Lectures on the Poisson Process PDF
Preview Lectures on the Poisson Process
‘An understanding of the remarkable properties of the Poisson process is essentialforanyoneinterestedinthemathematicaltheoryofprobabilityor initsmanyfieldsofapplication.Thisbookisalucidandthoroughaccount, rigorousbutnotpedantic,andaccessibletoanyreaderfamiliarwithmod- ernmathematicsatfirst-degreelevel.Itspublicationismostwelcome.’ —J.F.C.Kingman,UniversityofBristol ‘IhavealwaysconsideredthePoissonprocesstobeacornerstoneofapplied probability.Thisexcellentbookdemonstratesthatitisawholeworldinand ofitself.Thetextisexcitingandindispensabletoanyonewhoworksinthis field.’ —DietrichStoyan,TUBergakademieFreiberg ‘LastandPenrose’sLecturesonthePoissonProcessconstitutesasplendid addition to the monograph literature on point processes. While emphasis- ing the Poisson and related processes, their mathematical approach also covers the basic theory of random measures and various applications, especially to stochastic geometry. They assume a sound grounding in measure-theoretic probability, which is well summarised in two appen- dices (on measure and probability theory). Abundant exercises conclude eachofthetwenty-two“lectures”whichincludeexamplesillustratingtheir “course” material. It is a first-class complement to John Kingman’s essay onthePoissonprocess.’ —DarylDaley,UniversityofMelbourne ‘Pick n points uniformly and independently in a cube of volume n in Euclidean space. The limit of these random configurations as n → ∞ is the Poisson process. This book, written by two of the foremost experts onpointprocesses,givesamasterfuloverviewofthePoissonprocessand someofitsrelatives.Classicaltenetsofthetheory,likethinningproperties and Campbell’s formula, are followed by modern developments, such as Liggett’sextraheadstheorem,Fockspace,permanentalprocessesandthe Boolean model. Numerous exercises throughout the book challenge read- ersandbringthemtotheedgeofcurrenttheory.’ —YuvalPeres,PrincipalResearcher,MicrosoftResearch, andForeignAssociate,NationalAcademyofSciences LecturesonthePoissonProcess ThePoissonprocess,acoreobjectinmodernprobability,enjoysarichertheorythanis sometimesappreciated.Thisvolumedevelopsthetheoryinthesettingofageneral abstractmeasurespace,establishingbasicresultsandpropertiesaswellascertain advancedtopicsinthestochasticanalysisofthePoissonprocess.Alsodiscussedare applicationsandrelatedtopicsinstochasticgeometry,includingstationarypoint processes,theBooleanmodel,theGilbertgraph,stableallocationsandhyperplane processes.Comprehensive,rigorous,andself-contained,thistextisidealforgraduate coursesorforself-study,withasubstantialnumberofexercisesforeachchapter. Mathematicalprerequisites,mainlyasoundknowledgeofmeasure-theoretic probability,arekeptinthebackground,butarereviewedcomprehensivelyinan appendix.Theauthorsarewell-knownresearchersinprobabilitytheory,especially stochasticgeometry.Theirapproachisinformedbothbytheirresearchandbytheir extensiveexperienceinteachingatundergraduateandgraduatelevels. GÜNTER LASTisProfessorofStochasticsattheKarlsruheInstituteofTechnology. Heisadistinguishedprobabilistwithparticularexpertiseinstochasticgeometry,point processesandrandommeasures.Hehascoauthoredaresearchmonographonmarked pointprocessesonthelineaswellastwotextbooksongeneralmathematics.Hehas givenmanyinvitedtalksonhisresearchworldwide. MATHEW PENROSEisProfessorofProbabilityattheUniversityofBath.Heisan internationallyleadingresearcherinstochasticgeometryandappliedprobabilityand istheauthoroftheinfluentialmonographRandomGeometricGraphs.Hereceivedthe FriedrichWilhelmBesselResearchAwardfromtheHumboldtFoundationin2008, andhasheldvisitingpositionsasguestlecturerinNewDelhi,Karlsruhe,SanDiego, BirminghamandLille. INSTITUTE