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Lecture-27: Properties of Poisson Point Processes

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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 all x 2/ S and dN(x) = d dx1fx2Sg. Hence, for any function f : R ! R, we have Z f (x)dN(x) = f (S )1 = f (S ). ∑ i fSi2Ag ∑ i x2A i2N Si2A Definition 1.1. The Laplace functional L of a point process S and associated counting process N is defined for all non-negative function f : Rd ! R as Z LS( f ) , Eexp − f (x)dN(x) . Rd Remark . f (x) = k t 1 For simple function ∑i=1 i1fx2Aig, we can write the Laplace functional k Z ! k ! LS( f ) = Eexp − ∑ ti dN(x) = Eexp − ∑ ti N(Ai) , i=1 Ai i=1 as a function of the vector (t1,t2,...,tk), a joint Laplace transform of the random vector (N(A1),..., N(Ak)). This way, one can compute all finite dimensional distribution of the counting process N. Proposition 1.2. The Laplace functional of the Poisson process with intensity measure L is Z − f (x) LS( f ) = exp − (1 − e )dL(x) . Rd d Proof. For a bounded Borel measurable set A 2 B(R ), consider g(x) = f (x)1fx2Ag. Then, Z Z LS(g) = Eexp(− g(x)dN(x)) = Eexp(− f (x)dN(x)). Rd A ( ) = L ( ) = E − ( ) Clearly dN x dx1fx2Sg and hence we can write S g exp ∑Si2S\A f Si . We know that the probability of N(A) = jS(A)j = n points in set A is given by L(A)n PfN(A) = ng = e−L(A) . n! Given there are n points in set A, the density of n point locations are independent and given by n d (x ) L i 1fxi2Ag f (x1,..., xn) = . S1,...,Sn N(A)=n ∏ ( ) i=1 L A Hence, we can write the Laplace functional as L(A)n n Z dL(x ) Z −L(A) − f (xi) i −g(x) LS(g) = e ∑ ∏ e = exp − (1 − e )dL(x) . n! A L(A) Rd n2Z+ i=1 d Result follows from taking increasing sequences of sets Ak " R and monotone convergence theorem. 1 1.1 Superposition of point processes For simple point processes Sk with intensity measures Lk and counting process Nk, the superposition of point processes is defined as S = [kSk a simple point process with the counting process N = ∑k Nk iff ∑k Nk is locally finite. If Sk are Poisson then, we know that L = ∑k Lk. Next, we will show that the superposition S of Poisson processes is Poisson iff ∑k Lk is locally finite. Theorem 1.3. The superposition of independent Poisson point processes with intensities Lk is a Poisson point process with intensity measure ∑k Lk if and only if the latter is a locally finite measure. d Proof. Consider the superposition S = ∑k Sk of independent Poisson point processes Sk 2 R with in- tensity measures Lk. We will prove only side of this theorem. We assume that ∑k Lk is locally finite measure. It is clear that N(A) = ∑k Nk(A) is finite by locally finite assumption, for all bounded sets d A 2 B. In particular, we have dN(x) = ∑k dNk(x) for all x 2 R . From the monotone convergence theorem and the independence of counting processes, we have for a non-negative function f : Rd ! R, Z ! Z ! − f (x) LS( f ) = Eexp − f (x) dNk(x) = LS = exp − (1 − e ) Lk(x) . d ∑ ∏ k d ∑ R k k R k 1.2 Thinning of point processes Consider a probability retention function p : Rd ! [0,1] and a point process S. The thinning of point process S : W ! (Rd)N with the retention function p is a point process Sp : W ! (Rd)N such that p S = (Sn 2 S : Y(Sn) = 1), where Y(Sn) is an independent indicator random variable at each point Sn and E[Y(Sn) Sn] = p(Sn). Theorem 1.4. The thinning of the Poisson point process of intensity measure L with the retention probability function p yields a Poisson point process of intensity measure L(p) with Z L(p)(A) = p(x)dL(x) A for all bounded Borel measurable A ⊆ Rd. Proof. Let A ⊆ Rd be a bounded Boreal measurable set, and let f : Rd ! R be a non-negative function. Let Np be the associated counting process to the thinned point process Sp. Hence, for any set A 2 B, we Np(A) = Y(S ) have ∑Si2S(A) i . That is, dNp(x) = d 1 Y(S ). ∑ x fx=Sig i i2N Therefore, for any non-negative function g(x) = f (x)1fx2Ag, we can write Z Z p p g(x)dN (x) = f (x)dN (x) = ∑ f (Si)Y(Si). x2Rd x2A Si2A Consider the Laplace functional of the thinned point process Sp for the non-negative function g(x) = f (x)1fx2Ag, we can write the corresponding Laplace functional as Z n p − f (Si)Y(Si) LSp (g) = EE[exp − f (x)dN (x) N(A)] = ∑ PfN(A) = ng ∏E[ Si 2 A]. A n2Z+ i=1 Here, we denote the points of the point process in subset A as S(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 are iid. Since Y is a Bernoulli process independent of the underlying process S with E[Y(Si)] = p(Si), we get − f (Si)Y(Si) − f (Si) E[e Si 2 S(A)] = E[e p(Si) + (1 − p(Si)) Si 2 S(A)]. 2 0 L (x) 2 ( ) From the distribution L(A) for x S A for the Poisson point process S, we get Z n Z −L(A) 1 − f (x) −g(x) LSp (g) = e ∑ (p(x)e + (1 − p(x))dL(x) = exp − (1 − e )p(x)dL(x) . n! A Rd n2Z+ d Result follows from taking increasing sequences of sets Ak " R and monotone convergence theorem. 3.
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