[PPT] - Functional Steins Institut method Mines-Telecom L. Decreusefond PowerPoint Presentation

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Institut Mines-Telecom

Functional Stein’s method

L. Decreusefond

Borchard symposium

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Roadmap

Probability metrics

Types

Optimal Transportation Rubinstein Wasserstein Prohorov Entropy 2/39 Institut Mines-Telecom Functional Stein’s method

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Roadmap

Probability metrics

Types

Optimal Transportation Rubinstein Wasserstein Prohorov Entropy

Applications

Functional ineq. Convergence rate Ergodicity 2/39 Institut Mines-Telecom Functional Stein’s method

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Roadmap

Probability metrics

Computations

Girsanov ν ≪ µ Stein ν = T ∗m

Types

Optimal Transportation Rubinstein Wasserstein Prohorov Entropy

Applications

Functional ineq. Convergence rate Ergodicity 2/39 Institut Mines-Telecom Functional Stein’s method

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Roadmap

Probability metrics

Computations

Girsanov ν ≪ µ Stein ν = T ∗m

Types

Optimal Transportation Rubinstein Wasserstein Prohorov Entropy

Applications

Functional ineq. Convergence rate Ergodicity 2/39 Institut Mines-Telecom Functional Stein’s method

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Rubinstein distance

Definition

◮ (F, d) a Polish space

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Rubinstein distance

Definition

◮ (F, d) a Polish space ◮ c a distance on F

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Rubinstein distance

Definition

◮ (F, d) a Polish space ◮ c a distance on F ◮ F ∈ Lipc iff |F(x) − F(y)| ≤ c(x, y)

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Rubinstein distance

Definition

◮ (F, d) a Polish space ◮ c a distance on F ◮ F ∈ Lipc iff |F(x) − F(y)| ≤ c(x, y) ◮ µ and ν′ 2 proba. measures on F

dR(µ, ν′) = sup

F∈Lipc

Fdµ −
Fdν′

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Some examples

◮ F = Rn ◮ d = c=Euclidean distance ◮ Convergence in Rubinstein is equivalent to convergence in

distribution (Dudley)

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Some examples

◮ F = Rn ◮ d = c=Euclidean distance ◮ Convergence in Rubinstein is equivalent to convergence in

distribution (Dudley)

Wiener space

◮ F= space of continuous functions on [0, 1] ◮ d=uniform distance ◮ c=distance in the Cameron-Martin space

c(f , g) = 1 |f ′(s) − g′(s)|2ds 1/2

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Configuration space

Definition

A configuration is a locally finite set of particles on a Polish spaceY

f dω =
x∈ω

f (x)

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Configuration space

Definition

A configuration is a locally finite set of particles on a Polish spaceY

f dω =
x∈ω

f (x)

Vague topology

ωn

vaguely

− − − − → ω ⇐ ⇒

f dωn

n→∞

− − − →

f dω

for all f continuous with compact support from Y to R

5/39 Institut Mines-Telecom Functional Stein’s method

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Configuration space

Definition

A configuration is a locally finite set of particles on a Polish spaceY

f dω =
x∈ω

f (x)

Vague topology

ωn

vaguely

− − − − → ω ⇐ ⇒

f dωn

n→∞

− − − →

f dω

for all f continuous with compact support from Y to R d is the associated distance

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Lipschitz functionals

Distance between configurations

c(ω, η) = distTV(ω, η) = number of different points

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Lipschitz functionals

Distance between configurations

c(ω, η) = distTV(ω, η) = number of different points

Definition

F : NY → R is TV–Lip1 if |F(ω) − F(η)| ≤ distTV(ω, η)

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Lipschitz functionals

Distance between configurations

c(ω, η) = distTV(ω, η) = number of different points

Definition

F : NY → R is TV–Lip1 if |F(ω) − F(η)| ≤ distTV(ω, η)

Definition (Rubinstein distance)

dR(P, Q) := sup

F∈TV–Lip1

EPF − EQF
,

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Convergence in configuration space

Theorem (D- Schulte-Th¨ ale)

dR(Pn, Q) n→∞ − − − → 0 = ⇒ Pn

distr.