OF MATHEMATICAL STATISTICS TEXTBOOKS EditorialBoard D.R.Cox(UniversityofOxford) B.Hambly(UniversityofOxford) S.Holmes(StanfordUniversity) J.Wellner(UniversityofWashington) IMSTextbooksgiveintroductoryaccountsoftopicsofcurrentconcernsuitablefor advancedcoursesatmaster’slevel,fordoctoralstudentsandforindividualstudy.They aretypicallyshorterthanafullydevelopedtextbook,oftenarisingfrommaterial createdforatopicalcourse.Lengthsof100–290pagesareenvisaged.Thebooks typicallycontainexercises. OtherBooksintheSeries 1. ProbabilityonGraphs,byGeoffreyGrimmett 2. StochasticNetworks,byFrankKellyandElenaYudovina 3. BayesianFilteringandSmoothing,bySimoSärkkä 4. TheSurprisingMathematicsofLongestIncreasingSubsequences,byDanRomik 5. NoiseSensitivityofBooleanFunctionsandPercolation,byChristopheGarbanand JeffreyE.Steif 6. CoreStatistics,bySimonN.Wood 7. LecturesonthePoissonProcess,byGünterLastandMathewPenrose Lectures on the Poisson Process GÜNTER LAST KarlsruheInstituteofTechnology MATHEW PENROSE UniversityofBath UniversityPrintingHouse,CambridgeCB28BS,UnitedKingdom OneLibertyPlaza,20thFloor,NewYork,NY10006,USA 477WilliamstownRoad,PortMelbourne,VIC3207,Australia 314–321, 3rd Floor, Plot 3, Splendor Forum, Jasola District Centre, New Delhi – 110025, India 79AnsonRoad,#06–04/06,Singapore079906 CambridgeUniversityPressispartoftheUniversityofCambridge. ItfurtherstheUniversity’smissionbydisseminatingknowledgeinthepursuitof education,learning,andresearchatthehighestinternationallevelsofexcellence. www.cambridge.org Informationonthistitle:www.cambridge.org/9781107088016 DOI:10.1017/9781316104477 (cid:4)c GünterLastandMathewPenrose2018 Thispublicationisincopyright.Subjecttostatutoryexception andtotheprovisionsofrelevantcollectivelicensingagreements, noreproductionofanypartmaytakeplacewithoutthewritten permissionofCambridgeUniversityPress. Firstpublished2018 PrintedintheUnitedStatesofAmericabySheridanBooks,Inc. AcataloguerecordforthispublicationisavailablefromtheBritishLibrary. LibraryofCongressCataloging-in-PublicationData Names:Last,Günter,author.|Penrose,Mathew,author. Title:LecturesonthePoissonprocess/GünterLast,KarlsruheInstituteof Technology,MathewPenrose,UniversityofBath. Description:Cambridge:CambridgeUniversityPress,2018.|Series: InstituteofMathematicalStatisticstextbooks|Includesbibliographical referencesandindex. Identifiers:LCCN2017027687|ISBN9781107088016 Subjects:LCSH:Poissonprocesses.|Stochasticprocesses.|Probabilities. Classification:LCCQA274.42.L362018|DDC519.2/4–dc23 LCrecordavailableathttps://lccn.loc.gov/2017027687 ISBN978-1-107-08801-6Hardback ISBN978-1-107-45843-7Paperback CambridgeUniversityPresshasnoresponsibilityforthepersistenceoraccuracy ofURLsforexternalorthird-partyinternetwebsitesreferredtointhispublication anddoesnotguaranteethatanycontentonsuchwebsitesis,orwillremain, accurateorappropriate. ToOurFamilies Contents Preface pagexv ListofSymbols xix 1 PoissonandOtherDiscreteDistributions 1 1.1 ThePoissonDistribution 1 1.2 RelationshipsBetweenPoissonandBinomialDistributions 3 1.3 ThePoissonLimitTheorem 4 1.4 TheNegativeBinomialDistribution 5 1.5 Exercises 7 2 PointProcesses 9 2.1 Fundamentals 9 2.2 Campbell’sFormula 12 2.3 DistributionofaPointProcess 14 2.4 PointProcessesonMetricSpaces 16 2.5 Exercises 18 3 PoissonProcesses 19 3.1 DefinitionofthePoissonProcess 19 3.2 ExistenceofPoissonProcesses 20 3.3 LaplaceFunctionalofthePoissonProcess 23 3.4 Exercises 24 4 TheMeckeEquationandFactorialMeasures 26 4.1 TheMeckeEquation 26 4.2 FactorialMeasuresandtheMultivariateMeckeEquation 28 4.3 JanossyMeasures 32 4.4 FactorialMomentMeasures 34 4.5 Exercises 36 ix
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