Factorial moments of point processes

Factorial moments of oint rocesses Jean-Christohe Breton IRMAR - UMR CNRS 6625 Université de Rennes 1 Camus de Beaulieu F-35042 Rennes Cedex France Nicolas Privault Division of Mathematical Sciences School of Physical and Mathematical Sciences Nanyang Technological University 21 Nanyang Link Singaore 637371 May 19, 2014 Abstract We derive joint factorial moment identities for oint rocesses with Paangelou intensities. Our roof simlifies revious combinatorical aroaches to the comutation of moments for oint rocesses. We also obtain new exlicit sufficient conditions for the distributional invariance of oint rocesses with Paangelou intensities under random transformations. Key words: Point rocess, Paangelou intensity, factorial moment, moment identity. Mathematics Subject Classification 2010: 60G57; 60G55; 60H07. 1 Introduction Consider the comound Poisson random variable β 1 Z α1 + + β Z α 1.1 where β 1,..., β IR are constant arameters and Z α1,..., Z α is a sequence of indeendent Poisson random variables with resective arameters α 1,..., α IR +. The Lévy-Khintchine formula jean-christohe.breton@univ-rennes1.fr nrivault@ntu.edu.sg [e tβ 1Z α1 + +β Z α e α 1e β 1 t 1+ +α e βt 1 1

shows that the cumulant of order k 1 of 1.1 is given by α 1 β k 1 + + α β k. As a consequence, the moment of order n 1 of 1.1 is given by the Faà di Bruno formula as [ n β i Z αi i1 n m1 P 1 P m{1,...,n} i 1,...,i m1 where the above sum runs over all artitions P 1,..., P m of {1,..., n}. β P 1 i 1 α i1 β Pm i m α im, 1.2 Such cumulant-tye moment identities have been extended in [8 to Poisson stochastic integrals of random integrands through the use of the Skorohod integral on the Poisson sace, cf. [6, [7. The construction of the Skorohod integral has been extended to oint rocesses with Paangelou intensities in [9, and in [3, the moment identities of [8 have been extended to oint rocesses with Paangelou intensities via a simler combinatorial argument based on induction. In this aer we deal with factorial moments, which are known to be easier to handle than standard moments, cf. for examle, the direct relation between factorial moments and the correlation functions of oint rocesses 2.3 below. See also [2 for the use of factorial moments to light-traffic aroximations in queueing rocesses. In the case of random sets we obtain natural factorial moment identities by a direct induction argument, see Proosition 2.1 and Proosition 2.2. The moment indentities of [3 can then be recovered from standard relations between factorial moments and classical moments. On the other hand, our results allow us to derive new racticable sufficient conditions for the distributional invariance of oint rocesses, with Paangelou intensities, cf. Condition 3.8 in Proosition 3.2. Such conditions are shown to be satisfied on tyical examles including transformations acting within the convex hull generated by the oint rocess. 2

This aer is organized as follows. In Section 2, we derive factorial moment identities for random oint measure of random sets in Proositions 2.1 and 2.2, and in Section 3 we aly those identities to oint rocess transformations in Proosition 3.2. In Section 4, we show that the corresonding moment identities can be recovered by combinatorial arguments, cf. Proosition 4.2. In Section 5, we recover some recent results on the invariance of Poisson random measures under interacting transformations, with simlified roofs. Notation and reliminaries on Paangelou intensities Let be a Polish sace equied with a σ-finite measure σdx. Let Ω denote the sace of configurations whose elements ω Ω are identified with the Radon oint measures ω ɛ x, where ɛ x denotes the Dirac measure at x. A oint rocess x ω is a robability measure P on Ω equied with the σ-algebra F generated by the toology of vague convergence. Point rocesses can be characterized by their Cambell measure C defined on B F by CA B : 1 A x1 B ω \ {x} ωdx, A B, B F. Recall the Georgii-Nguyen-Zessin identity ux, ωωdx ux, ω xcdx, dω, 1.3 for all measurable function u : Ω Ω IR such that both sides of 1.3 make sense. In articular, a Poisson oint rocess with intensity σ is a oint rocess with Cambell measure C σ P, and the Poisson measure with intensity σdx will be denoted by π σ. In the sequel we will deal with oint rocesses whose Cambell measure Cdx, dω is absolutely continuous with resect to σ P, i.e. Cdx, dω cx, ωσdxp dω, 3

