Table Of Content‘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
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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
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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