− − − → Q

Convergence in NY 7/39 Institut Mines-Telecom Functional Stein’s method

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Generic scheme

Initial space Target space (E, ν) (F, µ)

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Generic scheme

Initial space Target space (E, ν) (F, µ) (F, T ∗ν) T

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Generic scheme

Initial space Target space (E, ν) (F, µ) (F, T ∗ν) T dist.(T ∗ν, µ) ?

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Motivation : Poisson polytopes

1 2 3 4 5 6 1 234 5 6

η ξ(η)

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Motivation : Poisson polytopes

1 2 3 4 5 6 1 234 5 6

η ξ(η)

Question

What happens when the number of points goes to infinity ?

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Poisson point process

Hypothesis

Points are distributed to a Poisson process of control measure tK

Definition

◮ The number of points is a Poisson rv (t K(Sd−1)) ◮ Given the number of points, they are independently drawn

with distribution K

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Framework

Initial space Target space (NX, πtK) (NY, πM)

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Framework

Initial space Target space (NX, πtK) (NY, πM) (NY, T ∗

t πtK)

Tt

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Framework

Initial space Target space (NX, πtK) (NY, πM) (NY, T ∗

t πtK)

Tt Tt(η) = tγ

2

x,y∈η(2)

= x − y 11/39 Institut Mines-Telecom Functional Stein’s method

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Rescaling : γ = 2/(d − 1)

Campbell-Mecke formula

E

x1,··· ,xk∈ω(k)

=

f (x) = tk

f (x1, · · · , xk)K(dx1, · · · , dxk)

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Rescaling : γ = 2/(d − 1)

Campbell-Mecke formula

E

x1,··· ,xk∈ω(k)

=

f (x) = tk

f (x1, · · · , xk)K(dx1, · · · , dxk)

Mean number of points (after rescaling)

1 2E

x=y∈ω

1tγx−tγy≤β = t2 2

Sd−1⊗Sd−1 1x−y≤t−γβdxdy

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Rescaling : γ = 2/(d − 1)

Campbell-Mecke formula

E

x1,··· ,xk∈ω(k)

=

f (x) = tk

f (x1, · · · , xk)K(dx1, · · · , dxk)

Mean number of points (after rescaling)

1 2E

x=y∈ω

1tγx−tγy≤β = t2 2

Sd−1⊗Sd−1 1x−y≤t−γβdxdy

Geometry

Vd−1(Sd−1∩Bd

βt−γ(y)) = κd−1(βt−γ)d−1+(d − 1)κd−1

2 (βt−γ)d+O(t−γ(

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Rescaling : γ = 2/(d − 1)

Campbell-Mecke formula

E

x1,··· ,xk∈ω(k)

=

f (x) = tk

f (x1, · · · , xk)K(dx1, · · · , dxk)

Mean number of points (after rescaling)

1 2E

x=y∈ω

1tγx−tγy≤β = t2 2

Sd−1⊗Sd−1 1x−y≤t−γβdxdy

Geometry

Vd−1(Sd−1∩Bd

βt−γ(y)) = κd−1(βt−γ)d−1+(d − 1)κd−1

2 (βt−γ)d+O(t−γ(

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Schulte-Th¨ ale (2012) based on Peccati (2011)

P(t2/(d−1)Tm(ξ) > x) − e−βx(d−1) m−1
i=0

(βxd−1)i i!

≤ Cxt− min(1/2,2/(d−1))

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Schulte-Th¨ ale (2012) based on Peccati (2011)

P(t2/(d−1)Tm(ξ) > x) − e−βx(d−1) m−1
i=0

(βxd−1)i i!

≤ Cxt− min(1/2,2/(d−1))

What about speed of convergence as a process ?

