Probability_theory_-_An_Analytic_View.pdf

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1、ChapterISumsofIndependentRandomVariablesInonewayoranother,mostprobabilisticanalysisentailsthestudyoflargefamiliesofrandomvariables.Thekeytosuchanalysisisanunderstandingoftherelationsamongthefamilymembers;andofallthepossiblewaysinwhichmembersofafamilycanberelated,byfarthesimplestiswh

2、entherelationshipdoesnotexistatall!Forthisreason,wewillbeginbylookingatfamiliesofindependentrandomvariables.x1.1IndependenceInthissectionwewillintroduceKolmogorov'swayofdescribingindependenceandproveafewofitsconsequences.x1.1.1.IndependentSigmaAlgebras.Let(;F;P)beaprobabilityspace(i

3、.e.,isanonemptyset,Fisa-algebraover,andPisamea-sureonthemeasurablespace(;F)havingtotalmass1);and,foreachifromthe(nonempty)indexsetI,letFibeasub-algebraofF.Wesaythatthe-algebrasFi;i2I,aremutuallyP-independentor,lessprecisely,P-independent,if,foreverynitesubsetfi1;:::;ingofdistinc

4、telementsofIandeverychoiceofAim2Fim;1mn,


(1.1.1)PAi1


Ain=PAi1


PAin:Inparticular,iffAi:i2IgisafamilyofsetsfromF,wesaythatAi;i2I;areP-independentiftheassociated-algebrasFi=f;;Ai;Ai{;g;i2I;are.Togainanappreciationfortheintuitiononwhichthisdenitionisbased,itisimportantton

5、oticethatindependenceofthepairA1andA2inthepresentsenseisequivalentto


PA1A2=PA1PA2;theclassicaldenitionwhichoneencountersinelementarytreatments.Thus,thenotionofindependencejustintroducedisnomorethanasimplegeneralizationoftheclassicalnotionofindependentpairsofsetsencounteredinn

6、on-measuretheoreticpresentations;andtherefore,theintuitionwhichunderliestheelemen-tarynotionappliesequallywelltothedenitiongivenhere.(SeeExercise1.1.8belowformoreinformationabouttheconnectionbetweenthepresentdenitionandtheclassicalone.)12ISumsofIndependentRandomVariablesAswillbeco

7、meincreasingevidentasweproceed,innitefamiliesofindepen-dentobjectspossesssurprisingandbeautifulproperties.Inparticular,mutuallyindependent-algebrastendtollupspaceinasensewhichismadeprecisebythefollowingbeautifulthoughtexperimentdesignedbyA.N.Kolmogorov.LetIbeanyindexset,takeF;=f;

8、;g,andforeachnonemp