On the Strong Law of Large Numbers for Weighted Sums of Random Elements in Banach Spaces

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On the Strong Law of Large Numbers for Weighted Sums of Random Elements in Banach Spaces
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english
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Liao, Yuan
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Doctorate ( Ph.D.)
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University of Florida
Degree Disciplines:
Statistics
Committee Chair:
Rosalsky, Andrew J
Committee Members:
Ghosh, Malay
Khare, Kshitij
Cantrell, Amy

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probability -- slln
Statistics -- Dissertations, Academic -- UF
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Let {Vn, n = 1} be a sequence of random elements in a real separable Banachspace and suppose that {Vn, n = 1} is stochastically dominated by a random element V. Let {an, n = 1} and {bn, n = 1} be real sequences with 0 < bn ? 8. The main results are strong laws of large numbers (SLLNs) obtained for the following two broad cases;the results are new even when the underlying Banach space is the real line. (i) Conditions are provided under which {an(Vn - EVn), n = 1} obeys a general SLLN of the form Sni=1 ai (Vi - EVi )/bn ? 0 almost certainly where the {Vn, n = 1} are independent. The underlying Banach space is assumed to satisfy the geometric condition that it is of Rademacher type p (1 = p = 2). Special cases include resultsof Woyczynski (1980), Teicher (1985), Adler, Rosalsky, and Taylor (1989), and Sung (1997). (ii) Conditions are provided under which {anVn, n = 1} obeys a general SLLN of the formSni=1 aiVi/bn ? 0 almost certainly irrespective of the joint distributions of the{Vn, n = 1}. No geometric conditions are imposed on the underlying Banach space.The results are general enough to include as special cases results of Petrov (1973), Teicher (1985), Sung (1997), and Rosalsky and Stoica (2010). Numerous examples are provided which illustrate, compare, or demonstrate the sharpness of the results.
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by Yuan Liao.
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Thesis (Ph.D.)--University of Florida, 2012.
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Adviser: Rosalsky, Andrew J.
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ONTHESTRONGLAWOFLARGENUMBERSFORWEIGHTEDSUMSOFRANDOMELEMENTSINBANACHSPACESByYUANLIAOADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2012

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c2012YUANLIAO 2

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Tomyteacherswithgratitude Tomyparentswithlove 3

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ACKNOWLEDGMENTS Firstandmostimportantly,ImustconveymysincerestgratitudetomyPh.Dadvisor,ProfessorAndrewRosalsky,forhisinvaluableguidanceandconstantsupportthroughoutmygraduatestudies.Thisdissertationwouldnothavebeenpossiblewithouthisstep-by-stepguidance.Healwaysgenerouslyshareshisideasandmakesgreatefforttoexplainthemintheclearestwaypossible.Ifeelveryfortunatetogettoknowhim.Heisanamiablementorfullofenthusiasmforprobabilitytheory.Heisalwayspatienttocorrectthefaultsinmyworkandprovidethemostinformativeandinspiringfeedback.ImustadmitthatIhavelearnedalotfromhismeticulousacademicattitude.Next,Iwouldliketothankeveryoneelseonmysupervisorycommittee:DistinguishedProfessorMalayGhosh,Dr.KshitijKhare,andDr.AmyCantrell.Iamgratefulforalloftheirsupportandhelp.Finally,Iwouldextendmyearnestthankstomyparents,whohavealwaysbeencondentandprideofmeandencouragingmetochasemydreams.Theyhavealwaysandforeverbeenmyinspiration! 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................. 4 LISTOFFIGURES ..................................... 6 ABSTRACT ......................................... 7 CHAPTER 1INTRODUCTION ................................... 8 1.1StrongLawofLargeNumbersforRandomVariables ............ 8 1.2StrongLawofLargeNumbersforBanachSpaceValuedRandomElements 11 1.3MotivationandOrganizationofDissertation ................. 14 2PRELIMINARIES:DEFINITIONS,LEMMAS,ANDNOTATION .......... 22 2.1BasicConceptsofBanachSpaces ...................... 22 2.2ProbabilityinBanachSpaces ......................... 25 2.3UsefulLemmas ................................. 35 3STRONGLAWSOFLARGENUMBERSINRADEMACHERTYPEp(1p2)BANACHSPACESFORINDEPENDENTSUMMANDS ............. 42 3.1Objective .................................... 42 3.2MainResults .................................. 42 4STRONGLAWSOFLARGENUMBERSFORRANDOMELEMENTSINGENERALBANACHSPACESIRRESPECTIVEOFTHEIRJOINTDISTRIBUTIONS ... 68 4.1Objective .................................... 68 4.2MainResults .................................. 68 5FUTURERESEARCHANDCONCLUSIONS ................... 91 5.1FutureResearch ................................ 91 5.2Conclusions ................................... 94 REFERENCES ....................................... 96 BIOGRAPHICALSKETCH ................................ 100 5

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LISTOFFIGURES Figure page 2-1ExpectedValueofaRandomElementinLp(R),1p<1 ........... 32 6