where the density cx, ω is called the Paangelou density. In this case the identity 1.3 reads ux, ωωdx ux, ω xcx, ωσdx, 1.4 and cx, ω 1 for Poisson oint rocess with intensity σ. Denoting by Ω 0 the set of finite configurations in Ω we will use the comound Cambell density ĉ : Ω 0 Ω IR + which is defined inductively by ĉ{x 1,..., x n, y}, ω : cy, ωĉ{x 1,..., x n }, ω {y}, n 0. 1.5 Given x n x 1,..., x n n, we will use the notation ε + x n for the oerator ε + x n F ω F ω {x 1,..., x n }, ω Ω, where F is any random variable on Ω. With this notation we also have ĉx n, ω cx 1, ωcx 2, ω {x 1 }cx 3, ω {x 1, x 2 } cx n, ω {x 1,..., x n 1 }. In addition, we define the random measure ˆσ n dx n on n by ˆσ n dx n ĉx n, ωσ n dx n ĉx n, ωσdx 1 σdx n, with σ n dx n σdx 1 σdx n. Finally, given a ossibly random set A we let NAω 1 Ax ωdx denote the cardinality of ω Aω. Recall that for a Poisson oint rocess with intensity σ, ωa has the Poisson distribution with arameter σa for any non random A F, and NA is indeendent of NB whenever A, B F are disjoint. 4

2 Factorial moments In this section we deal with the factorial moment defined by µ f nna : [NA n, where x n xx 1 x n + 1, x IR, n IN, is the falling factorial roduct and A is a ossibly random measurable subset of. Proosition 2.1 Let A Aω be a random set. For all n 1 and sufficiently integrable random variable F, we have [ F NA n ε + x n F 1 A nx 1,..., x n ω ˆσ n dx 1,..., dx n. n Proof. We show by induction on n 1 that [ F NA n ε + x n F 1 A x 1 1 A x n ω ˆσ n dx n. 2.1 n Clearly the formula [F NA ε + x F 1 A xω cx, ω σdx holds at the rank n 1 due to 1.4 alied to ux, ω F ω1 Aω x. Next, assuming that 2.1 holds at the rank n, we aly it with F relaced by F NA n and get [ F NA NA n ε + x n F NA n1 A x 1 1 A x n ω ˆσ n dx n n ε + x n F NA1 A x 1 1 A x n ω ˆσ n dx n n n ε + x n F 1 A x 1 1 A x n ω ˆσ n dx n n Nε + x n Aωε + x n F 1 A x 1 1 A x n ω ˆσ n dx n, 2.2 n where in 2.2 we used the relation ε + x n NAω Nε + x n Aω + 5 n δ xi ε + x n Aω i1

n Nε + x n Aω + ε + x n 1 A x i ω. Next, with x n+1 x 1,..., x n, x n+1, recalling that NA 1 Ax ωdx and alying 1.4 to ux, ω : ε + x n F 1 A x 1 1 A x n 1 A xω ĉx n, ω F ω {x 1,..., x n }1 Aω {x1,...,x n}x 1 1 Aω {x1,...,x n}x n 1 Aω {x1,...,x n}x ĉx n, ω i1 for fixed x 1,..., x n with the relation ε + x n+1 ε + x n+1 ε + x n we find [F NA NA n ε + x n+1 F 1A x 1 1 A x n+1 ω ĉx n, ω {x n+1 } cx n+1, ω n+1 σdx 1 σdx n+1 ε + x n+1 F 1 A x 1 1 A x n+1 ω ˆσ n dx n+1, n+1 where on the last line we used 1.5. By induction, in the next Proosition 2.2 we also obtain a joint factorial moment identity for a.s. disjoint random sets A 1,..., A. It extends the classical identity [ NA 1 n1 NA n ρ n x 1,..., x n σdx 1 σdx n, 2.3 A n 1 1 An for deterministic disjoint sets A 1,..., A, where ρ n x 1,..., x n is the correlation function of the oint rocess and n n 1 + + n. Proosition 2.2 Let n n 1 + + n disjoint for almost all ω Ω, then [ F NA 1 n1 NA n ε + x n F 1 n A 1 Proof. and A 1 ω,..., A ω be measurable and n x n ˆσ n dx n. 2.4 We roceed by induction on 1. For 1, the identity reduces to that of Proosition 2.1. We assume that the identity holds true for and show it for + 1. Letting n n 1 + + n and m n + n +1 we have: [ F NA 1 n1 NA +1 n+1 6

