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1、SequentialEstimationbyMalayGhosh,NitisMukhopadhyayandPranabK.SenCopyright©1997JohnWiley&Sons,Inc.Chapter2ProbabilisticResultsinSequentialAnalysis2.1INTRODUCTIONThischaptercontainsseveralusefulprobabilityresultsthatareusedrepeat-edlyinthelaterchapters.Westartwiththedefinitionandcertainusefulpro
2、pertiesofdiscretetimemartingalesinSection2.2.Section2.3introducesstoppingtimesandtheirrelationshipwithmartingalesandcontainstheop-tionalstoppingtheorem.SomeimportantmartingaleinequalitiesandWald'slemmasaregiveninSection2.4.ThissectionalsocontainstheFundamentalIdentityofSequentialAnalysis.Doob'
3、ssubmartingaleconvergencetheoremisgiveninSection2.5.MartingalecentrallimittheoremsaregiveninSection2.6.RandomcentrallimittheoremsandBerry-EsseenresultsaregiveninSection2.7startingwiththepioneeringworkofAnscombe(1952).Nextinthischapter,weincludecertainresultsfromlinearandnonlinearrenewaltheory.
4、ThefundamentalrenewaltheoremofBlackwell(1948)isgiveninSection2.8.ItsapplicationtofirstpassageandresidualwaitingtimesasdemonstratedinWoodroofe(1982)isalsogiveninthissection.ThenonlinearrenewaltheoryasdevelopedbyWoodroofe(1976,1977)andLaiandSiegmund(1977,1979)isgiveninSection2.9.Thesenonlinearre
5、newaltheoryresultsareusedrepeatedlyinlaterchapters.2.2MARTINGALESLet(Ω,A,P)beaprobabilityspace,andletACAiC···beasequenceofσ-fieldsallcontainedinA.ConsiderasequenceYn,n>1ofrandomvariableson(Ω,Α,P).ItisassumedthatY„is^-measurableandthatEY„<00.Thenwehavethefollowingdefinition.Definition2.2.1.{
6、Y„,An;n>1}issaidtobeamartingaleifE(Yn+iAn)=y„almostsurely(a.s.).19202.PROBABILISTICRESULTSINSEQUENTIALANALYSISRemark2.2.1.Agoodwaytorememberthisdefinitionistorecallthenotionofafairgame.Inthiscase,Y„denotesaplayer'sfortuneuponcompletionofthenthplay,andafairgameisdeterminedbythefactthatifonekno
7、wsone'sfortunesY,...,Yn,thenone'sexpectedfortuneuponcompletionofthe(m+n)thplayisone'spresentfortune,i.e.,E(Yn+mY,...,Yn)=Yna.s.Example2.2.1.LetA",'sbei.i.d.withΕ(Χή=0.DefineYn=ΣΪ=ιΧ