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AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophyONTHESTRONGLAWOFLARGENUMBERSFORWEIGHTEDSUMSOFRANDOMELEMENTSINBANACHSPACESByYUANLIAOMay2012Chair:AndrewRosalskyMajor:Statistics LetfVn,n1gbeasequenceofrandomelementsinarealseparableBanachspaceandsupposethatfVn,n1gisstochasticallydominatedbyarandomelementV.Letfan,n1gandfbn,n1gberealsequenceswith0
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CHAPTER1INTRODUCTION 1.1StrongLawofLargeNumbersforRandomVariables Probabilitytheory,asamathematicaldisciplineconcernedwiththeanalysisofrandomorchancephenomena,hasdevelopednotonlyprofoundlyinitsallclassicalbranchesbutalsowidelyfromproblemsarisingfromotherbranchesofsciencesuchasmathematicalstatisticsandphysics.Theessentialcomponentsofprobabilitytheoryareexperimentaloutcomes(calledsamplepoints),events,randomvariables,andstochasticprocesses.Thelattertwoaremathematicalabstractionsofnon-deterministicmeasuredquantitiesthatmayeitherbeasinglevalueorevolveovertimeinarandomfashion.Ifarandomexperimentisrepeatedmanytimes,asequenceofrandomeventswilldemonstratecertainpatternswhichcanbestudiedandpredicted.Tworepresentativemathematicalresultsdescribingsuchpatternsarethelawoflargenumbersandthecentrallimittheorem,whichhavebeencrownedasbeingthersttwoofthethreepearlsofprobabilitytheory.(Thelawofiteratedlogarithm,thethirdpearlofprobabilitytheory,hasnotyethadasbiganimpactonapplicationsthatcanbecomparedwiththeothertwo,sinceitcannotbeobservedeveninalargenumberofreplicationsoftheexperiment.) Thelawsoflargenumbershavebecomethesteppingstonebetweenprobabilitytheoryandmathematicalstatistics.Ononehand,thebasicgoalofprobabilitytheoryistocalculatetheprobabilitiesofeventsunderagivenprobabilisticmodel.Ontheotherhand,mathematicalstatisticsinacertainsensehandlestheinverseoftheproblemsofprobabilitytheory.Inotherwords,mathematicalstatisticspreparesitselftoclarifythestructureofprobabilistic-statisticalmodelsbasedonactualobservationsofvariousevents.Whileitisdifculttoforecastthegeneralprinciplesgoverningthebehaviorofasmallsetofrandomvariables,thelawsoflargenumbersencapsulatethenotionthatlargesetsofrandomvariablestendtolosevariousaspectsoftheirrandomness. 8

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Theystabilize,patternsemerge,andtheirgeneralbehaviorbecomesfairlypredictable.Indeed,thelawoflargenumbersprovidesarigorousmathematicaldescriptionforthestatisticallawsabstractedfromtheempiricalobservationthattheaverageoftheresultsobtainedfromalargenumberoftrialsshouldbeclosetoaxedvalue(calledtheexpectedvalueormeaninprobabilitytheoryandmathematicalstatistics),andwilltendtobecomecloserasmoretrialsareperformed. TherstspecialformofthelawoflargenumberswithrigorousmathematicalproofwasgivenbytheprominentSwissmathematicianJacobBernoulliinhisrenownedbookArsConjectandi(TheArtofConjecturing)publishedposthumouslyin1713.HistheoremispresentlycalledtheweaklawoflargenumbersforBernoullitrials.AccordingtoBernoulli'stheorem,ifSnisthenumberofoccurrencesofaneventAinnindependenttrialsandptheconstantprobabilityofoccurrenceofeventAineachoftheindependenttrials,thenforallpositiverealnumbers",limn!1PSn n)]TJ /F8 11.955 Tf 11.95 0 Td[(p<"=1; thatis,inprobabilityterminology,Sn=nconvergestopinprobability.Thistheoremwasextendedinthenext100yearsorsobythegreatFrenchmathematicianandphysicistSimeonD.PoissonandtheeminentRussianmathematicianPafnutyL.Chebychev.ItwasPoissonwhocoinedthephrasethelawoflargenumbers(inFrench,laloidesgrandsnombres).In200yearsorsoafterBernoulli'sweaklawoflargenumbers,theFrenchmathematicianEmileBorelobtainedthestronglawoflargenumbers(SLLN)forBernoullitrials,whichconcludesinprobabilityterminologythatSn=nconvergestopalmostcertainly(a.c.);thatis,Plimn!1Sn n=p=1. In1933,thepreeminentSovietRussianmathematicianAndreyN.KolmogorovinauguratedthemodernerainprobabilitytheoryinhisclassicmonographFoundations 9

