Probability Theory: STAT310/MATH230; September 12, 2010

1 Probability theory : STAT310/MATH230; September 12, Mathematics,StanfordUniversity,Stanford, , spaces,measuresand (mathematical) :thelaw of of , ,likelihood , , andStrongMarkov , :regularity, localmaximaandlevel sets369 Bibliography377310a:HomeworkSets2007379 PrefaceThesearethelecturenotesfora yearlong,PhDlevel coursein Probability TheorythatI taught atStanfordUniversity in 2004, thiscourseis to prepareincomingPhDstudents in Stanford smathematicsandstatisticsdepartments to do research in Probability theory . Morebroadly, thegoalof thetextis to helpthereadermasterthemathematicalfounda tionsof Probability theoryandthetechniquesmostcommonlyusedin provingtheoremsin therigorousstudyof themostfundamentalclassesof ,we introducein Chapter1 therelevant elements frommeasureandintegrationtheory, namely, theprobability spaceandthe -algebrasof eventsin it,randomvariablesviewedas measurablefunctions,theirexpectationas thecorrespondingLebesgueintegral,andthei mportant conceptof.

2 We studyin Chapter2 thevariousnotionsof convergenceof randomvariablesandderive theweakandstronglawsof is devotedtothetheoryof weakconvergence,therelatedconceptsof distributionandcharacteristicfunctionsan dtwo important specialcases:theCentralLimitTheorem(insh ortclt) Chapter1, we devoteChapter4 to thedefinition,existenceandpropertiesof theconditionalexpectationandtheassociate dregularconditionalprobability dealswithfiltrations,themathematicalnoti onof informationprogres-sionin time, a by productof thestudyof a collectionof ,aswellas maximalinequalities, ,we focushereon thediscretetimesettingsdeferringtheconti nuoustimecounterpartto providesa briefintroductionto thetheoryof Markov chains,a vastsubjectat thecoreof Probability theory .

3 To which many illustratesomeof theinterestingmathematicalpropertiesof such processesbyexaminingfewspecialcasesof setstheframeworkforstudyingright-continu ousstochasticprocessesindexedby a continuoustimeparameter,introducesthefam ilyof Gaussianpro-cessesandrigorouslyconstruct stheBrownianmotionasa Gaussianprocessofcontinuoussamplepathand zero-mean,stationaryindependent expandsourearliertreatment of martingalesandstrongMarkov pro-cessesto thecontinuoustimesetting,emphasizingther oleof such processesis thenillustratedbothinthecontextof Brownianmotionandthatof Markov ,in Chapter9 we re-constructtheBrownianmotionviathein-va rianceprincipleas thelimitof furtherdelveinto therich propertiesof itssamplepathandthemany applicationsof Brownianmotionto thecltandtheLaw of theIteratedLogarithm(inshort,lil).

4 Theintendedaudienceforthiscourseshouldha ve priorexposureto stochasticprocesses,at areassumedto have takena realanalysisclassdealingwithRiemanninteg ration,andmasteredwellthismaterial,prior knowledgeof measuretheoryis is quiteclearthatthesenotesaremuch influencedby thetextbooks[Bil95,Dur03, Wil91, KaS97] I have thankmy students outof whoseworkthistextmaterializedandmy teachingas-sistants SuChen,KshitijKhare,GuoqiangHu,JuliaSalz man,KevinSunandHuaZhoufortheirhelpin theassemblyof thenotesof morethaneighty students intoa coherent document. I amalsomuch indebtedto KevinRoss,AndreaMontanariandOanaMocioalc afortheirfeedback onearlierdraftsof thesenotes,to KevinRossforprovidingallthefiguresin thistext,andtoAndreaMontanari,DavidSiegm undandTzeLaiforcontributingsomeof theexercisesin ,CaliforniaApril2010 CHAPTER1 Probability , measureand integrationThischapteris devotedto themathematicalfoundationsof Probability ,namely, theprobabilityspaceandthe -algebrasof events in ,introducedin measurablefunctions 7 X( )

5 To definein conceptof expectationas thecorrespondingLebesgueintegral,extendi ngthehorizonof ourdiscussionbeyondthespecialfunctionsan dvariableswithdensity to which elementaryprobabilitytheoryis consideringindependence,themostfundament alaspectthatdifferentiatesprobability from(general)measuretheory, spaces,measuresand -algebrasWe shalldefineheretheprobability space( ,F,P) usingtheterminologyof is a setof allpossibleoutcomes of somerandomexper-iment. ProbabilitiesareassignedbyA7 P(A) toAin a subsetFof allpossiblesetsof boththeamount of informationavailableas a resultof theexperiment conductedandthecollectionof allevents ofpossibleinterestto pleasant mathematicalframeworkresultsby imposingonFthestructuralconditionsof a -algebra,as donein Themostcommonandusefulchoicesforthis -algebraarethenexploredin frommeasuretheory, namelyDynkin sandCarath eodory stheoremsandtheirapplicationto space( ,F, P).

