Poisson Limits for Pairwise and Area Interaction Point Processes Author(S): S
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Poisson Limits for Pairwise and Area Interaction Point Processes Author(s): S. Rao Jammalamadaka and Mathew D. Penrose Source: Advances in Applied Probability, Vol. 32, No. 1 (Mar., 2000), pp. 75-85 Published by: Applied Probability Trust Stable URL: http://www.jstor.org/stable/1428227 Accessed: 03/05/2009 10:27 Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at http://www.jstor.org/page/info/about/policies/terms.jsp. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, non-commercial use. Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at http://www.jstor.org/action/showPublisher?publisherCode=apt. Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. JSTOR is a not-for-profit organization founded in 1995 to build trusted digital archives for scholarship. We work with the scholarly community to preserve their work and the materials they rely upon, and to build a common research platform that promotes the discovery and use of these resources. For more information about JSTOR, please contact [email protected]. Applied Probability Trust is collaborating with JSTOR to digitize, preserve and extend access to Advances in Applied Probability. http://www.jstor.org Adv.Appl. Prob. (SGSA)32, 75-85 (2000) Printed in NorthernIreland ? AppliedProbability Trust 2000 POISSON LIMITS FOR PAIRWISE AND AREA INTERACTION POINT PROCESSES S. RAO JAMMALAMADAKA,*University of California MATHEWD. PENROSE,** Universityof Durham Abstract Suppose n particles xi in a region of the plane (possibly representing biological individuals such as trees or smaller organisms) have a joint density proportionalto - exp{- 2i<j 4(n(xi xj))}, with q a specified function of compact support. We obtain a Poisson process limit for the collection of rescaled interparticledistances as n becomes large. We give correspondingresults for the case of severaltypes of particles, representingdifferent species, and also for the area-interaction(Widom-Rowlinson) point process of interpenetratingspheres. Keywords: Area-interactionprocess; Gibbs distribution; limit laws; point process; spatial statistics;U-statistics AMS 1991 SubjectClassification: Primary 60G55 Secondary60F05; 60D05 1. Introduction Suppose the locations of n particles in d-dimensional space (such as biological individuals in a region of the plane or molecules in a region of space) are represented by points xi in Id, with d > 1. A simple stochastic model for their positions has them independently distributed with common probability density function f (for example, f could be a measure of the richness of the soil at a given point on the ground). We call this the null distribution. A more complex model is the pairwise interaction point process, also known as a form of Gibbs distribution. An energy function /)n : d -- I is specified and the null density is weighted multiplicatively, by exp(-n (xi - xj)) for each pair of distinct particles xi, xj, giving a joint density function of the form f ()(X ... Xn) = Z eXp- n(Xi - j) f(i[ ) (1.1) i<j<n i=1 with Zn a normalizing constant. For a discussion and bibliography on distributions of this form, see Diggle et al. [5], or Stoyan et al. [15, Section 5.5]. Here we consider only distributions which are conditioned to have n points. We consider the case where n(X) takes the form 0 (n2 x d), with ( a specified function of compact support. This means that as n becomes large the range of interaction becomes small in Received 27 July 1998; revision received 8 January1999. * Postal address:Department of Statisticsand Applied Probability,University of California,Santa Barbara, CA 93106, USA. Email address:[email protected]. ** Postal address:Department of MathematicalSciences, Universityof Durham,South Road, DurhamDH1 3LE, UK. Email address:[email protected]. 75 76 - SGSA S. R. JAMMALAMADAKAAND M. D. PENROSE such a way thatthe expectednumber of pairsof particleswhich interact,under the null distribu- tion, tends to a finitelimit. This correspondsto the 'sparseness'regime consideredby Saunders et al. in [12, 13], who obtained limit distributionsfor the numberof interactingpairs in the special case of a uniformnull distributionand with certainspecific step functionforms for 4. In this note we obtainlimiting distributionsfor the set of small interparticledistances, for a large class of null distributionsand functions 0 (Theorems3.1 and 3.2). We also considerthe case where there are severaltypes of particles(representing, for example, differentbiological species) with a different energy function for each pair of types of particles (see Theorem 4.1). Our proofs use Laplace functional methods and general results in the literatureon