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oftheTheoryofProbability.Kolmogorovtheresuccessfullygivesprobabilitytheoryarigorousaxiomaticbasis,harnessingthefullpowerofmeasuretheorybyregardingaprobabilityeventfunctionasameasureofmassonedenedonthe-algebraofevents.HisSLLNdeclaresthat,forasequenceofi.i.d.randomvariablesX1,X2,...andarealnumber,thefollowingareequivalent: (i) TheexpectedvalueofX1existsandis;thatis,EjX1j<1andEX1=. (ii) Thesamplemeanconvergestowithprobabilityone;thatis,1 nnXi=1Xi!a.c. ThiscompletesthelineofworkstartedbyJacobBernoulli;i.e.,theprecedingBernoulli'sweaklawoflargenumbers.Kolmogorov'sSLLNisthepreciseformofthefolkloreideaofthelawofaverages,andshowsconvincinglythattheKolmogorov'saxiomaticsystemhassuccessfullycapturedthetrueessenceofprobabilitytheory.Kolmogorov'sSLLNwasextendedbyMarcinkiewicz-Zygmund(1937)andFeller(1946)whoprovedSLLNsfori.i.d.randomvariablesusingmoregeneralnormingsequences. TheclassicSLLNscanbeextendedinvariousdirectionsandprovidesintuitionformanyothertheories.SomeSLLNscanbeobtainedunderweakenedassumptions,suchasforrandomvariableswhichareindependentbutnotnecessarilyidenticallydistributed,orforrandomvariableswhicharepairwiseindependent(ChowandTeicher(1997,Section5.2)).Somecanholdinmoregeneralforms,forexample,weightedsumsofrandomvariables.Stout(1974,Chapter4)givesanexcellentsurveyofknownresultsupto1974ontheSLLNproblemforweightedsumsofindependentrandomvariables.MartingaletheoryhasSLLNtypetheoremsderivedviaKolmogorov'sinequality(Feller(1971,SectionsVII.8andVII.9)).Ergodictheory,motivatedbyproblemsofstatisticalphysics,hasitsfoundationswiththeSLLNtypetheorems.Theunderlyingideaisthatforcertainsystemsthetimeaverageoftheirpropertiesconvergestotheaverageovertheentirespace(theso-calledensembleaverage).Twoofthemostimportantexamples 10

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arepointwiseergodictheoremsofBirkhoffandvonNeumann(Shiryaev(1996,ChapterV)).Inmathematicalstatistics,theprecedingSLLNtypetheoremsprovidenumerousconsistentestimatorsandstatistics. 1.2StrongLawofLargeNumbersforBanachSpaceValuedRandomElements Intheearly1950s,ProbabilityinBanachSpaces,asabranchofmodernmathematics,wasinitiatedbytheconsiderationofastochasticprocessasarandomelementinafunctionspace(ameasurablefunctionfromaprobabilityspacetoafunctionspace)and,inparticular,withthepioneeringworkbyFortetandMourier(1953)onthelawoflargenumbersandthecentrallimittheoremforsumsofindependentidenticallydistributedBanachspacevaluedrandomvariables(henceforthtobereferredtoasrandomelements).AlltechnicaldenitionsmentionedinSections1.2and1.3willbereviewedinChapter2. Thelawsoflargenumbersforidenticallydistributed(real-valued)randomvariableswereextendedtonormedlinearspacesbyMourier(1953)andTaylor(1972).Mourier(1953)establishedananalogueoftheclassicalKolmogorov'sSLLN.Specically,Mouriershowedthat,forasequenceofi.i.d.randomelementsfVn,n1ginarealseparableBanachspace,iftheexpectedvalueofV1,denotedbyEV1,exists(theexpectedvalueofarandomelementisdenedtobeitsPettisintegral),then1 nnXi=1(Vi)]TJ /F8 11.955 Tf 11.95 0 Td[(EV1)!0a.c.Taylor(1972)providedconditionsforidenticallydistributedrandomelementsinnormedlinearspacestoobeytheweaklawoflargenumbers. Toobtainthecorrespondingresultsforthenon-identicallydistributedrandomelements,additionalconditionsonthedistributionsoftherandomelementsand/orontheBanachspaceitselfareneeded.AdecisivesteptothemoderndevelopmentofprobabilityinBanachspaceswastheintroductionbyBeck(1962)ofaconvexityconditiononnormedlinearspacesequivalenttothevalidityoftheextensionofa 11

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classicalSLLNofKolmogorov.Hoffmann-JrgensenandPisier(1976)establishedaSLLNbyassumingtheunderlyingrealseparableBanachspaceisofRademachertypep(1p2).Actually,theyshowedthatforasequenceofindependentrandomelementsfVn,n1gwithzeroexpectedvaluesinarealseparableBanachspace,theBanachspaceisofRademachertypep(1p2)ifandonlyifthefollowingholds:1Xn=1EkVnkp np<1implies1 nnXi=1Vi!0a.c. ThusHoffmann-JrgensenandPisier(1976)providedanactualcharacterizationofRademachertypep(1p2)BanachspacesintermsofSLLN.AdetaileddiscussionmaybefoundinTaylor(1978,ChapterIV). ThestudyoftheSLLNforweightedsumsofindependentrandomvariablescontributesmuchtoitsextensiontotheSLLNforweightedsumsofindependentrandomelements.AdlerandRosalsky(1987a)and(1987b)presentedageneralSLLNforweightedsumsofstochasticallydominatedrandomvariables,whichisgeneralenoughtoinclude,asaspecialcase,Feller's(1946)celebratedextensionoftheMarcinkiewicz-ZygmundSLLN(e.g.,ChowandTeicher(1997,p.125)).AdlerandRosalsky(1987a)didnotrequirethesummandstobeindependent.Thehypothesesinvolveboththebehaviorofthetailofthedistributionofthedominatingrandomvariableandthegrowthbehavioroftheweightsandnormingconstants.Furthermore,thecenteringsequenceisrandom.AresultofAdlerandRosalsky(1987b)forweightedsumsofi.i.d.randomvariablesissubstantiallyimprovedbySung(1997)whoobtainedthesametheorembutunderlessstringentconditions. BasedontheworkofAdlerandRosalsky(1987a)and(1987b),Adler,Rosalsky,andTaylor(1989)establishedaSLLNforweightedsumsofindependentrandomelementsinnormedlinearspaces.Thehypothesesinvolvethedistributionsoftheindependentrandomelements,thegrowthbehaviorsoftheweightsandnormingconstants,andforsomeoftheresultsageometricconditionisimposedonthenormed 12