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Stein method in one picture

µ µ m = T ∗ν P∗

t µ = µ

P∗

t m

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Stein method

The main tool

Construct a Markov process (X(s), s ≥ 0)

◮ with values in F ◮ ergodic with µ as invariant distribution

X(s) distr. − − − → µ for all initial condition X(0)

◮ for which µ is a stationary distribution

X(0) distr. = µ = ⇒ X(s) distr. = µ, ∀s > 0

◮ Equivalently

LFdm = 0, ∀F iff m = µ

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Dirichlet structure

µ, L

LFdµ = 0

Pt =etL Pt F→

Fdµ

E(F,G)= d dt Pt F,GL2(µ)

at t = 0

LF= dPt F dt

t=0

E(F,G) =LF,GL2(µ) LF/ G tq E(F,H)= G,HL2(mµ), ∀H

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Dirichlet-Malliavin structure

µ, L

LFdµ = 0

Pt =etL Pt F→

Fdµ

E(F,G)= d dt Pt F,GL2(µ)

at t = 0

LF= dPt F dt

t=0

E(F,G) =LF,GL2(µ) LF/ G s.t. E(F,H)= G,HL2(µ), ∀H D s.t. E(F,G)= DF,DGL2(µ)⊗L2(µ)

L = D∗D 17/39 Institut Mines-Telecom Functional Stein’s method

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Gaussian measure on Rn

Standard Gaussian measure

◮ F = Rn, µ = N(0, Id) ◮ LF(u) = u.∇F(u) − ∆F(u) ◮ Semi-group

PtF(x) =

Rn F(e−tu +
1 − e−2tv) dµ(v)

◮ X = (X1, · · · , Xn) where Xk=Ornstein-Uhlenbeck process on

R dXk(t) = −Xk(t)dt + √ 2dBk(t)

◮ D = ∇

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Poisson distribution on N

Poisson

◮ F = N, µ=Poisson [λ] ◮ LF(n) = λ(F(n + 1) − F(n)) + n(F(n − 1) − F(n)) ◮ X(t) = nb of occupied servers in M/M/∞ ◮ Dist. X(t) = Poisson[θ(t, X(0))] where

θ(t, n) = e−tn + (1 − e−t)λ

◮ Semi-group

PtF(n) =

∞

k=0

F(k)e−θ(t,n) θ(t, n)k k!

◮ DF(n) = F(n + 1) − F(n)

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Poisson point process

PPP over Y

◮ F = configuration space over Y ◮ µ=dist. of PPP(M) ◮ Generator

LF(N) :=

R+ F(N + ǫy) − F(ω) dM(y)

+

y∈N

F(N − ǫy) − F(ω)

◮ X : Glauber process ◮ Dist. X(t)=PPP((1 − e−t)λ) + e−t-thinning of the I.C.

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Realization of a Glauber process

Y Time X(s) s S1 S2

◮ S1, S2, · · · : Poisson process of intensity M(Y) ds ◮ Lifetimes : Exponential rv of param. 1 ◮ Remark : Nb of particles ∼ M/M/∞

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Discrete gradient

Definition

DτF(N) = F(N + ǫτ) − F(N), for any τ ∈ Y

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Integration by parts

Definition

δλ = D∗ defined by Eµ

F δλ(G)
= Eµ
Y

DτF G(τ)dM(τ)

.

Theorem

For G deterministic, δλG =

Y

G(τ)(δN(τ) − dM(τ)) and DδλG = G.

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Stein representation formula

Rubinstein distance between T ∗ν and µ

P∞F(x) − P0F(x) = ∞ LPtF(x)dt

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Stein representation formula

Rubinstein distance between T ∗ν and µ

P∞F(x) − F(x) = ∞ LPtF(x)dt

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Stein representation formula

Rubinstein distance between T ∗ν and µ

F

Fdµ − F(x) = ∞ LPtF(x)dt

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Stein representation formula

Rubinstein distance between T ∗ν and µ

F

Fdµ −

F

F(x) d(T ∗ν)(x) =

F

∞ LPtF(x)dt d(T ∗ν)(x)

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Stein representation formula