n ε + x n F NA +1 n n+1 1 A 1 1 1 A n 1,..., x n ˆσ n dx n Nε + x n A +1 n+1 ε + x n F 1 n A 1 ε + y n+1 ε + x n F 1 n A 1 n n +1 1 ε + xn A n +1 y 1,..., y n+1 +1 ε + x m F 1 n A 1 m 1 n x 1,..., x n ˆσ n dx n 2.5 1 1 A n x 1,..., x n ĉ{x 1,..., x n }, ω ĉ{y 1,..., y n+1 }, ω σdy 1 σdy n+1 σdx 1 σdx n 2.6 1 n A +1 x 1,..., x m +1 ˆσdx 1,..., dx m, 2.7 where in 2.5 we used ε + x n NA +1 n+1 Nε + x n A +1 + Nε + x n A +1 + n δ xi ε + x n A +1 i1 n ε + x n 1 A+1 x i n +1 i1 n +1 and observe that the contribution of the sum is zero since, for all 1 k and 1 i n, 1 A+1 x i 1 Ak x i 0. In 2.6, we noted y n+1 y 1,..., y n+1 and used Proosition 2.1 with, for a fixed x n x 1,..., x n, F ω ε + x n F 1A n 1 1 n x 1,..., x n ω ĉ{x 1,..., x n }, ω and the set ε + x n A +1. Finally in 2.7, we used the following consequence of 1.5 ĉ {x 1,..., x n }, ω {y 1,..., y n } ĉ {y 1,..., y n }, ω ĉ {x 1,..., x n, y 1,..., y n }, ω together with ε + y n ε + x n ε + y n x n. 3 Transformations of oint rocesses In this section we derive a sufficient condition for the distributional invariance of oint rocesses under random transformations. Consider the finite difference oerator D x F ω F ω {x} F ω 7

where F is any random variable on Ω. Note that multile finite difference oerator exresses D Θ F η Θ 1 Θ +1+ η F ω η 3.1 where the summation above holds over all ossibly emty subset η of Θ. Let x n {x 1,..., x n }, from the relation ε + x n u 1 x 1, ω u n x n, ω ε + x 1,...,x n u 1 x 1, ω u n x n, ω D Θ u1 x 1, ω u n x n, ω, 3.2 Θ {x 1,...,x n} where D Θ D x1 D xl when Θ {x 1,..., x l } and from 2.4 we have [ F NA 1 n1 NA n ε + x n F 1 n A 1 1 A n x n ˆσ n dx n Θ {x 1,...,x n} D Θ F 1 n A 1 n x n ˆσ n dx n 3.3 for n 1 + + n n and a.s. disjoint sets A 1 ω,..., A ω. The next lemma will be useful in Proosition 3.2 to characterize the invariance of transformations of oint rocesses from 3.3. Lemma 3.1 Let m 1 and assume that for all x 1,..., x m the rocesses u i : Ω IR, 1 i m satisfy the condition D Θ1 u 1 x 1, ω D Θm u m x m, ω 0, 3.4 for every family {Θ 1,..., Θ m } of non emty subsets such that Θ 1 Θ m {x 1,..., x m }, for all x 1,..., x m and all ω Ω. Then we have for all x 1,..., x m and all ω Ω. Proof. It suffices to note that D x1 D xn u1 x 1, ω u l x l, ω D x1 D xm u1 x 1, ω u m x m, ω 0 3.5 Θ 1 Θ l {x 1,...,x n} D Θ1 u 1 x 1, ω D Θl u l x l, ω, 3.6 where the above sum is not restricted to artitions, but includes all ossibly emty sets Θ 1,..., Θ l whose union is {x 1,..., x n }. 8