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linearspace.Moreover,Adler,Rosalsky,andTaylor(1989)showedthatFeller's(1946)famousresultgeneralizingtheMarcinkiewicz-ZygmundSLLNholdsforrandomelementsinarealseparableRademachertypep(1
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theory.Sincesomestochasticprocessescanberegardedasbeingarandomelementinparticularfunctionspaces,thelawsoflargenumbersforrandomelementsmaybeapplied.Indecisiontheory,thelawsoflargenumberscanbeappliedtodevelopconsistentstatisticaldecisionprocedures.Qualitycontrolisanimportantindustrialapplicationofstatistics.Consistentestimatorsoftheparametersinacontinuousproductionprocesscanbeconstructedbyusingalawoflargenumbersforweightedsumsofrandomelements.Inthedensityestimationproblem,theintuitivefrequencyhistogramideacanbeextendedtoafunctionspaceapproach,andthelawsoflargenumbersinBanachspacecanbeappliedundersuitableconditions.AdetaileddiscussionmaybefoundinTaylor(1978,ChapterVIII). 1.3MotivationandOrganizationofDissertation LetfVn,n1gbeasequenceofrandomelementsdenedonaprobabilityspace(,F,P)takingvaluesinarealseparableBanachspacewithnormkk.SupposethatEVnexistsforalln1.Letfan,n1gandfbn,n1gbesequencesofconstantswith0
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AdlerandRosalsky(1987a)establishsomeSLLNsforweightedsumsofrandomvariablesunderrathergeneralconditions.Therein,itisnotassumedthattheunderlyingrandomvariablesareindependentoridenticallydistributedorevenintegrable.AdlerandRosalsky(1987b)intheirfollow-uparticleprovidesetsofnecessaryand/orsufcientconditionsfortheSLLNtoholdforweightedsumsformedfromsequencesofindependentandidenticallydistributed(i.i.d.)randomvariables.Inparticular,FernholzandTeicher's(1980)maintheoremisaspecialcaseofAdlerandRosalsky's(1987b)Theorem2(Proposition 1.3.1 below)takingan=1,bn=(dn)andcn=EXnforn1whereisafunctiondenedforpositivexsuchthat(x)=xisdecreasingforsome>1and0
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Sung(1997)thusimprovedTheorem2ofAdlerandRosalsky(1987b)(Proposition 1.3.1 )byshowingthatcondition( 1.3 )isnotneededwhen1tgDPfkDVk>tg,t0,n1.Moreover,supposethatEkVkq<1forsome1
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Acosta(1981)establishedinhisTheorem4.1thefollowingMarcinkiewicz-ZygmundtypeSLLNcharacterizationforarealseparableBanachspacebeingofRademachertypep(1p<2).Theimplication((i))(ii))wasalsoobtainedbyAzlarovandVolodin(1981). Proposition1.3.4(Theorem4.1ofdeAcosta(1981)). Let1p<2.LetXbearealseparableBanachspace.Thenthefollowingareequivalent: (i) TheBanachspaceXisofRademachertypep. (ii) Foreverysequenceofi.i.d.randomelementsfVn,n1ginXwithEkV1kp<1,Pni=1(Vi)]TJ /F8 11.955 Tf 11.95 0 Td[(EVi) n1=p!0a.c. (1.5) However,deAcosta(1981)didnotprovideanexplicitexamplewhereintheSLLNfailsforaBanachspacewhichisnotofRademachertypep.ThismotivatedustotakeadvantageofExample7.11ofLedouxandTalagrand(1991,p.190)(Example 3.2.2 inChapter3).WealsonotedinRemark3.2.6(iv)thatinthespecialcasewherefVn,n1gisasequenceofi.i.d.randomelementswithEkV1kq<1,an=1,bn=n1=q,n1where1
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3.2.3 asequenceofindependentbutnotidenticallydistributedrandomelementstoservethesamepurposeaproposofTheorem3.2.1. Aswasmentionedabove,deAcosta(1981)providedacharacterizationinhisTheorem4.1ofRademachertypep(1p<2)BanachspacesviaaMarcinkiewicz-ZygmundtypeSLLN.ThekeyresultusedbydeAcosta(1981)toprovetheSLLN( 1.5 )isthefollowingresultofdeAcosta(1981,Theorem3.1). Proposition1.3.5(Theorem3.1ofdeAcosta(1981)). Let1p<2andletXbearealseparableBanachspace.Thenforeverysequenceofi.i.d.randomelementsfVn,n1ginXwithEkV1kp<1,theSLLN( 1.5 )holdsifandonlyifPni=1(Vi)]TJ /F8 11.955 Tf 11.96 0 Td[(EVi) n1=pP!0. (1.7) DeAcosta(1981,Theorem3.1)anddeAcosta(1981,Theorem4.1)togetherassertthatRademachertypep(1p<2)BanachspacescanbecharacterizedbytheMarcinkiewicz-Zygmundtypeweaklawoflargenumbers( 1.7 ). Insummary,weestablishinChapter3theworkofpart(i)obtainingSLLNsassumingthefVn,n1gareindependentandtheunderlyingBanachspaceisofRademachertypep(1p2).Moreover,Theorem 3.2.1 andTheorem 3.2.2 arenewresultsevenwhentheunderlyingBanachspaceisthereallineR. Theworkofpart(ii),whichispresentedinChapter4,isaparalleldevelopmentoftheworkofpart(i)presentedinChapter3buttheargumentsaredistinctlydifferent. Withtheworkofpart(i)inhand,itseemednaturaltodevelopa0
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suchasaRademachertypep(1
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Petrov(1973,Theorem1),RosalskyandStoica(2010,Theorem2.1),andRosalskyandStoica(2010,Theorem2.2)obtainedreallineSLLNsoftheformPni=1Xi bn!0a.c.irrespectiveofthejointdistributionsoftherandomvariablesfXn,n1g.Wewerethusenlightenedtoextendtheirresultstothegeneralform( 1.8 )forBanachspacevaluedrandomelements. Petrov(1973,Theorem1)motivatedustoreplacecondition( 1.9 )bythecondition1Xn=1janj bnq<1 (1.10) underwhichweobtainaSLLNinTheorem4.2.2,oursecondmainresultinChapter4.Whenjanj=bn#,( 1.10 )isstrongerthan( 1.9 ).Thoughthe0
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showhowtheresultsimproveuponoraredifferentfromotherresultsintheliterature.Examplesarealsoprovidedtoshowthattheresultsaresharp.PriortothepresentationofthemainresultsinChapters3and4,notation,denitions,andsomerelevantresultsaboutBanachspacesarepresentedinChapter2,Section1.ProbabilisticconceptsinBanachspacesarepresentedinChapter2,Section2.Chapter2,Section3containsthelemmasneededinChapters3and4. WeendthischapterbymentioningthatmeanconvergenceversionsoftheMarcinkievicz-ZygmundSLLNhavebeeninvestigatedforboththecasesofasequenceofrandomvariablesandasequenceofBanachspacevaluedrandomelements.Klass(1973,Corollary12)provedforasequenceofi.i.d.randomvariablesfXn,n1gwithEjX1jp<1forsomepin[1,2)thatlimn!1EjPni=1(Xi)]TJ /F8 11.955 Tf 11.95 0 Td[(EXi)j n1=p=0. Korzeniowski(1984)extendedthisresultofKlass(1973)tothecaseofasequenceofrandomelementstakingvaluesinarealseparableRademachertypeq(1
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CHAPTER2PRELIMINARIES:DEFINITIONS,LEMMAS,ANDNOTATION 2.1BasicConceptsofBanachSpaces Somedenitions,lemmas,andnotationneedtobepresentedpriortostatingandprovingthemainresults. AnonemptysetXissaidtobea(real)linearspaceifthereisdenedabinaryoperationofadditionwhichmakesXanabeliangroupandanoperationofmultiplicationby(real)scalarswhichsatisfythedistributiveandidentitylaws;thisisstatedmorepreciselyasfollows. (a) Toeverypairofelement(u,v)2XX,therecorrespondsanelementw2Xsuchthatw=u+v. (b) Toeveryu2Xandt2R,therecorrespondsanelementtu2X. (c) Theoperationsdenedin(a)and(b)satisfy,forallu,v,w2Xandalls,t2R,thefollowingsevenproperties: (i) u+v=v+u, (ii) (u+v)+w=u+(v+w), (iii) u+v=u+wimpliesv=w, (iv) 1u=u, (v) (st)u=s(tu), (vi) (s+t)u=su+tu, (vii) s(u+v)=su+sv. ThezeroelementofXisdenotedby0.Whilethisisthesamesymbolastherealnumber0,itshouldbeclearfromthecontextastowhether0refersto02Xor02R. AreallinearspaceXissaidtobenormedifthereisareal-valuedfunctiondenedonXanddenotedbykksuchthatkksatises,forallu,v2Xandallt2R,thefollowingthreeproperties: (i) kuk0andkuk=0ifandonlyifu=0, (ii) ku+vkkuk+kvk, (iii) ktuk=jtjkuk. 22

