Stein's method for functional Poisson approximation

39
Institut Mines-Telecom Functional Poisson convergence L. Decreusefond (joint work with M. Schulte, C. Th¨ ale) Stein, Malliavin, Poisson and co.

description

Stein's method is applied to show convergence towards Poisson point process

Transcript of Stein's method for functional Poisson approximation

Page 1: Stein's method for functional Poisson approximation

InstitutMines-Telecom

Functional Poissonconvergence

L. Decreusefond(joint work with M. Schulte, C. Thale)

Stein, Malliavin, Poisson and co.

Page 2: Stein's method for functional Poisson approximation

Motivation : Poisson polytopes

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Motivation : Poisson polytopes

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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

I The number of points is a Poisson rv (t K(Sd−1))

I Given the number of points, they are independently drawnwith distribution K

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Rescaling : γ = 2/(d − 1)

Campbell-Mecke formula

E∑

x1,··· ,xk∈ω(k)6=

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)6=

f (x) = tk∫

f (x1, · · · , xk)K(dx1, · · · , dxk)

Mean number of points (after rescaling)

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2E∑

x 6=y∈ω1‖tγx−tγy‖≤β =

t2

2

∫∫Sd−1⊗Sd−1

1‖x−y‖≤t−γβdxdy

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Rescaling : γ = 2/(d − 1)

Campbell-Mecke formula

E∑

x1,··· ,xk∈ω(k)6=

f (x) = tk∫

f (x1, · · · , xk)K(dx1, · · · , dxk)

Mean number of points (after rescaling)

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2E∑

x 6=y∈ω1‖tγx−tγy‖≤β =

t2

2

∫∫Sd−1⊗Sd−1

1‖x−y‖≤t−γβdxdy

Geometry

Vd−1(Sd−1∩Bdβt−γ (y)) = κd−1(βt−γ)d−1+

(d − 1)κd−12

(βt−γ)d+O(t−γ(d+1))

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Rescaling : γ = 2/(d − 1)

Campbell-Mecke formula

E∑

x1,··· ,xk∈ω(k)6=

f (x) = tk∫

f (x1, · · · , xk)K(dx1, · · · , dxk)

Mean number of points (after rescaling)

1

2E∑

x 6=y∈ω1‖tγx−tγy‖≤β =

t2

2

∫∫Sd−1⊗Sd−1

1‖x−y‖≤t−γβdxdy

Geometry

Vd−1(Sd−1∩Bdβt−γ (y)) = κd−1(βt−γ)d−1+

(d − 1)κd−12

(βt−γ)d+O(t−γ(d+1))

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Schulte-Thale (2012) based on Peccati (2011)

∣∣∣∣∣P(t2/(d−1)Tm(ξ) > x)− e−βx(d−1)

m−1∑i=0

(βxd−1)i

i !

∣∣∣∣∣≤ Cx t

−min(1/2,2/(d−1))

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Schulte-Thale (2012) based on Peccati (2011)

∣∣∣∣∣P(t2/(d−1)Tm(ξ) > x)− e−βx(d−1)

m−1∑i=0

(βxd−1)i

i !

∣∣∣∣∣≤ Cx t

−min(1/2,2/(d−1))

What about speed of convergence as a process ?

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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

ωnvaguely−−−−→ ω ⇐⇒

∫f dωn

n→∞−−−→∫

f dω

for all f continuous with compact support from Y to R

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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

ωnvaguely−−−−→ ω ⇐⇒

∫f dωn

n→∞−−−→∫

f dω

for all f continuous with compact support from Y to R

Be careful !

δnvaguely−−−−→ ∅

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Convergence in configuration space

Theorem (DST)

dR(Pn,Q)n→∞−−−→ 0 =⇒ Pn

distr.−−−→ Q

Convergence in NY

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Example

Our settings

I Pt : PPP of intensity tK on C ⊂ XI f : dom f = C k/Sk −→ Y

Definition

T (∑x∈η

δx) =∑

(x1,··· ,xk )∈ηk6=

δt2/d f (x1,··· ,xk ) := ξ(η)

I f ∗K : image measure of (tK)k by f

I M: intensity of the target Poisson PP

Example

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Main result

Theorem (Two moments are sufficient)

supF∈TV–Lip1

E[F(PPP(M)

)]− E

[F(T ∗(PPP(tK)

)]≤ distTV(L, M) + 2(var ξ(Y)− Eξ(Y))

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Stein method

The main tool

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

I with values in configuration space

I ergodic with PPM(M) as invariant distribution

X (s)distr .−−−→ PPM(M)

for all initial condition X(0)

I for which PPM(M) is a stationary distribution

X (0)distr.= PPM(M) =⇒ X (s)

distr.= PPM(M), ∀s > 0

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In one picture

PPP(M) PPP(M)

T#(PPP(tK))

P#s (PPP(M))

P#s (T#(PPP(tK)))

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Realization of a Glauber process

Y

Time0

X (s)

sS1 S2

I S1,S2, · · · : Poisson process of intensity M(Y) ds

I Lifetimes : Exponential rv of param. 1

I Remark : Nb of particles ∼ M/M/∞

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Properties

Theorem

I Distr. X (s)=PPP((1− e−s)M) + e−s -thinning of the I.C.