In the next result, we recover Theorem 5.1 of [3 in a more direct way due to the use of factorial moments, but using a different cyclic tye condition. Condition 3.8 below is interreted by saying that D Θ1 h 1 τx 1, ω D Θm h m τx m, ω 0, 3.7 for any family h 1,..., h m of bounded real-valued Borel functions on Y. The merit of our condition 3.8 is to be satisfied in a tyical examle based on convex hull generated by a oint rocess, see age 11. Proosition 3.2 Let, Y be Polish saces equied resectively with σ-finite measures σ and µ and let τ : Ω Y be a random transformation such that τ, ω : Y mas bijectively σ to µ for all ω Ω, i.e. σ τ, ω 1 µ, ω Ω, and satisfying the condition D Θ1 τx 1, ω D Θm τx m, ω 0, 3.8 for every family {Θ 1,..., Θ m } of non emty subsets such that Θ 1 Θ m {x 1,..., x m }, for all x 1,..., x m and all ω Ω, m 1. Then τ : Ω Ω Y defined by τ ω ɛ τx,ω ω τ, ω 1, x ω ω Ω, transforms a oint rocess ξ with Paangelou intensity cx, ω with resect to σ P into a oint rocess on Y with correlation function ρ τ y 1,..., y n [ ĉ{τ 1 y 1, ω,..., τ 1 y n, ω}, ω, y 1,..., y n Y, with resect to µ. Proof. Consider B 1,..., B disjoint deterministic subsets of Y such that µb 1,..., µb are finite. From interretation 3.7, Condition 3.8 ensures 3.4 for u k x, ω 1 Bik τx, ω and, in turn, Lemma 3.1 shows that D x1 D xk 1Bi1 τx 1, ω 1 Bik τx k, ω 0, x 1,..., x k, 3.9 9

for all i 1,..., i k {1,..., } and ω Ω. For i 1,...,, let A i ω τ, ω 1 B i and τ NB i be the cardinal of τ ω B i, i.e. τ NB i ɛ τx,ω B i ɛ x A i NA i. x ω x ω Then alying 3.3 with F 1 and the disjoint random sets A i ω, i 1,...,, yields [ [ τ NB 1 n1 τ NB n NA1 n1 NA n Θ {x 1,...,x n} + D Θ 1 n A 1 D x1 D xn 1 n B 1 Θ{x 1,...,x n} Θ{x 1,...,x n} D Θ 1 n B 1 D Θ 1 n B 1 n x n ˆσ n dx n 1 B n τx n, ω ˆσ n dx n 1 B n τx n, ω ˆσ n dx n 1 B n τx n, ω ˆσ n dx n 3.10 n 1 + + n n 1, where τx n, ω stands for τx 1, ω,..., τx n, ω and where we used 3.9. Next, without loss of generality the generic term of 3.10 can be reduced to the term with Θ {x 1,..., x n 1 } and using 3.6, we have D x1 D xn 1 1 n B 1 1 B n τx n, ωˆσ n dx n D x1 D xn 1 1 n B 1 1 n 1 1 B n τx n 1, y n, ω ĉ{x 1,..., x n 1, τ 1 y n, ω}, ω σ n 1 dx n 1 µdy n with the change of variable y n τx n, ω. Finally, by alying the above argument recursively we obtain that [ τ NB 1 n1 τ NB n 1 n B 1 1 B n y n [ ĉ{τ 1 y 1, ω,..., τ 1 y n, ω}, ω µ n dy n, n 1,..., n n 1, which recovers the definition of the correlation function of τ ξ see 2.3. 10

