arXiv:math/0702820v1 [math.PR] 27 Feb 2007 ac,lclydpnetpitpoes adcr process, core hard process. process, renewal cesses, point dependent locally tance, 1) Systems. IE[ utai,e-mail: Australia, e-mail: Singapore, of Republic 118402, Singapore DOI: 236–244 (2006) Topics 52 Related Vol. and Series Series Time Notes–Monograph Lecture IMS o l bounded all for te ad soritrs sotno h ieec IP( difference the on often is interest our as hand, other (1) equation: Stein the to ae ntefloigosrain ongtv nee v integer C nonnegative by a developed observation: W following was the approximation on Poisson based for method Stein’s approximation Poisson 1. aua orlt h function the relate to natural fteeuto emt one solution bounded a permits equation the If and c 1 nttt fMteaia ttsis 2006 , Mathematical of Institute ewrsadphrases: and Keywords − † ∗ 2 1 M 00sbetclassifications: subject 2000 AMS olw oso itiuinwt mean with distribution Poisson follows upre yteACCnr fEclec o ahmtc an Mathematics for Excellence of Centre ARC the by Supported eateto ahmtc n ttsis nvriyo Me of University Statistics, and Mathematics of Department University National Sciences, Mathematical for Institute upre yrsac rn -5-0-5-1 fteNatio the of R-155-000-051-112 grant research by Supported A 10.1214/074921706000001076 f W rmPl hoyt ti’ method Stein’s to theory Palm From ( W Abstract: n i 20) n s tt td oso rcs approxima process superposition Poisson dependent p for study point and to of processes it dependence point use dependent local and define (2004)] Xia to t Palm and us and enables process immigration-death also appr the approach of process view Poisson of and point the approximation variable random son ) ( W − oso rcs approximation: process Poisson d Po( ) V T ≈ } IP( ainlUiest fSnaoeadUiest fMelbour of University and Singapore of University National [email protected] ( f λ L W )( : ( hsepsto xlistebscieso ti’ ehdf method Stein’s of ideas basic the explains exposition This o l bounded all for 0 Z W A oi .Y Chen Y. H. Louis ∈ + λf ] where )], ) A , → mirto-et rcs,ttlvraindsac,Was distance, variation total process, immigration-death ( Po( ) w − R IE 1) + λ where , Po( { ):=sup = : )) λf λf A λ − sup = ( rmr 00;scnay6F5 60G55. 60F05, secondary 60–02; primary )( W ( ⊂ wf w A Z A A Z IE = ) 1) + 1) + + ( ⊂ ⊂ + w Z Z f = : + + = ) and 1 : 236 − | | − , Z IE IP( ∗ { [email protected] + λf { λ f W 1 wf { n iu Xia Aihua and 1 0 f W eoe sPo( as denoted , A λf A , A A → ( 1 ( then , ( w ( A w , ∈ steidctrfnto on function indicator the is W W 2 ( ) with ) R W . . . , A − eedn uepsto fpitpro- point of superposition dependent ) then , 1) + } ) fSnaoe rneGog’ Park, George’s Prince 3 Singapore, of Po( 1) + − 0 = } ersial,i IE if Heuristically, . Po( a nvriyo Singapore. of University nal − 1 λ bun,Prvle I 3010, VIC Parkville, lbourne, A )( − f W L λ W ( A fpitprocesses. point of ( w f W )( le admvariable random alued W er.Telatter The heory. 2 ) λ ) xmto from oximation A . A ttsiso Complex of Statistics d , infrlocally for tion oess[Chen rocesses ,i n nyif only and if ), ∈ † − ) ( A ) W | ( A ≈ Po( e 1]wihis which [13] hen W ne ) ) } Po( rPois- or ) − ; λ }| )( Po( . λ A esendis- serstein .O the On ). ,leading ), { λ λf )( A A ( tis it , W = ) + Poisson process approximation 237

