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Stein's method is applied to show convergence towards Poisson point process
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InstitutMines-Telecom
Functional Poissonconvergence
L. Decreusefond(joint work with M. Schulte, C. Thale)
Stein, Malliavin, Poisson and co.
Motivation : Poisson polytopes
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η ξ(η)
2/25 Institut Mines-Telecom Functional Poisson convergence
Motivation : Poisson polytopes
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η ξ(η)
Question
What happens when the number of points goes to infinity ?
2/25 Institut Mines-Telecom Functional Poisson convergence
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
3/25 Institut Mines-Telecom Functional Poisson convergence
Rescaling : γ = 2/(d − 1)
Campbell-Mecke formula
E∑
x1,··· ,xk∈ω(k)6=
f (x) = tk∫
f (x1, · · · , xk)K(dx1, · · · , dxk)
4/25 Institut Mines-Telecom Functional Poisson convergence
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
4/25 Institut Mines-Telecom Functional Poisson convergence
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))
4/25 Institut Mines-Telecom Functional Poisson convergence
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))
4/25 Institut Mines-Telecom Functional Poisson convergence
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))
5/25 Institut Mines-Telecom Functional Poisson convergence
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 ?
5/25 Institut Mines-Telecom Functional Poisson convergence
Configuration space
Definition
A configuration is a locally finite set of particles on a Polish spaceY∫f dω =
∑x∈ω
f (x)
6/25 Institut Mines-Telecom Functional Poisson convergence
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
6/25 Institut Mines-Telecom Functional Poisson convergence
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−−−−→ ∅
6/25 Institut Mines-Telecom Functional Poisson convergence
Convergence in configuration space
Theorem (DST)
dR(Pn,Q)n→∞−−−→ 0 =⇒ Pn
distr.−−−→ Q
Convergence in NY
7/25 Institut Mines-Telecom Functional Poisson convergence
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
8/25 Institut Mines-Telecom Functional Poisson convergence
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))
9/25 Institut Mines-Telecom Functional Poisson convergence
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
10/25 Institut Mines-Telecom Functional Poisson convergence
In one picture
PPP(M) PPP(M)
T#(PPP(tK))
P#s (PPP(M))
P#s (T#(PPP(tK)))
11/25 Institut Mines-Telecom Functional Poisson convergence
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/∞
12/25 Institut Mines-Telecom Functional Poisson convergence
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)
13/25 Institut Mines-Telecom Functional Poisson convergence
Stein representation formula
Definition
PtF (η) = E[F (X (t)) |X (0) = η]
14/25 Institut Mines-Telecom Functional Poisson convergence
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
14/25 Institut Mines-Telecom Functional Poisson convergence
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
14/25 Institut Mines-Telecom Functional Poisson convergence
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
14/25 Institut Mines-Telecom Functional Poisson convergence
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
15/25 Institut Mines-Telecom Functional Poisson convergence
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
16/25 Institut Mines-Telecom Functional Poisson convergence
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))
]
17/25 Institut Mines-Telecom Functional Poisson convergence
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
18/25 Institut Mines-Telecom Functional Poisson convergence
A bit of geometry
η + δx + δy ξ(η + δx + δy )
f (x,y)
19/25 Institut Mines-Telecom Functional Poisson convergence
Last step
dR(PPP(M), ξ(η))
= supF∈TV–Lip1
(∫ ∞0
∫Y
Eζ [PtF (ζ + δy )− PtF (ζ)] (M− f ∗Kk)(dy) dt
+Remainder
)
20/25 Institut Mines-Telecom Functional Poisson convergence
A key property (on the target space)
Definition
DxF (ζ) = F (ζ + δx)− F (ζ)
21/25 Institut Mines-Telecom Functional Poisson convergence
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)
21/25 Institut Mines-Telecom Functional Poisson convergence
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)
22/25 Institut Mines-Telecom Functional Poisson convergence
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 })
23/25 Institut Mines-Telecom Functional Poisson convergence
Example (cont’d)
Theorem
dR(PPP(M)|[0,a] , ξ(η)|[0,a]) ≤ Cat−1
24/25 Institut Mines-Telecom Functional Poisson convergence
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))
25/25 Institut Mines-Telecom Functional Poisson convergence
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))
25/25 Institut Mines-Telecom Functional Poisson convergence
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))
25/25 Institut Mines-Telecom Functional Poisson convergence
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))
25/25 Institut Mines-Telecom Functional Poisson convergence