The roof of Proosition 3.2 also shows that if A 1,..., A are disjoint random subsets of such that D Θ1 1 A1 ωx 1 D Θm 1 Amωx m 0, for every family {Θ 1,..., Θ m } of non emty subsets such that Θ 1 Θ m {x 1,..., x m }, for all x 1,..., x m and all ω Ω, m 1, then we have [ NA 1 n1 NA n 1 n A 1 1 A n x nˆσ n dx n, n 1 + + n n. xamle Here we take IR d with norm and we consider an examle of transformation satisfying Condition 3.8, defined conditionally on the extremal vertices ω e ω of the convex hull of ω B0, 1. Denote by Cω the convex hull of ω, and by Cω its interior, we consider a maing τ : Ω such that for all ω Ω, τ, ω : leaves \ Cω e invariant thus including the extremal vertices ω e of Cω e while the oints inside Cω e are shifted deending on the data of ω e, i.e. we have τx, ω e, x Cω e, τx, ω x, x \ Cω e. 3.11 As shown in Proosition 3.3 below, such a transformation τ satisfies Condition 3.8. The next figure shows an examle of behaviour such a transformation, with a finite set of oints for simlicity of illustration. 11

τ Proosition 3.3 The maing τ : Ω given in 3.11 satisfies Condition 3.8. Proof. Let x 1,..., x m. Clearly, we can assume that some x i lies outside of Cω Cω e, otherwise D xi τx j, ω τx j, ω {x i } τx j, ω τx j, ω {x i } e τx j, ω e τx j, ω e τx j, ω e 0 for all i, j 1,..., m. Similarly, we can assume that Cω {x 1,..., x m } has at least one extremal oint x i {x 1,..., x m }. Now we have τx i, ω η τx i, ω x i for all η {x 1,..., x m }, hence D Θ τx i, ω 0, for all Θ {x 1,..., x m }, due to the following consequence of 3.1 D Θ τx i, ω η Θ 1 Θ +1 η τx i, ω η τx i, ω η Θ 1 Θ +1 η τx i, ω η1 1 Θ +1 0, where the summation above holds over all ossibly emty subset η of Θ. consequence, one factor of 3.8 necessarily vanishes. As a 12

4 Moment identities As a consequence of the our factorial moment identities, we recover the moment identities recently derived for random integrands in [8 for Poisson stochastic integrals and extended in to oint rocesses in [3. Let Sn, k 1 k! d 1 + +d k n d 1! d k! 4.1 denote the Stirling number of the second kind, i.e. the number of artitions of a set of n objects into k non-emty subsets, cf. also Relation 3 age 2 of [1. As a consequence we recover the following elementary moment identity from Proosition 2.1. Lemma 4.1 Let A 1 A 2 ω,..., A A ω be disjoint random sets. We have [ F NA 1 n 1... NA n n 1 k 1 0 n k 0 Sn 1, k 1... Sn, k k 1 + +k ε + x k1 + +k F 1A k 1 1 As a articular case, for a random set A Aω, we have Proof. [F NA n n k0 Using the classical identity kx k1 + +k ˆσ k 1+ +k dx k1 + +k. Sn, k ε + x k F 1 A x 1 1 A x k ˆσdx 1,..., dx k. k x n n Sn, k xx 1 x k + 1, k0 cf. e.g. [5 or age 72 of [4, we have [ F NA 1 n 1... NA n n1 n F Sn 1, k 1 NA 1 k1... Sn, k NA k n 1 k 1 1 k 1 0 n k 0 k 0 [ Sn 1, k 1... Sn, k F NA 1 k1... NA k 13

n 1 k 1 0 n k 0 Sn 1, k 1... Sn, k k 1 + +k ε + x k1 + +k F 1A k 1 1 kx k1 + +k ˆσ k 1+ +k dx k1 + +k where we use 2.4 in the last line. More generally, Lemma 4.1 allows us to recover the following moment identity, cf. Theorem 3.1 of [3, and Proosition 3.1 of [8 for the Poisson case. Proosition 4.2 Let u : Ω IR be a measurable rocess. We have [ n ux, ω ωdx n k1 B1 n,...,bn k where the sum runs over the artitions B n 1,..., B n k that all terms are integrable. ε + x ux1 k, Bn 1 ux k, Bn k ˆσdx k k 4.2 of {1,..., n}, for any n 1 such Proof. First we establish 4.2 for simle rocesses of the form ux, ω i1 F iω1 Ai ωx with a.s. disjoint random sets A i ω, 1 i. Alying Lemma 4.1 inductively we have [ F i i1 n 1 + +nn n 1 + +nn n m0 n [ n 1 Ai x ωdx F i NA i i1 [ F 1 NA 1 n1 F NA n k 1 + +k ε + x k1 + +k n 1 + +nn m ε + x m n 1 k 1 0 F n 1 1 F n 1 A k 1 1 n k 0 F n 1 1 F n 1 A k 1 1 Sn 1, k 1 Sn, k kx 1,..., x k1 + +k ĉ{x 1,..., x k1 + +k }, ω σdx 1 σdx k1 + +k k 1 + +km 0 k 1 n 1,...,0 k n Sn 1, k 1 Sn, k kx 1,..., x m 14 ĉ{x 1,..., x m }, ω σdx m

