paul gerhard hoel - introduction to stochastic processes (the houghton mifflin series in statistics)...

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Hoel Por Stone Iintroduction toStochastic Processes , Th.HoughtonMifinSeriesinStatistics undlertheEitorshipof HerlmanChernof LEOBREIMAN ProbabilityandStochasticJProcesses:WithaViewTowardApplications Statistics:WithaViewTowardApplications PAULG.HOEL,SIDNEYC.PORT,ANDCHARLESJ. STONE IntroductiontoProbabilityTheory IntroductiontoStatisticalT'eory IntroductiontoStochastic}rocesses PAULF.LAZARSFELDANDN1EIL. HENRY LatentStructureAnalysis G01IFRIEDE.NOETHER IntroductiontoStatistics-.AFreshApproach 3. b. CHO,HERBERTROBBns,W DAVSmGMUI) GreatExpectations:TheTheoryof OptimalStopping I.RICHARDSA NAGE Statistics:Uncertaintyand.Behavior nlDuCOD DbDCH5ClDC055C5Paul G.Hoel SidneyC.Port CharlesJ.Stone UniversityofCalifornia,LosAngeles HOUGHTONMIFFLIINCOMPANYBOSTON NewYorkAtlantaGeneva,IllinoisDaillasPaloAlto LOEYKLH1 l 972 UY HOLLH1ON MllLN LOMEANY.Al rightsresered.No Jpartofthisworkmaybt!reproducedortransmittedin any formorby anymeans,electronicormechanical,includingphotocopying andrecording,orbyany informationstorageorretrievalsystem,wlthout Jpermissioninwritingfromthepublisher. PRINTEDINTHEU.S.A. LmRARYOFCONGRESSCATALOCARDNUMBER:79-165035 ISBN:0-395-1 2076-4 General Preface Thisthree-volumeseriesgre'Noutofathree-quartercourseinproba.bility, statistics,andstochastic process(!staughtfor anumberof yearsatUCLA. VVefelt aneedforaseriesof booksthatwouldtreatthesesubjectsinaway thatiswell coordinate: d, but which would also give adequate emphasis to each subject as being interestingandusefulon itsownmerits. Thefrstvolume,oc|o o iooco|/|y I/eoy, presentsthefundarnental ideasofprobabilitytheoryandalsopreparesthestudentbothforcoursesin statistics and for further study in probability theory, including stochastic pro(;esses.Thesecondvolume,oc|o o se|s|ce/ Deoy, evelopsthebasic theoryofmathematicalstatisticsinasystematic,unifedmanner.Togethe:r,the frst two volumes contain the material that is often covered in a two-semester course inmathemlaticalstatistics. Thethirdvolume,Ioc|o o soc/es|c iocesses, treatsMarkovchains, Poissonprocesses,birthanddeathprocesses,Gaussianprocesses,Bro'wnian motion,andprocessesdefnedintermsofBrownianmotionbymeansofelementary stochastic diferentialequations.VPreface Inrecentyearstherehas been aneverincreasinginterestinthestudyofsystemswhichvaqtintimeinarandomnanner. Mathematica!mode|sofsuchsyste|sareknownas stochasticrocesses.Inthis bookwe resentane|ementaryaccountofsome ofthe important toics in thetheory ofsuch rocesses. We have tried tose|ect toics that are concetua||y interesting and that have found f|uitfu|a|icationin variousbranches ofscience and techno|ogy.A soc/:u|cocesscanbedehnedquitegenera||y as anyco||ectionofrandomvanab|es.:(/), EI, dehnedonacommonrobabi|itysace,where Iisasubsetof( 0 co)andisthoughtofasthetimearameter set. The rocessisca||edaco/|oseeeeocessif1'isaninterva|havingositive|engthanda/sce/eeeeeocessifIis asubset ofthe integers. Ifthe randomvariab|esx()a||take onva| uesfromthehxed set f,thenf isca||edthes/eeseceoftherocess.Many si.ochastic rocesses oftheoretica| and a|ied interest ossess th: ro-enythat, given the resent state ofthe rocess, the ast history does not adectconditiona| robabi|ities ofevents dehned in terms ofthe future. Suchrocessesareca||edMeko:ocesses. InChaters | andzwe studymeko:c/e|s,whichare discretearameter Markov processes whose state sace is hnite orcountab|yinhnite. InChater+westudythecorresondingcontinuousarameterrocesses,withthe"l'oissonrocess"as asecia|case.InChaters+wediscusscontinuousarameterrocesseswhosestatesace istyica||y the rea| |ine. In Chater 4we introduce 6ess|eocesses, which arecharacterized by the roerty that every |inear con.bination i nvo|ving a hnitenumber of the random variab|es x(/),/EI, is norma||y distributed. t\s animortant secia| case, we discuss the 0|eeocess, which arises as a n:athe-matica|mode| forthehysica|henomenon known as "Brownianmotion. "In Chater we discuss integ:ationand dierentiation ofstochastic rocesses. There we a|so usethe Wiener rocess to give a mathematica| mode| for 'whitenoise. "In Chater we discuss so|utions to nonhomogeneous ordinary dierentia|equations havingconstantcoemcients whose right-hand side iseither a stochasticrocess orwhite noise. We a|so discussestimationrob|ems invo|ving stochasticrocesses,andbrieyconsiderthe"sectra|distnbution"ofarocess.vv 70f900This text has been designed for a one-semester courseinstochasticrocesses.Written in c|ose conjunction v-ith a/oc|o /o ooeo4|/y I/eoy, the hrstvo|ume ofourthree-vo|ume series, itassumesthatth e studentisacquaintedwiththe materia| covered in aone-s:mestercourseinrobabi|ityforwhiche|ementaryca|cu|us is arerequisite.Some ofthe roofs in Chat:rs | and zare somevhat more dimcu|t than therestofthetext, andtheyaearinaendicestothesechaters. Theseroofsandthestarred materia| in Sectionz. robab|yshou|d b| omitted or discussed on|ybriey in an e|ementary courseAninstructorusingthis textinaone-quartercoursewi||robab|ynothavetimeto cover the entire text. He may wishto cover the hrst three chaters thorough|yandtheremainderastimeermits,erhasdiscussingthosetoicsinthe|astthreechaters that invo|ve the Wienerrocess. Anotherotion, however, isoe|ha-size continuous arameter roc:sses by omitting or skimming Chaters | and zand concentrating on Chaters 1-. (For exam|e , the instructor cou|d skiSections | . . | , | . . z, | . , z. z. z, z. . | , z. . | , and z. s)With some aid from theinstructor, the student shou|d be ab|e to read Chat|r 1 without having studiedthe hrst tvo chaters thorough|y. Chaters +are indeendent ofthe hrst twochatersanddeendonCater1on|y inminorways, main|yinthatthePoissonprocess introduced in Chater+is used in exam|es in the |ater chaters. Theroerties ofthe Poissoniocessthatareneeded|aterare summarizedinChater4andcanberegardedas axiomsforthePoissonproc:ss.The authors wish to thank the UCLA students who to|erated re|in:inaryversions of this text and whose comments resu|ted in numerous imrovernents.Mr. Luis0orostiza obtained the answers to the exercises and a|so made manysuggestions that resu|ted in signihcant imrovements. Fina||y, we wish to thankMrs.RuthGo|dsteinforherexce||enttying.T able of Contents 1MlarkovChains l1.1Markovchains havingtwo states 21.2Transition functionandinitia|distribution 51.3Exam|es 61.4Comutations with transitionfunctions l 21.4.1Hittingtimes l 41.4.2Transitionmatrix l 6 1.5;Transientandrecurrentstates l 7 1.6;Decomositionofthe state sace 2l 1.6.1Absortion robabi|ities 251.6.2Martinga|es 271.7'Birth and death chains 291.8:Branchingand queuingchains 331.8.1Branchingchain 341.8.2Queuingchain 36Aendix1.91Proofofresu|ts for the