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ThefunctionkkisthencalledanormonX.Property(ii)aboveiscalledthetriangleinequality. Asequencefvn,n1ginanormedlinearspaceXissaidtoconvergetoanelementvofXiflimn!1kvn)]TJ /F8 11.955 Tf 12.5 0 Td[(vk=0.Thiswillbedenotedbylimn!1vn=vorbyvn!vasn!1.Asequencefvn,n1ginanormedlinearspaceXissaidtobeaCauchysequenceifforevery">0,thereexistsanintegerNsuchthatkvn)]TJ /F8 11.955 Tf 12.14 0 Td[(vmk<"whenevernNandmN;i.e.,limn!1supm>nkvm)]TJ /F8 11.955 Tf 11.95 0 Td[(vnk=0.AnormedlinearspaceXissaidtobecompleteifeveryCauchysequenceofXconvergestoanelementofX.AcompletenormedlinearspaceiscalledaBanachspace. AsubsetSofanormedlinearspaceXissaidtobedenseinXifitsclosure(thatis,thesmallestclosedsubsetofXcontainingS)equalsX.IfXhasacountabledensesubset,thenXissaidtobeseparable. InthefollowingexampleswelistseveralparticularrealBanachspaces. Example2.1.1. Thespace`p,1p<1,istheclassofallrealsequencesv=(v1,v2,...)suchthatP1k=1jvkjp<1.Withthenormdenedbykvkp= 1Xk=1jvkjp!1=p,eachofthespaces`p,1p<1,isarealseparableBanachspace. Example2.1.2. Thespace`1isthecollectionofallboundedrealsequencesv=(v1,v2,...).Withthenormdenedbykvk1=supfjvkj,k1g,`1isarealBanachspacewhichisnotseparable(e.g.,Taylor(1978,p.10)).Letc0denotethesubspaceof`1whichconsistsoftherealsequencesthatconvergetozero.Withthesamenormas`1,c0isarealseparableBanachspace. 23