n m0 n n 1 + +nn m ε + x m F n 1 1 F n m0 P 1 P m{1,...,n} i 1,...,i m1 I 1 I{1,,m} I 1 n 1,..., I n Sn 1, I 1 Sn, I I 1! I! m! 1 A1 x j 1 A x j ĉ{x 1,..., x m }, ω σdx m j I 1 j I ε + x m m F P 1 i 1 1 Ai1 x 1 F Pm i m 1 Aim x m 4.3 ˆσdx m, 4.4 where in 4.3 we made changes of variables in the integral and, in 4.4, we used the combinatorial identity of Lemma 4.3 below with α i,j 1 Ai x j, 1 i, 1 j m, and β i F i. The roof is concluded by using the fact that the A i s in 4.5 are disjoint, as follows: [ n F i 1 Ai x ωdx i1 n m0 P 1 P m{1,...,n} n m0 P 1 P m{1,...,n} [ m ε + x m m ε + x m i1 F P 1 i 1 Ai x 1 F i 1 Ai x 1 i1 P1 i1 F Pm i 1 Ai x m ˆσdx m F i 1 Ai x m Pm ˆσdx m. i1 4.5 The general case is obtained by aroximating ux, ω with simle rocesses. Using 3.2, we can also write [ n ux, ω ωdx n k1 B1 n,...,bn k Θ {x 1,...,x k } k D Θ u Bn 1 x 1 u Bn k x k ˆσdx k The next lemma has been used above in the roof of Proosition 4.2. Lemma 4.3 Let m, n, N, α i,j 1 i,1 j m and β 1,..., β IR. We have n 1 + +nn I 1 I{1,...,m} I 1 n 1,..., I n 15 Sn 1, I 1 Sn, I.

Proof. P 1 P m{1,...,n} i 1,...,i m1 Observe that 4.1 ensures Sn, I β n α j j I for all α j, j I, β IR, n N. We have n 1 + +nn n 1 + +nn I 1 I{1,...,m} I 1 n 1,..., I n a I P 1 1 a {1,...,n 1 } n 1 + +nn n 1 + +nn n 1 + +nn P 1 1 P 1 k 1 {1,...,n 1 } I 1! I! m! β n 1 1 α 1,j j I 1 β n α,j j I β P 1 i 1 α i1,1 β Pm i m α im,m. 4.6 I 1 I{1,...,m} I 1 n 1,..., I n a I Pa{1,...,n} αj β P j j I Sn 1, I 1 Sn, [I I 1! I! m! β n 1 1 α 1,j j I 1 I 1! I! m! α1,j1 β P 1 j 1 1 j 1 I 1 I 1 I{1,...,m} I 1 n 1,..., I n I 1 I{1,...,m} I 1 n 1,..., I n I 1! I! m! I 1! I! m! k 1 + +km 1 k 1 n 1,...,1 k n a I 1 P 1 a {1,...,n 1 } a I P a {1,...,n } αl,jl β P l j l l l1 j l I l a I 1 P 1 a {1,...,n 1 } i 1,...,i m1 β n α,j j I α,j β P j J a I P a {1,...,n } a I P a {1,...,n } α l,jl β P l j l l l1 j l I l l1 j l I l m P k 1 + +k 1 +1 P k 1 + +k {1,...,n} j1 j αij,jβ P i 1 im j + + Pj i j 16