As a special case in applications, we consider independent Bernoulli random vari- ables X1, · · · , Xn with IP(Xi = 1) = 1 − IP(Xi = 0) = pi, 1 ≤ i ≤ n, and n n W = i=1 Xi, λ = IE(W )= i=1 pi. Since P Pn n IE[Wf(W )] = IE[Xif(W )] = piIEf(Wi + 1), i=1 i=1 X X where Wi = W − Xi, we have

n IE{λfA(W + 1) − WfA(W )} = piIE[fA(W + 1) − fA(Wi + 1)] i=1 Xn 2 = pi IE∆fA(Wi + 1), i=1 X 1−e−λ where ∆fA(i)= fA(i + 1) − fA(i). Further analysis shows that |∆fA(w)|≤ λ (see [6] for an analytical proof and [26] for a probabilistic proof). Therefore

n 1 d (L(W ), Po(λ)) ≤ 1 ∧ p2. T V λ i i=1   X Barbour and Hall [7] proved that the lower bound of dT V (L(W ), Po(λ)) above is of the same order as the upper bound. Thus this simple example of Poisson approx- imation demonstrates how powerful and effective Stein’s method is. Furthermore, it is straightforward to use Stein’s method to study the quality of Poisson approx- imation to the sum of dependent random variables which has many applications (see [18] or [8] for more information).

2. Poisson process approximation

Poisson process plays the central role in modeling the data on occurrence of rare events at random positions in time or space and is a building block for many other models such as Cox processes, marked Poisson processes (see [24]), compound Poisson processes and L´evy processes. To adapt the above idea of Poisson random variable approximation to Poisson process approximation, we need a probabilistic interpretation of Stein’s method which was introduced by Barbour [4]. The idea is to split f by defining f(w)= g(w) − g(w − 1) and rewrite the Stein equation (1) as

(2) Ag(w): = λ[g(w + 1) − g(w)] + w[g(w − 1) − g(w)] = 1A(w) − Po(λ)(A), where A is the generator of an immigration-death process Zw(t) with immigration rate λ, unit per capita death rate, Zw(0) = w, and stationary distribution Po(λ). The solution to the Stein equation (2) is

∞ (3) gA(w)= − IE[1A(Zw(t)) − Po(λ)(A)]dt. Z0 This probabilistic approach to Stein’s method has made it possible to extend Stein’s method to higher dimensions and process settings. To this end, let Γ be a compact metric space which is the carrier space of the being approximated. Suppose d0 is a metric on Γ which is bounded by 1 and ρ0 is a pseudo-metric on Γ 238 L. H. Y. Chen and A. Xia which is also bounded by 1 but generates a weaker topology. We use δx to denote k the point mass at x, let X : = { i=1 δαi : α1,...,αk ∈ Γ, k ≥ 1}, B(X ) be the Borel σ–algebra generated by the weak topology ([23], pp. 168–170): a sequence P {ξn} ⊂ X converges weakly to ξ ∈ X if Γ f(x)ξn(dx) → Γ f(x)ξ(dx) as n → ∞ for all bounded continuous functions f on Γ. Such topology can also be generated R R by the metric d1 defined below (see [27], Proposition 4.2). A on Γ is defined as a measurable mapping from a probability space (Ω, F, IP) to (X , B(X )) (see [23], p. 13). We use Ξ to stand for a point process on Γ with finite intensity measure λ which has total mass λ: = λ(Γ), where λ(A) = IEΞ(A), for all Borel set A ⊂ Γ. Let Po(λ) denote the distribution of a Poisson process on Γ with intensity measure λ. Since a point process on Γ is an X -valued random element, the key step of extending Stein’s method from one dimensional Poisson approximation to higher dimensions and process settings is, instead of considering Z+-valued immigration- death process, we now need an immigration-death process defined on X . More precisely, by adapting (2), Barbour and Brown [5] define the Stein equation as

