗P.

  • S : space of cylindrical random variables of the form

F = f(X(h1), . . . , X(hn)) ; f ∈ Cc(Rn, R), hi ∈ H. For F ∈ S, DF =

n

  • i=1

∂f ∂xi (X(h1), . . . , X(hn)) hi.

  • δ adjoint of D and L = −δD ”Laplacian operator”.

Theorem (Nourdin, Peccati)

For any F ∈ D1,2 with E [F] = 0, distW

  • F ∗ ˜

Q, P

  • ≤
  • E
  • 1 − DF , −DL−1FH
  • 2 1

2 .

slide-19
SLIDE 19

7/ 14 Stein’s method Motivation Discrete Malliavin calculus Convergence results

Sketch of the proof

  • Lϕ(x) = xϕ(x) − ϕ′(x) =: L1ϕ(x) + L2ϕ(x).
  • F = {ϕ ∈ C2(R, R) : ϕ′∞ ≤ 1, ϕ′′∞ ≤ 2}.

E   

L1ϕ(F )

  • F
  • L(L−1F )

ϕ(F)   

slide-20
SLIDE 20

7/ 14 Stein’s method Motivation Discrete Malliavin calculus Convergence results

Sketch of the proof

  • Lϕ(x) = xϕ(x) − ϕ′(x) =: L1ϕ(x) + L2ϕ(x).
  • F = {ϕ ∈ C2(R, R) : ϕ′∞ ≤ 1, ϕ′′∞ ≤ 2}.

E   

L1ϕ(F )

  • F
  • L(L−1F )

ϕ(F)    = E

  • −δ(DL−1F) ϕ(F)
slide-21
SLIDE 21

7/ 14 Stein’s method Motivation Discrete Malliavin calculus Convergence results

Sketch of the proof

  • Lϕ(x) = xϕ(x) − ϕ′(x) =: L1ϕ(x) + L2ϕ(x).
  • F = {ϕ ∈ C2(R, R) : ϕ′∞ ≤ 1, ϕ′′∞ ≤ 2}.

E   

L1ϕ(F )

  • F
  • L(L−1F )

ϕ(F)    = E

  • −δ(DL−1F) ϕ(F)

I.P.P. = E

  • Dϕ(F) , −DL−1FH
slide-22
SLIDE 22

7/ 14 Stein’s method Motivation Discrete Malliavin calculus Convergence results

Sketch of the proof

  • Lϕ(x) = xϕ(x) − ϕ′(x) =: L1ϕ(x) + L2ϕ(x).
  • F = {ϕ ∈ C2(R, R) : ϕ′∞ ≤ 1, ϕ′′∞ ≤ 2}.

E   

L1ϕ(F )

  • F
  • L(L−1F )

ϕ(F)    = E

  • −δ(DL−1F) ϕ(F)

I.P.P. = E

  • Dϕ(F) , −DL−1FH
  • = E
  • ϕ′(F)DF , −DL−1FH
  • (chain rule)
slide-23
SLIDE 23

7/ 14 Stein’s method Motivation Discrete Malliavin calculus Convergence results

Sketch of the proof

  • Lϕ(x) = xϕ(x) − ϕ′(x) =: L1ϕ(x) + L2ϕ(x).
  • F = {ϕ ∈ C2(R, R) : ϕ′∞ ≤ 1, ϕ′′∞ ≤ 2}.

E   

L1ϕ(F )

  • F
  • L(L−1F )

ϕ(F)    = E

  • −δ(DL−1F) ϕ(F)

I.P.P. = E

  • Dϕ(F) , −DL−1FH
  • = E
  • ϕ′(F)DF , −DL−1FH
  • (chain rule)

= E

  • −L2ϕ(F )

ϕ′(F)

remainder

  • −ϕ′(F)
  • 1 + DF , −DL−1FH
slide-24
SLIDE 24

7/ 14 Stein’s method Motivation Discrete Malliavin calculus Convergence results

Sketch of the proof

  • Lϕ(x) = xϕ(x) − ϕ′(x) =: L1ϕ(x) + L2ϕ(x).
  • F = {ϕ ∈ C2(R, R) : ϕ′∞ ≤ 1, ϕ′′∞ ≤ 2}.