6 We use2 to denotethesetof allpossiblesubsetsof .Theevent spaceis thusa subsetFof 2 , consistingof allallowed events,thatis, thoseevents to which we 2 is a -algebra(ora -field), if(a) F,(b)IfA FthenAc Fas well (whereAc= \A).(c)IfAi Ffori= 1,2,3,..thenalso iAi s law, we know that( iAci)c= iAi. Thusthefollowingis equivalent to property (c)of :(c )IfAi Ffori= 1,2,3,..thenalso iAi Probability , pair( ,F)withFa -algebra of subsetsof is called ameasurablespace. Givena measurablespace( ,F), ameasure is anycountablyadditivenon-negativesetfunct iononthisspace. Thatis, :F [0, ], havingtheproperties:(a) (A) ( ) = 0forallA F.

7 (b) ( nAn) = n (An)foranycountablecollectionof disjointsetsAn addition ( )= 1, wecall themeasure aprobability measure, andoftenlabel it byP(it is alsoeasyto see thatthenP(A) 1forallA F). (b)of relaxedto involve onlyfinitecollectionsof disjoint setsAn, we say that is afinitelyadditivenon-negative measuretheorywe sometimesconsidersigned measures, whereby is nolongernon-negative, henceitsrangeis [ , ], andsay thatsuch measureisfinitewhenitsrangeisR( assignedaninfinitemeasure). a triplet( ,F, ), with a measure on themeasurablespace( ,F). A measure space ( ,F,P) withPa probabilitymeasureis called aprobability thefundamentalpropertiessharedby allprob-ability ( ,F,P)be a probabilityspace andA,B, everyprobabilitymeasure.

8 (a) BthenP(A) P(B).(b) iAithenP(A) iP(Ai).(c) Continuity frombelow:IfAi A, thatis,A1 A2 ..and iAi=A,thenP(Ai) P(A).(d)Continuity fromabove:IfAi A, thatis,A1 A2 ..and iAi=A,thenP(Ai) P(A). measuretheory, notethatproperties(a)-(c)of measure , whereasthecontinuity fromaboveholdswhenever (Ai)< finitelyadditivenon-negativesetfunction onameasurablespace( ,F)withthe continuity propertyBn F, Bn , (Bn)< = (Bn) 0mustbe countablyadditiveif ( )< . Givean examplethatit is notnecessarilyso when ( )= .The -algebraFalways containsat leasttheset anditscomplement, theemptyset . Necessarily,P( )= 1 andP( ) = ,if we takeF0={ , }as our -algebra,thenwe areleftwithnodegreesof freedomin choiceofP.

9 For thisreasonwe callF0thetrivial -algebra. Fixing ,we may expectthatthelargerthe -algebrawe consider,themorefreedomwe have in choosingtheprobability someextent, thatis, as longas we have noproblemsatisfyingtherequirements in thedefinitionof a Probability naturalquestioniswhenshouldwe expectthemaximalpossible -algebraF= 2 to be useful? is countablewecan andtypically shalltakeF= 2 . Indeed, in suchsituationswe assigna probabilityp >0to each ,MEASURESAND -ALGEBRAS9makingsure that p = 1. Then,it is easyto see thattakingP(A) = Ap foranyA resultswitha probabilitymeasure on( ,2 ).

10 For instance, when is finite,wecantakep =1| |, theuniformmeasure on , wherebycomputingprobabilitiesis thesameas a singlecointoss,forwhichwehave 1={H,T}( = Hif thecoinlandsonitshead and = Tif itlandsonitstail),andF1={ , ,H,T}, or whenweconsidera finitenumber ofcointosses,sayn, in whichcase n={( 1,.., n) : i {H,T},i= 1,..,n}is thesetof all possiblen-tuplesof cointosses,whileFn= 2 nis thecollectionof all possiblesetsofn-tuplesof thesetof all non-negativeintegers ={0,1,2,..}andF= 2 , where wegetthePoissonprobabilitymeasure of parameter >0whenstartingfrompk= kk!e fork= 0,1,2,..When is uncountablesuch a strategyas in thatif we takep =P({ })>0 foruncountablymany valuesof , we shallendupwithP( )=.