Ag(ξ): = [g(ξ + δx) − g(ξ)]λ(dx)+ [g(ξ − δx) − g(ξ)]ξ(dx) (4) ZΓ ZΓ = h(ξ) − Po(λ)(h), where Po(λ)(h) = IEh(ζ) with ζ ∼ Po(λ). The operator A is the generator of an X -valued immigration-death process Zξ(t) with immigration intensity λ, unit per capita death rate, Zξ(0) = ξ ∈ X , and stationary distribution Po(λ). Its solution is

∞ (5) gh(ξ)= − IE[h(Zξ(t)) − Po(λ)(h)]dt, Z0 (see [5]). To measure the error of approximation, we use Wasserstein pseudo-metric which has the advantage of allowing us to lift the carrier space to a bigger carrier space. Of course, other metrics such as the total variation distance can also be considered and the only difference is to change the set of test functions h. Let

m n 1 if m 6= n, 1 m ρ δx , δy : = minπ ρ (xi,y ) if m = n ≥ 1, 1  i j   m i=1 0 π(i) i=1 j=1 0 if n = m =0, X X  P   where the minimum is taken over all permutations π of {1, 2,...,m}. Clearly, ρ1 is a metric (resp. pseudo-metric) if ρ0 is a metric (resp. pseudo-metric) on X . Set

H = {h on X : |h(ξ1) − h(ξ2)|≤ ρ1(ξ1, ξ2) for all ξ1, ξ2 ∈X} .

For point processes Ξ1 and Ξ2, define

ρ2(L(Ξ1), L(Ξ2)): = sup |IEh(Ξ1) − IEh(Ξ2)|, h∈H then ρ2 is a metric (resp. pseudo-metric) on the distributions of point processes if ρ1 is a metric (resp. pseudo-metric). In summary, we defined a Wasserstein pseudo- metric on the distributions of point processes on Γ through a pseudo-metric on Γ as shown in the following chart: Poisson process approximation 239

Carrierspace Γ Configurationspace X Space of the distributions of point processes ρ0 −→ ρ1 −→ ρ2 (≤ 1) (≤ 1) (≤ 1)

As a simple example, we consider a defined as n

Ξ= Xiδ i , n i=1 X where, as before, X1,...,Xn are independent Bernoulli random variables with IP(Xi = 1) = 1 − IP(Xi = 0) = pi, 1 ≤ i ≤ n. Then Ξ is a point process on n carrier space Γ = [0, 1] with intensity measure λ = piδ i . With the metric i=1 n ρ (x, y) = |x − y|: = d (x, y), we denote the induced metric ρ by d . Using the 0 0 P 2 2 Stein equation (4), we have IEh(Ξ) − Po(λ)(h)

= IE [gh(Ξ + δx) − gh(Ξ)]λ(dx)+ [gh(Ξ − δx) − gh(Ξ)]Ξ(dx)  Γ Γ  n Z Z = piIE [gh(Ξ + δ i ) − gh(Ξ)] − [gh(Ξi + δ i ) − gh(Ξi)] n n i=1 n o Xn 2 = p IE [gh(Ξi +2δ i ) − gh(Ξi + δ i )] − [gh(Ξi + δ i ) − gh(Ξi)] , i n n n i=1 X n o where Ξi = Ξ − Xiδ i . It was shown in [27], Proposition 5.21, that n 3.5 2.5 (6) sup |gh(ξ + δα + δβ) − gh(ξ + δα) − gh(ξ + δβ)+ gh(ξ)|≤ + , h∈H,α,β∈Γ λ |ξ| +1 λ n where, and in the sequel, |ξ| is the total mass of ξ, λ = (Γ) = i=1 pi. Hence