branchingandqueuingchains 361.9.1Branchingchain 3&1.9.2Queuingchain 392St.ltionaryDistribut:ions of a MarkovChain 472.1E|ementaryroerties ofstationary distributions 472.2,Exam|es 492.2.1Birth and deathchain 502.2.2Partic|es in abox 532.3Average number ofvisits to arecurrent state 562.4Nu|| recurrent and ositive recurrent states 602.5;Existence and uniqueness ofstationaq/ distributions 632.5.1Reducib|echains 672.6;Queuingchain 692.6.1Proof 70XX 8D0 0f0.0t0Dt82.'7Convergenceto the stationary distribution 72Aendix2.: 8Proofofconvergence 752.S.1Periodiccase 772.8.2A resu|tfromnumbertheory 793NlarkovPureJum.Processes &43.1Construction ofjumrocesses &43.2Birth and deathrocesses &93.2.1Two-statebirthanddeathrocess 923.2.2Poissonrocess 943.2.3Pure birthrocess 9&3.2.4Inhnite serer queue 993.:3Proerties ofa Markovurejumrocess l 023.3.1A|icationsto birth and death rocesses l 044S.condOrderProcesses l l l 4.1Meanandcovariancefunctions l l l 4.:2Gaussian rocesses l l 94..3The Wiener rocess l 225C,ontinuity,Integrltion, andDiferelntiationofSecon,d OrderProcessesl 2&5.1Continuity assumtionsl 2&5.1.1Continuity ofthe mean andcovariance functions l 2&5.1.2Continuity ofthe sam|e functions l 305.:2Integrationl 325.:3Dierentiation l 355.4White noise l4l6St:ochasticDiferelltialEquations,I:stimationTheor", alldSpectralDistriibutionsl 526.1First orderditrentia|equations l 546.:ZDierentia|equations ofordernl 596.2.1The case n=2 l 666.3 6.4 Estimationtheory6.3.1Genera|rinci|es ofestimation6.3.2Some exam|es ofotima|redictionSectra| distributionAnswers to ExercisesG|ossary ofNotationIndexl 70l 73l 74l 77l 90l 9920l1 MarkovChains (onsider asystemthat can beinany one of a fniteor countably infnite number of states. Letfdenote this set of states.We can assumethatfis a subset of the integers.Theset!iscalledthestatespaceofthesystem,.Letthesystembe observed at the discrete moments of time n=0,1, 2,. . . , and let Xn denote the state of thesystemattinlen. Sinceweareinterestedinnon-deterministicsystems,wethinkof Xmn>0,as random variables defned on a common probability space.Little can besaid about such random variablesunlessSOlneadditionalstructureis imposeduponthem.1rhesimplestpossiblestructureisthatof independentrandomvariables. This wouldbeagoodmodelforsuchsystemsasrepeatedexperimLentsin whichfuture statesof thesystemareindependentof pastandpresentstates. Inmostsystems that arise in practice,however,pastand presentstatesof thesysteminfuence the future states even if they do not uniquely determine them. vanysystemshave the propertythatgiventhe present state,thepaststateshave noinfuenceonthefuture.ThispropertyiscalledtheMarkovproperty,and systemshavingthispropertyar(calledMarkovchains.TheMarkovpropertyis definedpreciselyby therequirenlentthat for every choiceof thenonnegativeinteger1andthenumbers Xo,. . . ,xn+ 1,each in C.The conditional probabilities P(Xn+ 1=YIXn=x) arc called the transition probabilitiesofthechain.InthisbookwewillstudyMarkovchainshaving stationarytransitionprobabilities,i . e. , thosesuchthatP(Xn + 1=YIXn=x)is independentof n.Fromnow011,whenwesaythatXmn>0,formsaMarkov chain, we mean that these rando]t variablessatisfy theMarkov property andhave sta1tionary transition probabilities. lrhestudyofsuchMarkovchainsisworthwhilefromtwoviewpoints.First, theyhavearichtheory,muchof whichcanbepresentedatanelementar

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