E   

L1ϕ(F )

  • F
  • L(L−1F )

ϕ(F)    = E

  • −δ(DL−1F) ϕ(F)

I.P.P. = E

  • Dϕ(F) , −DL−1FH
  • = E
  • ϕ′(F)DF , −DL−1FH
  • (chain rule)

= E

  • −L2ϕ(F )

ϕ′(F)

remainder

  • −ϕ′(F)
  • 1 + DF , −DL−1FH

Tools

  • A caracterization of P in terms of 1st-order differential operators.
slide-25
SLIDE 25

7/ 14 Stein’s method Motivation Discrete Malliavin calculus Convergence results

Sketch of the proof

  • Lϕ(x) = xϕ(x) − ϕ′(x) =: L1ϕ(x) + L2ϕ(x).
  • F = {ϕ ∈ C2(R, R) : ϕ′∞ ≤ 1, ϕ′′∞ ≤ 2}.

E   

L1ϕ(F )

  • F
  • L(L−1F )

ϕ(F)    = E

  • −δ(DL−1F) ϕ(F)

I.P.P. = E

  • Dϕ(F) , −DL−1FH
  • = E
  • ϕ′(F)DF , −DL−1FH
  • (chain rule)

= E

  • −L2ϕ(F )

ϕ′(F)

remainder

  • −ϕ′(F)
  • 1 + DF , −DL−1FH

Tools

  • A caracterization of P in terms of 1st-order differential operators.
  • A Malliavin derivative operator D.
slide-26
SLIDE 26

7/ 14 Stein’s method Motivation Discrete Malliavin calculus Convergence results

Sketch of the proof

  • Lϕ(x) = xϕ(x) − ϕ′(x) =: L1ϕ(x) + L2ϕ(x).
  • F = {ϕ ∈ C2(R, R) : ϕ′∞ ≤ 1, ϕ′′∞ ≤ 2}.

E   

L1ϕ(F )

  • F
  • L(L−1F )

ϕ(F)    = E

  • −δ(DL−1F) ϕ(F)

I.P.P. = E

  • Dϕ(F) , −DL−1FH
  • = E
  • ϕ′(F)DF , −DL−1FH
  • (chain rule)

= E

  • −L2ϕ(F )

ϕ′(F)

remainder

  • −ϕ′(F)
  • 1 + DF , −DL−1FH

Tools

  • A caracterization of P in terms of 1st-order differential operators.
  • A Malliavin derivative operator D.
  • An integration by parts formula including δ.
slide-27
SLIDE 27

8/ 14 Stein’s method Motivation Discrete Malliavin calculus Convergence results

Stein-Malliavin criterion on a discrete setting Question : how to estimate distW ˜ Q, P

  • if ˜

Q = F ∗PA where PA = ⊗a∈APa (A countable set) and F is a functional of independent random variables ?

slide-28
SLIDE 28

8/ 14 Stein’s method Motivation Discrete Malliavin calculus Convergence results

Stein-Malliavin criterion on a discrete setting Question : how to estimate distW ˜ Q, P

  • if ˜

Q = F ∗PA where PA = ⊗a∈APa (A countable set) and F is a functional of independent random variables ? ... And then to construct a Malliavin calculus on a denumerable product of probability space (EA, EA, PA) where EA =

a∈A Ea, EA = ∨ a∈AEa and PA = ⊗a∈APa ?

slide-29
SLIDE 29

9/ 14 Stein’s method Motivation Discrete Malliavin calculus Convergence results

Gradient S : space of cylindrical random variables of the form F = F(X1, . . . , Xn).

Definition

For F ∈ S, DF ∈ L2(A × EA) is defined by : For all a ∈ A, DaF(XA) = F(XA) − E [F(XA) | Ga] = F(XA) − E′ [F(XAa, X′

a)]

where Ga = σ{Xb, b = a} and X′

a is an independent copy of Xa.

(2)

slide-30
SLIDE 30

9/ 14 Stein’s method Motivation Discrete Malliavin calculus Convergence results

Gradient S : space of cylindrical random variables of the form F = F(X1, . . . , Xn).

Definition

For F ∈ S, DF ∈ L2(A × EA) is defined by : For all a ∈ A, DaF(XA) = F(XA) − E [F(XA) | Ga] = F(XA) − E′ [F(XAa, X′

a)]

where Ga = σ{Xb, b = a} and X′

a is an independent copy of Xa.

  • The domain of D, D is the closure of S w.r.t. the norm

F2

D = F2 L2(EA) + DF2 L2(A×EA)

  • Property : for F, G ∈ S,

E

  • a∈A

DaF · G

  • = E
  • F
  • a∈A

DaG

  • (2)
slide-31
SLIDE 31

9/ 14 Stein’s method Motivation Discrete Malliavin calculus Convergence results

Gradient S : space of cylindrical random variables of the form F = F(X1, . . . , Xn).