β P 1 i 1 P 1 P m{1,...,n} i 1,...,i m1 α i1,1 β Pm i m α im,m, by a reindexing of the summations and the fact that the reunions of the artitions P j 1,..., P j I j, 1 j, of disjoint subsets of {1,..., m} run the artition of {1,..., m} when we take into account the choice of the subsets and the ossible length k j, 1 j, of the artitions. Note that the combinatorial result of Lemma 4.3 can also be shown in a robabilistic way when α i,j α i, 1 i, 1 j m. Next we have n m0 λ m n 1 + +nn n 1 + +nn n 1 + +nn [ n β i Z λαi n m0 λ m i1 k 1 + +km k 1 n 1,...,k n n 1 k 1 0 Sn 1, k 1 Sn, k β n 1 1 α k 1 Sn 1, k 1 λα 1 k1 βn 1 1 β n [Z n 1 λα 1 Z n λα P 1 P m{1,...,n} i 1,...,i m1 n k 0 1 β n α k Sn, k λα k β n 1 1 β n β P 1 i 1 α i1 β Pm i m α im, 4.7 since by the relation 1.2 between standard and factorial moments the moment of order n i of Z λαi is given by [ ni Z n i λα i Sn i, kλα i k. k0 The above relation 4.7 being true for all λ, this imlies 4.6 for this choice of α i,j s. 5 Poisson case In the Poisson case, we have cx, ω 1 and the results of the revious sections secialize immediately to new factorial moment identities for Poisson oint rocesses 17

with intensity σdx. For any random set A Aω and sufficiently integrable random variable F, we have [ F NA n ε + x n F 1 A x 1 1 A x n σdx 1 σdx n, n n 1. For all almost surely disjoint random sets A i ω, 1 i, and sufficiently integrable random variable F, we have [ F NA 1 n1 NA n ε + x n F 1 n A 1 n x 1,..., x n σdx 1 σdx n with n n 1 + + n. In addition, we have the following roosition whose roof is similar to that of Proosition 3.2 although it cannot be obtained as a direct consequence of Proosition 3.2 and is not available in the non-poisson oint rocess setting., Proosition 5.1 Consider A 1 ω,..., A ω a.s. σa i ω is deterministic, i 1,...,, and disjoint random sets such that D Θ1 1 Ai ωx 1 D Θm 1 Ai ωx m 0, 5.1 for every family {Θ 1,..., Θ m } of non emty subsets such that Θ 1 Θ m {x 1,..., x m }, all x 1,..., x m, all ω Ω. Then the family NA1,..., NA is a vector of indeendent Poisson random variables with arameters σa 1,..., σa. Proof. Let n n 1 + + n. Under Condition 5.1, Lemma 3.1 and 3.5 show that D x1 D xk 1Ai1 ωx 1 1 Aik ωx k 0, x 1,..., x k, 5.2 for all i 1,..., i k {1,..., n} and ω Ω. Since in addition σa i is deterministic, i 1,...,, then by 3.3 with F 1 we obtain [ NA 1 n1 NA n D Θ 1A n 1 Θ {x 1,...,x n} 18 n x n σ n dx n

D x1 D xn 1A n 1 + Θ{x 1,...,x n} Θ{x 1,...,x n} D Θ 1A n 1 D Θ 1A n 1 n x n σ n dx n n x n σ n dx n n x n σ n dx n 5.3 using 5.2. Next, without loss of generality the generic, term of 5.3 can be reduced to the term with Θ {x 1,..., x n 1 } and using 3.5, we have D x1 D xn 1 1A n 1 1 A n x n σ n dx n n k D 1 Θn1 + +n k 1 +j A k x n1 + +n k 1 +j σ n dx n n 1 n Θ 1 Θ n{x 1,...,x n 1 } k1 j1 n 1 Θ 1 Θ n{x 1,...,x n 1 } k1 j1 j1 D 1 Θn1 + +n 1 +j A x n1 + +n 1 +j n 1 Θ 1 Θ n 1 {x 1,...,x n 1 } k1 j1 n 1 j1 1 n k D 1 Θn1 + +n k 1 +j A k x n1 + +n k 1 +j 5.4 D Θn 1 A x n σdx n σ n 1 dx n 1 1 n k D 1 Θn1 + +n k 1 +j A k x n1 + +n k 1 +j D Θn1 + +n 1 +j 1 A x n1 + +n 1 +jσa σ n 1 dx n 1 σa D x1 D xn 1 1 n A 1 n 1 1 1 n 1 A x n 1 σ n 1 dx n 1 5.5 where 5.5 comes from the fact that in 5.4 only the term with Θ n is not zero since D Θ 1 A x σdx D Θ 1 A xσdx D Θ σa 0 using σa is deterministic. Finally, by alying the above argument recursively we obtain [ NA 1 n1 NA n 1 n A 1 1 A n x nσ n dx n σa 1 n1 σa n, 19

