Lecture-26: Properties of Poisson point processes

Lecture-26: Properties of Poisson point processes

1 Laplace functional

LetX=_R^d_{be the}d-dimensional Euclidean space. Recall that all the random points are unique for a simple point processS:Ω→X^N, and henceScan also be considered as a set of countable points inX.

LetN:Ω→_Z^B(X)₊ be the counting process associated with the simple point processS.

Remark1. We observe thatdN(x) =0 for allx∈/SanddN(x) =δx1{x∈S}. Hence, for any Borel measur- able function f :X→Rand boundedA∈B(X), we haveR

x∈Af(x)dN(x) =_∑x∈S∩Af(x).

Definition 1.1. TheLaplace functionalLS:R^X₊→_R+ of a point processS:Ω→_X^N and associated counting processN:Ω→Z^B₊^(X)is defined for all non-negative Borel measurable function f :X→R+

as

LS(f)≜Eexp

− Z

R^d f(x)dN(x)

.

Remark2. For a simple functionf =_∑^k_i=1t_i1A_i, we can write the Laplace functional as a function of the vector (t₁,t2, . . . ,t_k), LS(f) =_Eexp

−_∑^k_i=1t_iR

A_idN(x)=_Eexp

−_∑^k_i=1t_iN(A_i). We observe that this is 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 a Poisson point process S:Ω→X^N with intensity measure Λ: B(X)→R+evaluated at any non-negative Borel measurable function f :X→R+, is

LS(f) =exp

− Z

X(1−e^−f(x))_dΛ(x)

.

Proof. For a bounded Borel measurable setA∈_B(_X), consider the truncated functiong=f1A. Then, LS(_g) =_Eexp(−

Z

Xg(_x)_dN(_x)) =_Eexp(−

Z

Af(_x)_dN(_x))_.

ClearlydN(x) =δx1{x∈S}and hence we can writeLS(g) =_Eexp(−_∑x∈S∩Af(x)). 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(x_i) Λ(A) ^1{xi∈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(x_i) Λ(A) =exp

− Z

X(1−e^−g(x))_dΛ(x)

. Result follows from taking increasing sequences of setsAk↑Xand monotone convergence theorem.

1

1.1 Superposition of point processes

Definition 1.3. LetS^k:Ω→X^Nbe a simple point process with intensity measuresΛk:B(_X)→R+and counting processN_k:Ω→_Z^B(X)₊ , for eachk∈_{N. The}superpositionof point processes(S^k:k∈_N)is defined as a point processS≜∪_kS^k.

Remark 3. The counting process associated with superposition point processS:Ω→_X^N is given by N:Ω→Z^B₊^(X) defined by N≜∑kNk, and the intensity measure of point process S is given byΛ: B(X)→R+defined byΛ=_∑kΛkfrom monotone convergence theorem.

Remark4. The superposition processSis simple iff∑kNkis locally finite.

Theorem 1.4. The superposition of independent Poisson point processes(S^k:k∈N)with intensities(_Λk:k∈ N)is a Poisson point process with intensity measure∑kΛkif and only if the latter is a locally finite measure.

Proof. Consider the superpositionS=∪_kS^kof independent Poisson point processesS^k∈_Xwith inten- sity measuresΛk. We will prove just the sufficiency part this theorem. We assume that∑kΛkis locally finite measure. It is clear thatN(A) =_∑kN_k(A)is finite by locally finite assumption, for all bounded setsA∈_B(_X). In particular, we havedN(x) =_∑kdN_k(x)for allx∈X. From the monotone convergence theorem and the independence of counting processes, we have for a non-negative Borel measurable function f :X→_R+,

LS(f) =Eexp − Z

Xf(x)

∑

k

dN_k(x)

!

=

∏

k

L_Sk=exp − Z

X(1−e^−f(x))

∑

k

Λ_k(x)

! .

1.2 Thinning of point processes

Definition 1.5. Consider a probabilityretention function p:X→[0, 1]and an independent Bernoulli point retention processY:Ω→ {0, 1}^Xsuch thatEY(x) =p(x)for all x∈_{X. The}thinningof point processS:Ω→_X^Nwith the probability retention functionp:X→[0, 1]is a point processS^(p):Ω→_X^N defined by

S^(p)≜(Sn∈S:Y(Sn) =1),

whereY(Sn)is an independent indicator for the retention of each pointSnandE[Y(Sn)Sn] =p(Sn). Theorem 1.6. The thinning of a Poisson point process S:Ω →_X^N of intensity measureΛ:B(_X)→_R+ with the retention probability function p:X→[0, 1], yields a Poisson point process S^(p):Ω→_X^Nof intensity measureΛ^(p):B(_X)→_R+defined for all bounded A∈_B(_X)asΛ^(p)(A)≜R

Ap(x)_dΛ(x).

Proof. LetA∈B(X)be a bounded Boreal measurable set, and letf:X→R+be a non-negative function.

LetN^(p)be the associated counting process to the thinned point processS^(p). Hence, for any bounded set A∈B(X), we have N^(p)(A) = _∑x∈S∩AY(x). That is, dN^(p)(x) = δ_xY(x)1{x∈S}. Therefore, for any non-negative functiong(x) =f(x)_1{x∈A}, we can writeR

x∈Xg(x)dN^(p)(x) =R

x∈Af(x)dN^(p)(x) =

∑x∈S∩Af(x)Y(x). We can write the Laplace functional of the thinned point process S^(p)for the non- negative functiong(x) = f(x)1{x∈A}, as

L_S⁽p)(g) =_E

E[exp

− Z

Af(x)dN^p(x)

|N(A)]

=

∑

n∈Z+

P{N(A) =n}

∏

n i=1

E[^{−f(Si)Y(Si)}| {Si∈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 are i.i.d.. SinceYis a Bernoulli process independent of the underlying processSwithE[Y(S_i)] =p(S_i), we get

E[e^{−f(Si)Y(Si)}| {Si∈S∩A}] =E[e^−f(Si)p(Si) + (1−p(Si))| {Si∈S∩A}]. From the distribution^Λ_Λ(A)^′(x) forx∈S∩Afor the Poisson point processS, we get

L_S(p)(g) =e^−Λ(A)

∑

n∈Z+

1 n!

Z

A(p(x)e^−f(x)+ (1−p(x))dΛ(x) n

=exp

− Z

X(1−e^−g(x))p(x)dΛ(x)

. Result follows from taking increasing sequences of setsA_k↑_Xand monotone convergence theorem.

2