I X (s)s→∞−−−→ PPP(M)

I If X (0)distr.= PPP(M) then X (s)

distr.= PPP(M)

I Generator

LF (ω) :=

∫YF (ω + δy )− F (ω) M(dy)

+∑y∈ω

F (ω − δy )− F (ω)

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Stein representation formula

Definition

PtF (η) = E[F (X (t)) |X (0) = η]

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Stein representation formula

Definition

PtF (η) = E[F (X (t)) |X (0) = η]

Fundamental Lemma∫F (ω)πM(dω)− F (ξ) =

∫ ∞0

LPsF (ξ)ds

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Stein representation formula

Definition

PtF (η) = E[F (X (t)) |X (0) = η]

Fundamental Lemma∫F (ω)πM(dω)− F (ξ(η)) =

∫ ∞0

LPsF (ξ(η))ds

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Stein representation formula

Definition

PtF (η) = E[F (X (t)) |X (0) = η]

Fundamental Lemma∫F (ω)πM(dω)− EF (ξ(η)) = E

∫ ∞0

LPsF (ξ(η))ds

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What we have to compute

Distance representation

dR(PPP(M),T#(PPP(tK)))

= supF∈TV–Lip1

(E

∫ ∞0

∫Y

[PsF (ξ(η) + δy )− PsF (ξ(η))] M(dy) ds

+ E

∫ ∞0

∑y∈ξ(η)

[PsF (ξ(η)− δy )− PtsF (ξ(η))] ds

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Transformation of the stochastic integral

Bis repetita

dR(PPP(M), ξ(η))

= supF∈TV–Lip1

(E

∫ ∞0

∫Y

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

+ E

∫ ∞0

∑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) =

∫Y

Ef (y , ζ + δy ) M(dy)

is equivalent to

E

∫YDyU(ζ)f (y , ζ) M(dy) = E

[U(ζ)

∫Yf (y , ζ)(dζ(y)−M(dy))

]

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Consequence of the Mecke formula

Proof.

E∑

y∈ξ(η)

PtF (ξ(η)− δy )− PtF (ξ(η))

=1

k!

∫domf

E[PtF (ξ(η + δx1 + . . .+ δxk )− δf (x1,...,xk ))

− PtF (ξ(η + δx1 + . . .+ δxk ))]

Kk(d(x1, . . . , xk))

= − 1

k!

∫domf

E[PtF (ξ(η) + δf (x1,··· ,xk ))− PtF (ξ(η))

]Kk(d(x1, . . . , xk)) + Remainder

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A bit of geometry

η + δx + δy ξ(η + δx + δy )

f (x,y)

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Last step

dR(PPP(M), ξ(η))

= supF∈TV–Lip1

(∫ ∞0

∫Y

Eζ [PtF (ζ + δy )− PtF (ζ)] (M− f ∗Kk)(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

∫ ∞0

∫Y

Eζ [PtF (ζ + δy )− PtF (ζ)] (M− f ∗Kk)(dy) dt

=

∫ ∞0

∫Y

Eζ [DyPtF (ζ)] (M− f ∗Kk)(dy) dt

=

∫ ∞0

e−t∫Y

Eζ [PtDyF (ζ)] (M− f ∗Kk)(dy) dt

≤∫Y|M− f ∗Kk |(dy)

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Generic Theorem

Theorem

dR(PPP(M)|[0,a] , ξ(η)|[0,a])

≤ dTV(f ∗Kk ,M) + 3 · 2k+1 (f ∗Kk)(Y) r(domf )

where

r(domf ) := sup1≤`≤k−1,

(x1,...,x`)∈X`

Kk−`({(y1, . . . , yk−`) ∈ Xk−` :

(x1, . . . , x`, y1, . . . , yk−`) ∈ 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)∈µ2t, 6=

‖x − y‖b1‖x−y‖≤δt

Theorem (Reitzner-Schlte-Thle (2013) w.o. conv. rate)

Assume

I t2δbtt→∞−−−→ λ

I N ∼ Poisson(κdλ/2)

I (Xi , i ≥ 1) iid, uniform in Bd(λ1/d)

I Then

dTV(t2b/dL(b)t ,

N∑j=1

‖Xj‖b) ≤ c(|t2δbt − λ|+ t−min(2/d ,1))

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Compound Poisson approximation

The process

L(b)t (µ) =

1

2

∑(x ,y)∈µ2t, 6=

‖x − y‖b1‖x−y‖≤δt

Theorem (Reitzner-Schlte-Thle (2013) w.o. conv. rate)

Assume

I t2δbtt→∞−−−→ λ

I N ∼ Poisson(κdλ/2)

I (Xi , i ≥ 1) iid, uniform in Bd(λ1/d)

I Then

dTV(t2b/dL(b)t ,

N∑j=1

‖Xj‖b) ≤ c(|t2δbt − λ|+ t−min(2/d ,1))

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Compound Poisson approximation

The process

L(b)t (µ) =

1

2

∑(x ,y)∈µ2t, 6=

‖x − y‖b1‖x−y‖≤δt

Theorem (Reitzner-Schlte-Thle (2013) w.o. conv. rate)

Assume

I t2δbtt→∞−−−→ λ

I N ∼ Poisson(κdλ/2)

I (Xi , i ≥ 1) iid, uniform in Bd(λ1/d)

I Then

dTV(t2b/dL(b)t ,

N∑j=1

‖Xj‖b) ≤ c(|t2δbt − λ|+ t−min(2/d ,1))

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Compound Poisson approximation

The process

L(b)t (µ) =

1

2

∑(x ,y)∈µ2t, 6=

‖x − y‖b1‖x−y‖≤δt

Theorem (Reitzner-Schlte-Thle (2013) w.o. conv. rate)

Assume

I t2δbtt→∞−−−→ λ

I N ∼ Poisson(κdλ/2)

I (Xi , i ≥ 1) iid, uniform in Bd(λ1/d)

I Then

dTV(t2b/dL(b)t ,

N∑j=1

‖Xj‖b) ≤ c(|t2δbt − λ|+ t−min(2/d ,1))

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