d2(L(Ξ), Po(λ)) = sup |IEh(Ξ) − Po(λ)(h)| P h∈H n 2 3.5 2.5 (7) ≤ pi + IE λ Xj +1 i=1 1≤j≤n,j6=i ! X n 6 P 2 ≤ pi λ − max i n pi 1≤ ≤ i=1 X since 1 1 Xj IE = IE z 1≤j≤n,j6=i dz Xj +1 1≤j≤n,j6=i Z0 P 1 P = [zpj + (1 − pj )]dz 0 j n,j i Z 1≤ Y≤ 6= 1 1 1 ≤ e−pj (1−z)dz = e−(λ−pj )(1−z)dz ≤ , λ − pj 0 j n,j i 0 Z 1≤ Y≤ 6= Z (see [27], pp. 167–168). Since d2(L(Ξ), Po(λ)) ≥ dT V (L(|Ξ|), Po(λ)) and the lower 1 n 2 bound of dT V (L(|Ξ|), Po(λ)) is of the same order as λ i=1 pi [7], the bound in (7) is of the optimal order. P 240 L. H. Y. Chen and A. Xia

3. From Palm theory to Stein’s method

Barbour’s probabilistic approach to Stein’s method is based on the conversion of a first order difference equation to a second order difference equation. In this section, we take another approach to Stein’s method from the point of Palm theory. The connection between Stein’s method and Palm theory has been known to many others (e.g., T. C. Brown (personnel communication), [9]) and the exposition here is mainly based on [14] and [27]. There are two properties which distinguish a Poisson process from other process- es: independent increments and the number of points on any bounded set follows . Hence, a Poisson process can be thought as a process pieced together by lots of independent “Poisson components” (if the location is an atom, the “component” will be a Poisson random variable, but if the location is diffuse, then the “component” is either 0 or 1) ([27], p. 121). Consequently, to specify a Poisson process N, it is sufficient to check that “each component” N(dα) is Poisson and independent of the others, that is IE{[IEN(dα)]g(N + δα) − N(dα)g(N)} = 0, which is equivalent to

IE[g(N)N(dα)] (8) = IEg(N + δ ), IEN(dα) α for all bounded function g on X and all α ∈ Γ (see [27], p. 121). To make the heuristic argument rigorous, one needs the tools of Campbell measures and Radon- Nikodym derivatives ([23], p. 83). In general, for each point process Ξ with finite mean measure λ, we may define the Campbell measure C(B,M) = IE[Ξ(B)1Ξ∈M ] for all Borel B ⊂ Γ, M ∈B(X ). This measure is finite and admits the following disintegration:

(9) C(B,M)= Qs(M)λ(ds), ZB or equivalently,

IE[Ξ(ds)1 ] Q (M)= Ξ∈M ,M ∈B(X ), s ∈ Γ λ a.s., s λ(ds) where {Qs, s ∈ Γ} are probability measures on B(X ) ([23], p. 83 and p. 164) and are called Palm distributions. Moreover, (9) is equivalent to that, for any measurable function f : Γ × X → R+,

(10) IE f(α, Ξ)Ξ(dα) = f(α, ξ)Qα(dξ)λ(dα) ZB  ZB ZX for all Borel set B ⊂ Γ. A point process Ξα (resp. Ξα − δα) on Γ is called a Palm process (resp. reduced Palm process) of Ξ at location α if it has the Palm distribution Qα and, when Ξ is a simple point process (a point process taking values 0 or 1 at each location), the Palm distribution L(Ξα) can be interpreted as the conditional distribution of Ξ given that there is a point of Ξ at α. It follows from (10) that the Palm process satisfies

IE f(α, Ξ)Ξ(dα) = IE f(α, Ξα)λ(dα) ZΓ ZΓ Poisson process approximation 241 for all bounded measurable functions f on Γ × X . In particular, Ξ is a Poisson process if and only if L(Ξα)= L(Ξ + δα), λ a.s. where the extra point δα is due to the “Poisson property” of Ξ{α}, and Ξα|Γ\{α} has the same distribution as Ξ|Γ\{α} because of independent increments. Here ξ|A stands for the point measure restricted to A ⊂ Γ ([23], p. 12). In other words, Ξ ∼ Po(λ) if and only if