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Example2.1.3. ThespaceLp(R),1p<1,istheclassofallrealLebesguemeasurablefunctionsv()onRsuchthatRRjv(t)jpdt<1.Withthenormdenedbykvkp=ZRjv(t)jpdt1=p,eachofthespacesLp(R),1p<1,isarealseparableBanachspace. Example2.1.4. ThespaceL1(R)istheclassofallrealLebesguemeasurablefunctionsv()thatareboundedalmosteverywhere(a.e.)onRwithrespecttoLebesguemeasure.Withthenormdenedbykvk1=inff:jv(t)ja.e.g,thespaceL1(R)isaBanachspacewhichisnotseparable(e.g.,Taylor(1978,p.11)).Thenormkvk1iscalledtheessentialsupremumofjv()jandisalsodenotedby(jvj). Thecollectionofallcontinuouslinearfunctionals(thatis,continuousreal-valuedlinearfunctions)denedonanormedlinearspaceXiscalledthedualspaceofXandisdenotedbyX.Werecallthatalinearfunctionalisafunctionf:X!Rsatisfyingf(au+bv)=af(u)+bf(v)forallu,v2Xandalla,b2R. Asequencefbn,n1ginaBanachspaceXissaidtobeaSchauderbasisforXifforeachv2Xthereexistsauniquesequenceofscalarsftn,n1gsuchthatv=limn!1nXk=1tkbk. WhenXhasaSchauderbasisfbn,n1g,asequenceoflinearfunctionalsffk,k1gcanbedenedbyfk(v)=tk,k=1,2,...wherev2Xandv=limn!1Pnk=1tkbk.Thelinearfunctionalsffk,k1gXarecalledthecoordinatefunctionals. 24

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ThefollowingTheorem 2.1.1 istheRieszRepresentationTheorem(e.g.,Royden(1988,p.132))anditwillbeusedinExample 2.2.3 below.TheRieszRepresentationTheoremisacrowningachievementintwentiethcenturymathematics. Theorem2.1.1(RieszRepresentationTheorem). ForeachfinthedualspaceofLp(R),1p<1,thereexistsgf2Lq(R)where1=p+1=q=1(q=1ifp=1)suchthatf(h)=ZRh(x)gf(x)dxforallh2Lp(R). Remark2.1.1. TheRieszRepresentationTheoremistermedarepresentationtheorembecauseitprovidesaconcreterepresentationforthemembersofthedualspaceofLp(R),1p<1.Informally,Theorem 2.1.1 assertsthatthedualspaceofLp(R),1p<1isLq(R)where1=p+1=q=1(q=1ifp=1). 2.2ProbabilityinBanachSpaces Let(,F,P)beaprobabilityspace.LetXdenotearealseparableBanachspacewithanormkk.LetXbeequippedwithitsBorel-algebraB(X);i.e.,B(X)isthe-algebrageneratedbytheclassofopensubsetsofXdeterminedbythemetricd(u,v)=ku)]TJ /F8 11.955 Tf 11.35 0 Td[(vk,u,v2X.ArandomelementVinXisaF-measurabletransformationfromtothemeasurablespace(X,B(X));i.e.,V)]TJ /F11 7.97 Tf 6.59 0 Td[(1(A)2FforallA2B(X). Remark2.2.1. ArandomelementisageneralizationofarandomvariablesincetheBorel-algebrageneratedbyallintervalsofrealnumbersoftheform(,b)istheclassofBorelsubsetsofR.Therefore,VisarandomelementinRifandonlyifVisarandomvariable.Furthermore,randomelementsinann-dimensionalEuclideanspaceRnaren-dimensionalrandomvectors. ThefollowingProposition 2.2.1 showsthatsomepropertiesofrandomvariablescanbeextendedtothesettingofrandomelements.AfurtherdiscussionmaybefoundinTaylor(1978,ChapterII). 25