Rubinstein distance between T ∗ν and µ

F

Fdµ −

F

F(x) d(T ∗ν)(x) =

F

∞ LPtF(x)dt d(T ∗ν)(x)

IPP on initial space

F

Fdµ −

E

F ◦ Tdν =

E

∞ (LµPµ

t F) ◦ T

dν dt

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Stein representation formula

Rubinstein distance between T ∗ν and µ

F

Fdµ −

F

F(x) d(T ∗ν)(x) =

F

∞ LPtF(x)dt d(T ∗ν)(x)

IPP on initial space

F

Fdµ −

E

F ◦ Tdν =

E

∞ Dν((Pµ

t F) ◦ T) dν dt

+ Remainder

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Last steps

Commutation of gradients

F

Fdµ −

E

F ◦ Tdν =

E

∞ Dµ(Pµ

t F) ◦ T dν dt

+ Remainder + Remainder′

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Last steps

Commutation of gradients

F

Fdµ −

E

F ◦ Tdν =

E

∞ Dµ(Pµ

t F) ◦ T dν dt

+ Remainder + Remainder′

Intertwining

DµPµ

t F = e−Φµ(t)Pµ t DµF

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Example

Our settings

◮ Pt : PPP of intensity tK on C ⊂ X ◮ f : dom f = C 2/S2 −

→ Y

Definition

T(

x∈η

δx) =

(x1,x2)∈ηk

=

δt2/(d−1)f (x1,x2) := ξ(η)

◮ L : image measure of (tK)2 by f ◮ M: intensity of the target Poisson PP

Example 26/39 Institut Mines-Telecom Functional Stein’s method

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Main result

Theorem (Two moments are sufficient)

sup

F∈TV–Lip1

E

F
PPP(M)
− E
F
T ∗(PPP(tK)
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Institut Mines-Telecom Functional Stein’s method

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Main result

Theorem (Two moments are sufficient)

sup

F∈TV–Lip1

E

F
PPP(M)
− E
F
T ∗(PPP(tK)
≤ distTV(M, L) + 2(var ξ(Y) − Eξ(Y))

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What we have to compute

Distance representation

dR(PPP(M), T ∗(PPP(tK))) = sup

F∈TV–Lip1

E

∞

Y

[PtF(ξ(η) + δy) − PtF(ξ(η))] M(dy) dt + E ∞

y∈ξ(η)

[PtF(ξ(η) − δy) − PtF(ξ(η))] dt  

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Transformation of the stochastic integral

Bis repetita

dR(PPP(M), ξ(η)) = sup

F∈TV–Lip1

E

∞

Y

[PtF(ξ(η) + δy) − PtF(ξ(η))] M(dy) dt + E ∞

y∈ξ(η)

[PtF(ξ(η) − δy) − PtF(ξ(η))] dt  

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Mecke formula

Mecke formula ⇐ ⇒ IPP

E  

y∈ζ

f (y, ζ)   = E

Y

f (y, ζ + δy) M

dy)
is equivalent to

E

Y

DyU(ζ)f (y, ζ) M(dy)

= E
U(ζ)
Y

f (y, ζ)(dζ(y) − M(dy))

where

DyU(ζ) = U(ζ + δy) − U(ζ)

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Consequence of the Mecke formula

Proof.

Eη

y∈ξ(η)

PtF(ξ(η) − δy) − PtF(ξ(η)) =

domf

Eη

PtF(ξ(η)) − PtF(ξ(η) + δf (x1,x2))
K2(d(x1, x2))

+ Remainder

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Consequence of the Mecke formula

Proof.

