Stein's method for functional Poisson approximation

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

Motivation : Poisson polytopes
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3
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5
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1 23456
η ξ(η)
2/25 Institut Mines-Telecom Functional Poisson convergence

Motivation : Poisson polytopes
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2
3
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0
1 23456
η ξ(η)
Question
What happens when the number of points goes to inﬁnity ?

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

Rescaling : γ = 2/(d − 1)
Campbell-Mecke formula
E
x1,··· ,xk ∈ω
(k)
=
f (x) = tk
f (x1, · · · , xk)K(dx1, · · · , dxk)

E
x1,··· ,xk ∈ω
(k)
=
f (x) = tk
f (x1, · · · , xk)K(dx1, · · · , dxk)
Mean number of points (after rescaling)
1
2
E
x=y∈ω
1 tγx−tγy ≤β =
t2
2 Sd−1⊗Sd−1
1 x−y ≤t−γβdxdy

E
x1,··· ,xk ∈ω
(k)
=
f (x) = tk
f (x1, · · · , xk)K(dx1, · · · , dxk)
Mean number of points (after rescaling)
1
2
E
x=y∈ω
1 tγx−tγy ≤β =
t2
2 Sd−1⊗Sd−1
1 x−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(

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!
≤ Cx t− min(1/2,2/(d−1))

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!
≤ Cx t− min(1/2,2/(d−1))
What about speed of convergence as a process ?

Configuration space
Definition
A configuration is a locally finite set of particles on a Polish spaceY
f dω =
x∈ω
f (x)

Deﬁnition
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

Deﬁnition
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
Be careful !
δn
vaguely
−−−−→ ∅

Convergence in conﬁguration space
Theorem (DST)
dR(Pn, Q)
n→∞
−−−→ 0 =⇒ Pn
distr.
−−−→ Q
Convergence in NY

Example
Our settings
Pt : PPP of intensity tK on C ⊂ X
f : dom f = Ck/Sk −→ Y
Deﬁnition
T(
x∈η
δx ) =
(x1,··· ,xk )∈ηk
=
δt2/d f (x1,··· ,xk ) := ξ(η)
f ∗K : image measure of (tK)k by f
M: intensity of the target Poisson PP
Example

Main result
Theorem (Two moments are suﬃcient)
sup
F∈TV–Lip1
E F PPP(M) − E F T∗
(PPP(tK)
≤ distTV(L, M) + 2(var ξ(Y) − Eξ(Y))

Stein method
The main tool
Construct a Markov process (X(s), s ≥ 0)
with values in conﬁguration space
ergodic with PPM(M) as invariant distribution
X(s)
distr.
−−−→ PPM(M)
for all initial condition X(0)
for which PPM(M) is a stationary distribution
X(0)
distr.
= PPM(M) =⇒ X(s)
distr.
= PPM(M), ∀s > 0

In one picture
PPP(M) PPP(M)
T#(PPP(tK))
P#
s (PPP(M))
P#
s (T#(PPP(tK)))

Realization of a Glauber process
Y
Time0
X(s)
sS1 S2
S1, S2, · · · : Poisson process of intensity M(Y) ds
Lifetimes : Exponential rv of param. 1
Remark : Nb of particles ∼ M/M/∞

Properties
Theorem
Distr. X(s)=PPP((1 − e−s)M) + e−s-thinning of the I.C.
X(s)
s→∞
−−−→ PPP(M)
If X(0)
distr.
= PPP(M) then X(s)
distr.
= PPP(M)
Generator
LF(ω) :=
Y
F(ω + δy ) − F(ω) M(dy)
+
y∈ω
F(ω − δy ) − F(ω)

Stein representation formula
Deﬁnition
PtF(η) = E[F(X(t)) | X(0) = η]

Deﬁnition
PtF(η) = E[F(X(t)) | X(0) = η]
Fundamental Lemma
F(ω)πM(dω) − F(ξ) =
∞
0
LPsF(ξ)ds

Deﬁnition
PtF(η) = E[F(X(t)) | X(0) = η]
Fundamental Lemma
F(ω)πM(dω) − F(ξ(η)) =
∞
0
LPsF(ξ(η))ds

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

What we have to compute
Distance representation
dR(PPP(M), T#
(PPP(tK)))
= sup
F∈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



Transformation of the stochastic integral
Bis repetita
dR(PPP(M), ξ(η))
= sup
F∈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



Mecke formula
Mecke formula ⇐⇒ IPP
E
y∈ζ
f (y, ζ) =
Y
Ef (y, ζ + δy ) M dy)
is equivalent to
E
Y
Dy U(ζ)f (y, ζ) M(dy) = E U(ζ)
Y
f (y, ζ)(dζ(y) − M(dy))

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

A bit of geometry
η + δx + δy ξ(η + δx + δy )
f (x,y)

Last step
dR(PPP(M), ξ(η))
= sup
F∈TV–Lip1
∞
0 Y
Eζ [PtF(ζ + δy ) − PtF(ζ)] (M − f ∗
Kk
)(dy) dt
+Remainder

A key property (on the target space)
Deﬁnition
Dx F(ζ) = F(ζ + δx ) − F(ζ)

A key property (on the target space)
Deﬁnition
Dx F(ζ) = F(ζ + δx ) − F(ζ)
Intertwining property
For the Glauber point process
Dx PtF(ζ) = e−t
PtDx F(ζ)

Consequence
∞
0 Y
Eζ [PtF(ζ + δy ) − PtF(ζ)] (M − f ∗
Kk
)(dy) dt
=
∞
0 Y
Eζ[Dy PtF(ζ)] (M − f ∗
Kk
)(dy) dt
=
∞
0
e−t
Y
Eζ[PtDy F(ζ)] (M − f ∗
Kk
)(dy) dt
≤
Y
|M − f ∗
Kk
|(dy)

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 ) := sup
1≤ ≤k−1,
(x1,...,x )∈X
Kk−
({(y1, . . . , yk− ) ∈ Xk−
:
(x1, . . . , x , y1, . . . , yk− ) ∈ domf })

Example (cont’d)
Theorem
dR(PPP(M)|[0,a] , ξ(η)|[0,a]) ≤ Cat−1

Compound Poisson approximation
The process
L
(b)
t (µ) =
1
2
(x,y)∈µ2
t,=
x − y b
1 x−y ≤δt
Theorem (Reitzner-Schlte-Thle (2013) w.o. conv. rate)
Assume
t2δb
t
t→∞
−−−→ λ

The process
L
(b)
t (µ) =
1
2
(x,y)∈µ2
t,=
x − y b
1 x−y ≤δt
Assume
t2δb
t
t→∞
−−−→ λ
N ∼ Poisson(κd λ/2)

The process
L
(b)
t (µ) =
1
2
(x,y)∈µ2
t,=
x − y b
1 x−y ≤δt
Assume
t2δb
t
t→∞
−−−→ λ
(Xi , i ≥ 1) iid, uniform in Bd (λ1/d )

The process
L
(b)
t (µ) =
1
2
(x,y)∈µ2
t,=
x − y b
1 x−y ≤δt
Assume
t2δb
t
t→∞
−−−→ λ
(Xi , i ≥ 1) iid, uniform in Bd (λ1/d )
Then
dTV(t2b/d
L
(b)
t ,
N
j=1
Xj
b
) ≤ c(|t2
δb
t − λ| + t− min(2/d,1)
)

Stein's method for functional Poisson approximation

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