Lecture-27: Properties of Poisson point processes

Lecture-27: Properties of Poisson point processes

1 Laplace functional

For a realization of a simple point process S, we observe that dN(x) =0 for allx ∈/S anddN(x) = δx1{x∈S}. Hence, for any function f:R^d→R, we have

Z

x∈Af(x)dN(x) =

∑

i∈N

f(S_i)1_{Si∈A}=

∑

S_i∈A

f(S_i).

Definition 1.1. TheLaplace functionalLof a point processS and associated counting process Nis defined for all non-negative function f :R^d→Ras

LS(f),Eexp

− Z

R^d f(x)dN(x)

.

Remark1. For simple functionf(x) =_∑^k_i=1t_i1_{x∈Ai}, we can write the Laplace functional LS(f) =_Eexp −

∑

k i=1

t_i Z

A_idN(x)

!

=_Eexp −

∑

k i=1

t_iN(A_i)

! ,

as a function of the vector(t₁,t2, . . . ,t_k), a joint Laplace transform of the random vector(N(A₁), . . . ,N(A_k)). This way, one can compute all finite dimensional distribution of the counting processN.

Proposition 1.2. The Laplace functional of the Poisson process with intensity measureΛis LS(f) =exp

− Z

R^d(1−e^−f(x))_dΛ(x)

.

Proof. For a bounded Borel measurable setA∈ B(R^d), considerg(x) =f(x)1{x∈A}. Then, LS(g) =_Eexp(−

Z

R^dg(x)dN(x)) =_Eexp(−

Z

Af(x)dN(x)). ClearlydN(x) =δx1_{x∈S}and hence we can writeLS(g) =_Eexp −_∑S

i∈S∩Af(S_i). We know that the probability ofN(A) =|S(A)|=npoints in setAis given by

P{N(_A) =_n}=_e^−Λ(A)Λ(A)ⁿ n! .

Given there arenpoints in setA, the density ofnpoint locations are independent and given by fS₁,...,Sn

N(A)=n(x1, . . . ,xn) =

∏

n i=1

dΛ(xi)1{x_i∈A}

Λ(A) ^. Hence, we can write the Laplace functional as

LS(g) =e^−Λ(A)

∑

n∈Z+

Λ(A)ⁿ n!

∏

n i=1 Z

Ae^−f(xi)dΛ(xi) Λ(A) =exp

− Z

R^d(1−e^−g(x))_dΛ(x)

.

Result follows from taking increasing sequences of sets A_k↑_R^dand monotone convergence theorem.

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1.1 Superposition of point processes

For simple point processesSkwith intensity measuresΛk and counting processNk, thesuperposition of point processes is defined asS=∪_kSka simple point process with the counting processN=_∑kNk

iff∑kN_kis locally finite. IfS_kare Poisson then, we know thatΛ=_∑kΛk. Next, we will show that the superpositionSof Poisson processes is Poisson iff∑kΛkis locally finite.

Theorem 1.3. The superposition of independent Poisson point processes with intensitiesΛkis a Poisson point process with intensity measure∑kΛkif and only if the latter is a locally finite measure.

Proof. Consider the superpositionS=_∑kS_k of independent Poisson point processesS_k∈_R^dwith in- tensity measuresΛ_k. We will prove only side of this theorem. We assume that∑kΛ_k is locally finite measure. It is clear that N(A) =_∑kN_k(A)is finite by locally finite assumption, for all bounded sets A∈B. In particular, we havedN(x) =_∑kdNk(x)for all x ∈R^d. From the monotone convergence theorem and the independence of counting processes, we have for a non-negative function f :R^d→R,

LS(f) =_Eexp − Z

R^d f(x)

∑

k

dN_k(x)

!

=

∏

k

LS_k=_exp − Z

R^d(₁−e^−f(x))

∑

k

Λk(x)

! .

1.2 Thinning of point processes

Consider a probabilityretention functionp:R^d→[0, 1]and a point processS. Thethinningof point processS:Ω→(R^d)^Nwith the retention functionpis a point processS^p:Ω→(R^d)^Nsuch that

S^p= (Sn∈S:Y(Sn) =1),

whereY(Sn)is an independent indicator random variable at each pointSnandE[Y(Sn)Sn] =p(Sn). Theorem 1.4. The thinning of the Poisson point process of intensity measureΛwith the retention probability function p yields a Poisson point process of intensity measureΛ^(p)with

Λ^(p)(A) = Z

Ap(x)_dΛ(x) for all bounded Borel measurable A⊆_R^d.

Proof. LetA⊆R^dbe a bounded Boreal measurable set, and let f :R^d→Rbe a non-negative function.

LetN^pbe the associated counting process to the thinned point processS^p. Hence, for any setA∈ B, we haveN^p(A) =_∑Si∈S(A)Y(Si). That is,

dN^p(x) =

∑

i∈N

δ_x1_{x=Si}Y(Si). Therefore, for any non-negative functiong(x) = f(x)1_{x∈A}, we can write

Z

x∈R^dg(x)dN^p(x) = Z

x∈Af(x)dN^p(x) =

∑

S_i∈A

f(Si)Y(Si).

Consider the Laplace functional of the thinned point processS^pfor the non-negative functiong(x) = f(x)1{x∈A}, we can write the corresponding Laplace functional as

LS^p(g) =_EE[exp

− Z

Af(x)dN^p(x)

N(A)] =

∑

n∈Z+

P{N(A) =n}

∏

n i=1

E[^{−f(Si)Y(Si)}S_i∈A].

Here, we denote the points of the point process in subsetAasS(A) =S∩A. The first equality follows from the definition of Laplace functional and taking nested expectations. Second equality follows from the fact that the distribution of all points of a Poisson point process areiid. SinceYis a Bernoulli process independent of the underlying processSwithE[Y(Si)] =p(Si), we get

E[e^{−f(Si)Y(Si)}

S_i∈S(A)] =_E[e^−f(Si)p(S_i) + (1−p(S_i))S_i∈S(A)]. 2

From the distribution^Λ_Λ(A)^0(x) forx∈S(A)for the Poisson point processS, we get LS^p(g) =e^−Λ(A)

∑

n∈Z+

1 n!

_Z

A

(p(x)e^−f(x)+ (1−p(x))_dΛ(x) n

=exp

− Z

R^d(1−e^−g(x))p(x)_dΛ(x)

.

Result follows from taking increasing sequences of sets A_k↑R^dand monotone convergence theorem.

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