Eη

y∈ξ(η)

PtF(ξ(η) − δy) − PtF(ξ(η)) =

domf

Eη

PtF(ξ(η)) − PtF(ξ(η) + δf (x1,x2))
K2(d(x1, x2))

+ Remainder

Problem

ξ(η) + δf (x1,x2) = ξ(η + δx1 + δx2)

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A bit of geometry

η + δx + δy ξ(η + δx + δy)

f (x,y) 32/39 Institut Mines-Telecom Functional Stein’s method

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A bit of geometry

η + δx + δy ξ(η + δx + δy)

f (x,y)

Conclusion

ξ(η + δx + δy) = ξ(η) + ξ(δx + δy) + ˆ ξ(x, y; η)

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Last step

dR(PPP(M), ξ(η)) = sup

F∈TV–Lip1

∞

Y

Eζ [PtF(ζ + δy) − PtF(ζ)] (M − L)(dy) dt +Remainder

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A key property (on the target space)

Definition

DxF(ζ) = F(ζ + δx) − F(ζ)

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A key property (on the target space)

Definition

DxF(ζ) = F(ζ + δx) − F(ζ)

Intertwining property

For the Glauber point process DxPtF(ζ) = e−tPtDxF(ζ)

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Consequence

∞

Y

Eζ [PtF(ζ + δy) − PtF(ζ)] (M − L)(dy) dt = ∞

Y

Eζ[DyPtF(ζ)] (M − L)(dy) dt = ∞ e−t

Y

Eζ[PtDyF(ζ)] (M − L)(dy) dt ≤

Y

|M − L|(dy)

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Generic Theorem

Theorem

dR(PPP(M)|[0,a] , ξ(η)|[0,a]) ≤ dTV(L, M) + 3 · 2k+1 (L)(Y) r(domf ) where r(domf ) := sup

1≤ℓ≤k−1, (x1,...,xℓ)∈Xℓ

Kk−ℓ({(y1, . . . , yk−ℓ) ∈ Xk−ℓ : (x1, . . . , xℓ, y1, . . . , yk−ℓ) ∈ domf })

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Example (cont’d)

Theorem

dR(PPP(M)|[0,a] , ξ(η)|[0,a]) ≤ Cat−1

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Compound Poisson approximation

The process

L(b)

t (µ) = 1

2

(x,y)∈µ2

t,=

x − yb1x−y≤ǫt

Theorem (Reitzner-Schulte-Th¨ ale (2013) w.o. conv. rate)

Assume

◮ t2ǫb t t→∞

− − − → λ

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Compound Poisson approximation

The process

L(b)

t (µ) = 1

2

(x,y)∈µ2

t,=

x − yb1x−y≤ǫt

Theorem (Reitzner-Schulte-Th¨ ale (2013) w.o. conv. rate)

Assume

◮ t2ǫb t t→∞

− − − → λ

◮ N ∼ Poisson(κdλ/2)

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Compound Poisson approximation

The process

L(b)

t (µ) = 1

2

(x,y)∈µ2

t,=

x − yb1x−y≤ǫt

Theorem (Reitzner-Schulte-Th¨ ale (2013) w.o. conv. rate)

Assume

◮ t2ǫb t t→∞

− − − → λ

◮ N ∼ Poisson(κdλ/2) ◮ (Xi, i ≥ 1) iid, uniform in Bd(λ1/d)

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Compound Poisson approximation

The process

L(b)

t (µ) = 1

2

(x,y)∈µ2

t,=

x − yb1x−y≤ǫt

Theorem (Reitzner-Schulte-Th¨ ale (2013) w.o. conv. rate)

Assume

◮ t2ǫb t t→∞

− − − → λ

◮ N ∼ Poisson(κdλ/2) ◮ (Xi, i ≥ 1) iid, uniform in Bd(λ1/d) ◮ Then

dTV(t2b/dL(b)

t , N

j=1

Xjb) ≤ c(|t2ǫb

t − λ| + t− min(2/d,1))

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Bibliographie

Functional Stein’s method

◮ L. Coutin and L. Decreusefond, Stein’s method for Brownian

approximations, Communications on Stochastic Analysis, vol. 7, no. 3, pp. 349372, Sep. 2013.

◮ L. Coutin and L. Decreusefond, Higher order expansions via

Stein’s method, Communications on Stochastic Analysis, 2014.

◮ L. Decreusefond, M. Schulte and C. Th¨

ale, Functional Poisson approximation in Rubinstein distance, ArXiv 1406.5484, 2014.

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