IE f(α, Ξ+ δα)λ(dα) − f(α, Ξ)Ξ(dα) =0, ZΓ ZΓ  for a sufficiently rich class of functions f, so we define

Df(ξ): = f(x, ξ + δx)λ(dx) − f(x, ξ)ξ(dx). ZΓ ZΓ If IEDf(Ξ) ≈ 0 for an appropriate class of test functions f, then L(Ξα) is close to L(Ξ + δα), which means that L(Ξ) is close to Po(λ) under the metric or pseudo- metric specified by the class of test functions f. If fg is a solution of Df(ξ)= g(ξ) − Po(λ)(g), then a distance between L(Ξ) and Po(λ) is measured by |IEDfg(Ξ)| over the class of functions g. From above analysis, we can see that there are many possible solutions fg for a given function g. The one which admits an immigration-death process interpretation is by setting f(x, ξ)= h(ξ) − h(ξ − δx), so that Df takes the following form:

Df(ξ)= [h(ξ + δx) − h(ξ)]λ(dx)+ [h(ξ − δx) − h(ξ)]ξ(dx)= Ah(ξ), ZΓ ZΓ where A is the same as the generator defined in section 2.

4. Locally dependent point processes

We say a point process Ξ is locally dependent with neighborhoods {Aα ⊂ Γ: α ∈ Γ} c c λ if L(Ξ|Aα )= L(Ξα|Aα ), α ∈ Γ a.s. The following theorem is virtually from Corollary 3.6 in [14] combined with the new estimates of Stein’s factors in [27], Proposition 5.21. Theorem 4.1. If Ξ is a point process on Γ with finite intensity measure λ which has the total mass λ and locally dependent with neighborhoods {Aα ⊂ Γ: α ∈ Γ}. Then 3.5 2.5 ρ (L(Ξ), Po(λ)) ≤ IE + (Ξ(Aα) − 1)Ξ(dα) 2 λ |Ξ(α)| +1 Zα∈Γ   3.5 2.5 +IE + λ(dα)λ(dβ), λ (α) Zα∈Γ Zβ∈Aα |Ξβ | +1! α (α) ( ) c c where Ξ = Ξ|Aα and Ξβ = Ξβ|Aα . 1 n 2 Remark. The error bound is a “correct” generalization of λ i=1 pi with the Stein factor 1 replaced by a nonuniform bound. λ P 242 L. H. Y. Chen and A. Xia

5. Applications

5.1. Mat´ern hard core process on IRd

A Mat´ern hard core process Ξ on compact Γ ⊂ IRd is a model for particles with repulsive interaction. It assumes that points occur according to a Poisson process with uniform intensity measure on Γ. The configurations of Ξ are then obtained by deleting any point which is within distance r of another point, irrespective of whether the latter point has itself already been deleted [see Cox & Isham [17], p. 170]. The point process is locally dependent with neighborhoods {B(α, 2r): α ∈ Γ}, where B(α, s) is the ball centered at α with radius s. Let λ be the intensity measure of Ξ, d0(α, β) = min{|α − β|, 1}, then

µVol(B(0, 1))(2r)d d (L(Ξ), Po(λ)) = O , 2 Vol(Γ)   where µ is the mean of the total number of points of the original Poisson process (see [14], Theorem 5.1).