n 1,..., n 1, which characterizes the Poisson distribution with arameters σa 1,..., σa. As a consequence of Proosition 5.1 we recover Theorem 3.3 in [7 on the invariance for Poisson measures when, σ Y, µ. Note that Condition 5.6, which is interreted as in 3.7 above, is actually a stregthening of Condition 3.8 in [7 and fills a ga in the roof of Theorem 3.3 therein, based on the fact that D x uω 0 does not imly D Θ uω 0 when x Θ 0. Theorem 5.2 Let τ : Ω Y be a random transformation such that τ, ω : Y mas σ to µ for all ω Ω, i.e. σ τ, ω 1 µ, ω Ω, and satisfying the condition D Θ1 τx 1, ω D Θm τx m, ω 0, 5.6 for every family {Θ 1,..., Θ m } of non emty subsets such that Θ 1 Θ m {x 1,..., x m }, all x 1,..., x m, all ω Ω, and all i 1,...,. Then τ : Ω Ω Y defined by τ ω x ω ɛ τx,ω ω τ, ω 1, ω Ω, mas π σ to π µ, i.e. τ π σ is the Poisson measure π µ with intensity µdy on Y. Proof. For any family B 1,..., B of disjoint measurable subsets of Y with finite measure, we let A i ω τ 1 B i, ω, i.e. 1 Ai 1 Bi τ, ω, i 1,...,, and by Proosition 5.1, we find that ω τ ωb 1,..., τ ωb ωa 1,..., ωa is a vector of indeendent Poisson random variables with arameters µa 1,..., µa since σa i ω στ 1 B i, ω µb i is deterministic, i 1,...,, and 5.1 comes from the following consequence of 5.6: D Θ1 1 Ai1 ωx 1 D Θm 1 Aim ωx m 20

D Θ1 1 Bi1 τx 1, ω D Θm 1 Bim τx m, ω 0. The examle of random transformation given on age 11 at the end of Section 3 also satisfies Condition 5.6 in Theorem 5.2. References [1 M. Bernstein and N. J. A. Sloane. Some canonical sequences of integers. Linear Algebra Al., 226/228:57 72, 1995. [2 B. B laszczyszyn. Factorial moment exansion for stochastic systems. Stochastic Process. Al., 562:321 335, 1995. [3 L. Decreusefond and I. Flint. Moment formulae for general oint rocesses. Prerint ariv:1211.4811, 2012, to aear in J. Funct. Anal. [4 B. Fristedt and L. Gray. A modern aroach to robability theory. Probability and its Alications. Birkhäuser Boston Inc., Boston, MA, 1997. [5 T. Gerstenkorn. Relations between the crude, factorial and inverse factorial moments. In Transactions of the ninth Prague conference on information theory, statistical decision functions, random rocesses, Vol. A Prague, 1982, ages 247 252. Reidel, Dordrecht, 1983. [6 N. Privault. Moment identities for Poisson-Skorohod integrals and alication to measure invariance. C. R. Math. Acad. Sci. Paris, 347:1071 1074, 2009. [7 N. Privault. Invariance of Poisson measures under random transformations. Ann. Inst. H. Poincaré Probab. Statist., 484:947 972, 2012. [8 N. Privault. Moments of Poisson stochastic integrals with random integrands. Probability and Mathematical Statistics, 322:227 239, 2012. [9 G. L. Torrisi. Point rocesses with Paangelou conditional intensity: from the Skorohod integral to the Dirichlet form. Markov Process. Related Fields, 192:195 248, 2013. 21