U- statistics. Proposition2.1 on generalizedGibbs distributionsand Proposition4.1 on multitype U-statistics are key steps in our proofs, and are also of some interestin their own right. In Section 5 we apply the same methodsto the area-interactionpoint process, conditioned to have n points. For our purposes,this is definedby a joint density of the form n f (n(xl, ...,n) c (exp-ynVrn(xl,x2,..., Xn)}) f(xi), (1.2) i=l where Vr(xl, ...., Xn)denotes the volume of the union of balls of radiusr centredat x ..., Xn, and Yn and rn are parameters. This form of density (for d = 2, hence the nomenclature) was proposed by Baddeley and van Lieshout [2] in the context of spatial statistics, having previously appearedin the physics literature[17]. We investigate a limiting regime for the parameters(Yn, rn) which is analogous to that consideredfor pairwise interactionprocesses; there turns out to be an asymptotic equivalence with a pairwise interactionprocess with a specific non-step functionform for 0. Both for pairwise and area interactions,the 'sparse' limits consideredare not particularly naturalfrom the statisticalphysics perspective,but are quite reasonablein the representation of the locations, for example, of a rareplant or nesting sites of a rarebird. 2. A general result We start by looking at generalized Gibbs distributions,by which we mean distributions for which some tractablenull measure,tn is weighted by a Radon-Nikodym derivativepro- portional to e-U, where U is the value of an 'energy function' summed over the points of an induced point process. More formally, a generalized Gibbs distributionis a probability measure ,' definedby (2.1) below. We shall give a simple result (Proposition2.1) about gen- eralized Gibbs distributionswhich may by applicablein other settings than those considered here. See for example [16] for a varietyof possible applicationsof models of this form. Let S be a complete separablemetric space, and let B denote the Borel a-field on S. Let A be the space of locally finite countingmeasures on (S, B). For M E MAand g: S -> R we use the inner productnotation (M, g) for the integralfs g dM. Clearly M is identifiedwith a set of points and (M, g) is the sum of the values of g at those points. A pointprocess on S is a randomelement 4 of M. If X is a locally finite measureon (S, SB), a Poisson process with mean measure X is a point process for which the random variables ( , IAi ) are independent Poisson variables with respective means X(Ai) for any finite collection of disjoint sets Ai E S with X(Ai) < oo. In terms of Laplace functionals,Poisson processes are characterizedas follows (see [11, Proposition3.6]): Lemma 2.1. Let 4 be a pointprocess on S and let Xbe a locallyfinite measureon (S, S). Then t is a Poissonprocess with meanmeasure X, if and only iffor all measurableg: S -- (-oo, oo] boundedbelow and of compactsupport, Poissonlimits for pointprocesses SGSA? 77 Eexp(- (, g)) = exp (e-(x) - l)(dx). Here and throughoutthis paper,we use the conventionexp(-oo) = 0. Our general result is in the following setting. For n E N, let (On, 7n, /n) be probability spaces and let n : Qn -X AMbe measurablefor n = 1, 2, 3,.... Suppose :? S - (-oo, oo] is a measurable 'energy function' of compact support. The associated generalized Gibbs measuresare probabilitymeasures /1 on (On, 7n) definedby /4(dw) = Zn1exp(-(n()(w), 0)),dn(dw), (2.1) with zn a normalizingconstant. We write AOfor the set of discontinuitypoints of 4. We are interestedin the asymptoticdistributions of tthe unrunderhe measures/n and 'n. For A C S write AA for the boundaryof A. If 4 and 4n (n E N) are point processes on S, we say the sequence 4n converges weakly to 4, and write n =i 4, if the finite-dimensional distributionsconverge, i.e. if for any finite collection of bounded Borel sets Ai satisfying 4(oAi) = 0 almost surely,the joint distributionsof n(Ai) convergeweakly to those of 4(Ai). This is equivalentto other definitions of weak convergence;see e.g. [4, Section 9.1]. Weak convergenceof point processes is characterizedin terms of Laplacefunctionals as follows [11, Proposition3.19]: Lemma 2.2. Let4 and 4n (n = 1, 2, 3, ...) be point processes on S. Then n => 4 if and only if E exp(- (n, g)) -> Eexp(- (, g)) for all measurableg : S -> [0, oo] of compactsupport with (4, lAg) = 0 almost surely. The mainresult of this section says thatif underiLn the point processes4n are asymptotically Poisson, then under the generalized Gibbs distribution1/4 given by (2.1), they are again asymptoticallyPoisson, with mean measureinflated locally by exp(-0(x)): Proposition 2.1. Suppose p is non-negative. Suppose that under An the sequence of point processes 4nconverges weakly to a Poisson process with locally finite mean measureX satis- fying X\(A() = 0.