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Proposition2.2.1(Taylor(1978)). (i) LetVn,n1beasequenceofrandomelementsinaBanachspaceXsuchthatVn(!)convergestoV(!)foreach!2.ThenVisarandomelementinX. (ii) LetVbearandomelementinaBanachspaceXandletYbearandomvariable.ThenYVisarandomelementinX. (iii) IftherealBanachspaceXisseparable,thenkV)]TJ /F8 11.955 Tf 12.25 0 Td[(WkisarandomvariablewheneverVandWarerandomelementsinX.Inparticular,takingW=0,kVkisarandomvariableifVisarandomelement. (iv) IftheBanachspaceXisseparable,thenafunctionV:!XisarandomelementinXifandonlyiff(V)isarandomvariableforeachf2X. Remark2.2.2. (i) ThenecessityhalfinProposition 2.2.1 (iv)istruewithouttheassumptionthatXisseparable. (ii) Notallofthepropertiesofrandomvariablescanbeextendedtothesettingofrandomelements.Forexample,thesumoftworandomvariablesarerandomvariables,butthesumoftworandomelementsinaBanachspaceXmaynotbemeasurable.However,ifXisseparable,thenweseefromProposition 2.2.1 (iv)thatthesumoftworandomelementsinXisarandomelementinX. (iii) Taylor(1978,p.26)presentedanexampleshowingthatifXisnotseparable,thenkV)]TJ /F8 11.955 Tf 12.22 0 Td[(WkisnotnecessarilyarandomvariablewhereV,W,andV)]TJ /F8 11.955 Tf 12.22 0 Td[(WarerandomelementsinX.Consequently,Proposition 2.2.1 (iii)canfailwithouttheassumptionthatXisseparable. WenowdenemodesofconvergenceofasequenceofrandomelementsinarealseparableBanachspace.LetfVn,n1gbeasequenceofrandomelementsinarealseparableBanachspaceX.ThenfVn,n1gconvergestoarandomelementVinX (i) withprobabilityoneoralmostcertainly(a.c.)ifPnlimn!1kVn)]TJ /F8 11.955 Tf 11.96 0 Td[(Vk=0o=1,andthisisdenotedVn!Va.c.(orlimn!1Vn=Va.c.). (ii) inprobabilityiflimn!1PfkVn)]TJ /F8 11.955 Tf 11.95 0 Td[(Vk"g=0forall">0,andthisisdenotedVnP!V. (iii) intherthmeanforr>0ifEkVnkr<1foralln1andlimn!1EkVn)]TJ /F8 11.955 Tf 12.04 0 Td[(Vkr=0,andthisdenotedVnLr!V.Necessarily,wehaveEkVkr<1. 26

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ArandomelementinaBanachspaceandtheunderlyingprobabilitymeasureinduceaprobabilitymeasureontheBanachspaceanditsBorelsubsets.Theproba-bilitydistributionofarandomelementVinaBanachspaceXistheinducedmeasure,denotedbyPV,on(X,B(X));i.e.,PVfBg=PfV2Bg,B2B(X).TherandomelementsVandWinXaresaidtobeidenticallydistributedifPfV2Bg=PfW2BgforallB2B(X).AfamilyofrandomelementsinXissaidtobeidenticallydistributedifitseverypairisidenticallydistributed.AnitesetofrandomelementsfV1,...,VnginXissaidtobeindependentifforeverychoiceofB1,...,Bn2B(X),PfV12B1,...,Vn2Bng=PfV12B1gPfVn2Bng.AfamilyofrandomelementsinXissaidtobeindependentifitseverynitesubsetisindependent. TheexpectedvalueormeanofarandomelementVinarealseparableBanachspaceX,denotedEV,isdenedtobethePettisintegralprovideditexists;i.e.,VhastheexpectedEVinXifforeachf2X,wehaveE[f(V)]=f(EV) (2.1) whereXisthedualspaceofX.Notethattheleft-handsideof( 2.1 )makessensebecauseofProposition 2.2.1 (iv)andalsonotethatnecessarilyf(V)isintegrableforeachf2X.ThePettisintegralwasintroducedbyPettisin1938(Pettis(1938)).AcompletecharacterizationofwhenthePettisintegralexistswasprovidedbyBrooks(1969).AfurtherdiscussionanddetailsregardingthepropertiesofthePettisintegralmaybefoundinHilleandPhillips(1985,pp.76). 27

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Theexpectedvalueofrandomelementsenjoyssimilarpropertiesasdoestheexpectedvalueofrandomvariables(Proposition 2.2.2 below)andsometimescanbeobtainedasintherandomvariablecase(Proposition 2.2.3 andExample 2.2.3 below). Weillustratethedenitionoftheexpectedvalueofarandomelementwiththefollowingverysimpleexample(Example 2.2.1 ).Moreinvolvedexamples(Examples 2.2.2 and 2.2.3 )arepresentedbelow. Example2.2.1. LetXbeanL1randomvariableandletv2XwhereXisanarbitraryrealseparableBanachspace.LetV=Xv.ThentheexpectedvalueEVofVexistsandisgivenbyEV=(EX)v.(ThisisofcoursepreciselywhatonewouldexpecttobetheexpectedvalueofV.) Proof:ByProposition 2.2.1 (ii),V=XvisarandomelementinXsincevcanberegardedasadegeneraterandomelementinX.Thenf(V)=f(Xv)=Xf(v)forallf2XsinceX(!)canberegardedasarealscalarforeach!2.Thus,( 2.1 )holdssinceE[f(V)]=E[Xf(v)]=f(v)EX=f((EX)v)forallf2X.Therefore,theexpectedvalueEVofVexistsandisgivenbyEV=(EX)v.2 Proposition2.2.2(Taylor(1978)). LetV,V1andV2berandomelementsinarealseparableBanachspaceX,then (i) IfEV1andEV2exist,thenE(V1+V2)existsandE(V1+V2)=EV1+EV2. (ii) IfEVexistsandt2R,thenE(tV)existsandE(tV)=tEV. (iii) IfEkVk<1,thentheexpectedvalueEVofVexistsandkEVkEkVk. Proposition2.2.3. IfVisacountably-valuedrandomelementinXtakingvaluesfvi,i1g,thentheexpectedvalueEVofVexistsandisgivenbyEV=1Xi=1viPfV=vigprovided1Pi=1kvikPfV=vig<1. 28