5.2. Palindromes in a genome

Let {Ii : 1 ≤ i ≤ n} be locally dependent Bernoulli random variables, {Ui : 1 ≤ i ≤ n} be independent Γ-valued random elements which are also independent of n {Ii : 1 ≤ i ≤ n}, set Ξ= i=1 IiδUi , then Ξ is a point process on Γ. For Ui = i/n this point process models palindromes in a genome where Ii represents whether a palindrome occurs at i/nP. The point process can also be used to describe the vertices in a . In general, the Ui’s could take the same value and one cannot tell which Ui and therefore which Ii contributes to the value. To overcome this difficulty we ′ n ′ lift the process up to a point process Ξ = i=1 Iiδ(i,Ui) on a larger space Γ = {1, 2,...,n} × Γ. The metric d0 becomes a pseudo-metric ρ0, that is, ρ0((i,s), ′ P (j, t)) = d0(s,t), and Ξ a locally dependent process (see [14], section 4). It turns n out that the Poisson process approximation of Ξ = i=1 IiδUi is a special case of the following section. P

5.3. Locally dependent superposition of point processes

Since the publication of the Grigelionis Theorem [20] which states that the super- position of independent sparse point processes on carrier space R+ is close to a Poisson process, there has been a lot of study on the weak convergence of point processes to a Poisson process under various conditions (see, e.g., [16, 19, 21] and [10]). Extensions to dependent superposition1 of sparse point processes have been carried out in [1, 2, 3, 11, 22]. Schuhmacher [25] considered the Wasserstein dis- tance between the weakly dependent superposition of sparse point processes and a Poisson process. Let Γ be a compact metric space, {Ξi : i ∈ I} be a collection of point processes λ on Γ with intensity measures i, i ∈ I. Define Ξ = i∈I Ξi with intensity measure 1We use “(resp. locally, weakly) dependent superpositionP of point processes” to mean that the point processes are (resp. locally, weakly) dependent among themselves. Poisson process approximation 243

λ λ = i∈I i. Assume {Ξi : i ∈ I} are locally dependent: that is, for each i ∈ I, there exists a neighbourhood Ai ⊂ I such that i ∈ Ai and Ξi is independent of P {Ξj : j 6∈ Ai}. n The locally dependent point process Ξ = i=1 IiδUi can be regarded as a locally dependent superposition of point processes defined above. P Theorem 5.1 ([15]). With the above setup, λ = λ(Γ), we have

3.5 2.5 ′ d (L(Ξ), Po(λ)) ≤ IE + d (Vi, Vi,α)λi(dα) 2 λ |Ξ(i)| +1 1 i Γ X∈I   Z 3.5 2.5 ′ + + IE IE d (Ξi, Ξi,α)λi(dα), λ |Ξ(i)| +1 1 i Γ X∈I   Z (i) Ξ , V = Ξ , Ξ is the reduced Palm process of Ξ where Ξ = j6∈Ai j i j∈Ai\{i} j i,α i at α, P P IE[Ξi(dα)1Vi∈M ] IP(Vi,α ∈ M)= for all M ∈B(X ) IEΞi(dα) and n ′ d1(ξ1, ξ2) = min d0(yi,zπ(i)) + (m − n) π : permutations of {1,...,m} i=1 n m X for ξ1 = i=1 δyi and ξ2 = i=1 δzi with m ≥ n [Brown & Xia [12]].

CorollaryP 5.2 ([14]). For Ξ=P i∈I IiδUi and λ = i∈I pi defined in section 5.2, P 3.5 P2.5 d2(L(Ξ), Po(λ)) ≤ IE + IiIj λ Vi +1 i X∈I j∈AXi\{i}   3.5 2.5 + + IE Ij =1 pipj , λ Vi +1 i∈I j∈Ai    X X where V = I . i j6∈Ai j CorollaryP 5.3 ([15]). Suppose that {Ξi : 1 ≤ i ≤ n} are independent renewal processes on [0,T ] with the first arrival time of Ξi having distribution Gi and its λ inter-arrival time having distribution Fi, and let Ξ= i∈I Ξi and be its intensity measure, then P n 2 6 i=1[2Fi(T )+ Gi(T )]Gi(T )/(1 − Fi(T )) d2(L(Ξ), Po(λ)) ≤ . n Gj (T ) Gi(T ) − maxj P i=1 1−Fj (T ) P References

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