Definition

For F ∈ S, DF ∈ L2(A × EA) is defined by : For all a ∈ A, DaF(XA) = F(XA) − E [F(XA) | Ga] = F(XA) − E′ [F(XAa, X′

a)]

where Ga = σ{Xb, b = a} and X′

a is an independent copy of Xa.

  • The domain of D, D is the closure of S w.r.t. the norm

F2

D = F2 L2(EA) + DF2 L2(A×EA)

  • Property : for F, G ∈ S,

E

  • a∈A

DaF · G

  • = E
  • F
  • a∈A

DaG

  • (2)
slide-32
SLIDE 32

10/ 14 Stein’s method Motivation Discrete Malliavin calculus Convergence results

Divergence and integration by parts formula

Definition (Divergence)

For any U ∈ Dom δ, δU ∈ L2(EA) DF, UL2(A×EA) = F, δUL2(EA), for all F ∈ D. (3) By property (2), δU =

  • a∈A

DaUa.

Theorem (Integration by parts formula - LD, HH)

For any F ∈ D, U ∈ Dom δ, E

  • a∈A

DaF · Ua

  • = E [F · δU] .
slide-33
SLIDE 33

11/ 14 Stein’s method Motivation Discrete Malliavin calculus Convergence results

Normal Approximation

  • P = N(0, 1).
  • T = {h ∈ C1(R, R) : h′∞ ≤ 1}.

Theorem (LD, HH)

For any F : EA → R, s.t. F ∈ D and E [F] = 0, distW

  • F ∗PA, P
  • ≤ E
  • 1 −
  • a∈A

DaF (−DaL−1)F

  • +
  • a∈A

E

  • EA
  • F − F(XA¬a; x)

2 dPa(x) |DaL−1F|

  • .

where X′

¬a = XA¬a ∪ {X′ a} and L = −δ D is the number operator.

slide-34
SLIDE 34

11/ 14 Stein’s method Motivation Discrete Malliavin calculus Convergence results

Normal Approximation

  • P = N(0, 1).
  • T = {h ∈ C1(R, R) : h′∞ ≤ 1}.

Theorem (LD, HH)

For any F : EA → R, s.t. F ∈ D and E [F] = 0, distW

  • F ∗PA, P
  • ≤ E
  • 1 −
  • a∈A

DaF (−DaL−1)F

  • +
  • a∈A

E

  • EA
  • F − F(XA¬a; x)

2 dPa(x) |DaL−1F|

  • .

where X′

¬a = XA¬a ∪ {X′ a} and L = −δ D is the number operator.

slide-35
SLIDE 35

12/ 14 Stein’s method Motivation Discrete Malliavin calculus Convergence results

Application : Lyapounov theorem

Corollary (Lyapounov theorem - LD, HH)

Let (Xn, n ≥ 1) be a sequence of thrice integrable independent random variables defined on (Ω, A, P). Denote σ2

n = var(Xn), s2 n = n

  • j=1

σ2

j and Yn = 1

sn

n

  • j=1

(Xj − E [Xj]) . Assume that lim

n→∞

1 s3

n n

  • j=1

E

  • |Xj − E [Xj] |3

= 0. (4) Then, distW (PYn, P) ≤ 2( √ 2 + 1) s3

n n

  • j=1

E

  • |Xj − E [Xj] |3

.

slide-36
SLIDE 36

13/ 14 Stein’s method Motivation Discrete Malliavin calculus Convergence results

References

  • L. Decreusefond, The Stein-Dirichlet-Malliavin method.

ESAIM: Proceedings, 2015.

  • I. Nourdin, G. Peccati, Stein’s method on Wiener chaos, Probab.

Theory Related Fields.

  • G. Peccati, J.L. Sol´

e, M.S. Taqqu, F. Utzet, Stein’s method and normal approximation of Poisson functionals. Annals of Probability, 2010.

  • N. Privault, Stochastic analysis in discrete and continuous

settings with normal martingales. Springer-Verlag, Berlin, 2009.

  • L. Decreusefond, H. Halconruy, Malliavin and Dirichlet

structures for independent random variables. Stochastic Processes and their Applications, 2019.

slide-37
SLIDE 37

14/ 14 Stein’s method Motivation Discrete Malliavin calculus Convergence results

Thanks for your attention !