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Proof:Letv=1Pi=1viPfV=vig.Thenv2XsinceXiscomplete.Moreover,( 2.1 )holdssinceforeachf2X,E[f(V)]=1Xi=1f(vi)PfV=vig=limn!1nXi=1f(vi)PfV=vig=limn!1f nXi=1viPfV=vig!=f limn!1nXi=1viPfV=vig!(sincefiscontinuous)=f 1Xi=1viPfV=vig!=f(v). Thus,theexpectedvalueofVexistsandisgivenbyEV=v=1Pi=1viPfV=vig.2 Example2.2.2. IfaBanachspaceXhasaSchauderbasisfbn,n1gwithcoordinatefunctionalsffn,n1g,theneachrandomelementVinXcanbeexpressedasV=1Pn=1fn(V)bnpointwisein!2.IfVhasexpectedvalueEV2X,thenE[fn(V)]=fn(EV)sinceeachfnisinX.ThusEV=1Xn=1fn(EV)bn=1Xn=1E[fn(V)]bn. (2.2) Notethatthespaces`p,1p<1sharethesameSchauderbasisfv(n),n1gwherev(n)istheelementof`phaving1initsnthpositionand0elsewhere.Thus,eachrandomelementVin`p,1p<1canbeexpressedasasequenceofrandomvariablesffn(V),n1g;i.e.,V=(f1(V),f2(V),...)=(V1,V2,...)(say).Furthermore,if 29

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theexpectedvalueEVofVexists,thenby( 2.2 )wegetEV=1Xn=1E[fn(V)]v(n)=(E(f1(V)),E(f2(V)),...)=(EV1,EV2,...). (2.3) (Again,thisispreciselywhatonewouldexpecttobetheexpectedvalueofV.) ParallelingtheRieszRepresentationTheoremwhichconcernstherealseparableBanachspaceLp(R),1p<1,thefollowingrepresentationtheoremfor`p,1p<1(Wilansky(1964,p.91))willbeusedinRemark 2.2.3 belowwhichpertainstoExample 2.2.2 Theorem2.2.1. Foreachf2`p,1p<1,thereexistsb(f)=(b1(f),b2(f),...)2`qwhere1=p+1=q=1(q=1ifp=1)suchthatf(a)=1Xn=1anbn(f)foralla=(a1,a2,...)2`p. Remark2.2.3. LetV=(V1,V2,...)bearandomelementin`p(1p<1)asinExample 2.2.2 .Ifweassume1Xn=1EjVnjp<1 (2.4) (thatis,(EjV1jp,EjV2jp,...)2`1),thenwealsoobtaintheexpectedvalueofVwiththeform( 2.3 )viaTheorem 2.2.1 asfollows.Noteattheoutsetthat( 2.4 )impliesthatVnisintegrableforeachn1.Letv=(EV1,EV2,...).Thenv2`psincekvkp= 1Xn=1jEVnjp!1=p 1Xn=1EjVnjp!1=p(byJensen'sInequality)<1(by( 2.4 )). ByTheorem 2.2.1 ,f(V)=1Pn=1Vnbn(f)foreachf2`pwhereb(f)=(b1(f),b2(f)),...)2`qand1=p+1=q=1(q=1ifp=1). 30

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Then,foreachm1,mXn=1Vnbn(f)mXn=1jVnjjbn(f)j1Xn=1jVnjjbn(f)jkVkpkb(f)kq(byHolder'sInequality). Moreover,kVkpkb(f)kqisintegrablesinceE(kVkpkb(f)kq)=kb(f)kqEkVkp=kb(f)kqE24 1Xn=1jVnjp!1=p35kb(f)kq"E 1Xn=1jVnjp!#1=p(byJensen'sInequality)=kb(f)kq"1Xn=1EjVnjp#1=p(byLemma 2.3.6 )<1(by( 2.4 )). Thus,bytheLebesgueDominatedConvergenceTheorem,E[f(V)]=E 1Xn=1Vnbn(f)!=E limm!1mXn=1Vnbn(f)!=limm!1E mXn=1Vnbn(f)!=limm!1 mXn=1(EVn)bn(f)!=1Xn=1(EVn)bn(f)=f(v)(byTheorem 2.2.1 ) 31

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recallingthatv=(EV1,EV2,...).Hence,theexpectedvalueEVofVexistsandisgivenby( 2.3 ). Example2.2.3. LetVbearandomelementinX=Lp(R),1p<1(Example 2.1.3 )withRRRjV(!)(x)jpdxdP(!)<1.ThentheexpectedvalueofEVofVexistsandisgivenbyEV=RVdPviewedasafunctionofx2R;i.e.,EV:R!Risgivenbyx7!RV(!)(x)dP(!).(Onceagain,thisispreciselywhatonewouldexpecttobetheexpectedvalueofV.)Figure 2-1 belowisprovidedtohelpclarifythenotionoftheexpectedvalueofarandomelementVinLp(R),1p<1. Figure2-1. ExpectedValueofaRandomElementinLp(R),1p<1 Proof:Forxedx2R,V()(x)isarandomvariable.Deneafunctionv()onRbyv(x)=EV()(x)=ZV(!)(x)dP(!)forallx2R.Thenv()isLebesguemeasurable.Since1p<1,wehaveforeachx2Rthatjv(x)jp=jEV()(x)jpEjV()(x)jpbyJensen'sinequality.Sov()2Lp(R)sinceZRjv(x)jpdxZREjV()(x)jpdx 32

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=ZRZjV(!)(x)jpdP(!)dx=ZZRjV(!)(x)jpdxdP(!)<1. Ontheotherhand,forxed!2,V(!)()2Lp(R).BytheRieszRepresentationTheorem(Theorem 2.1.1 ),foreachfinthedualspaceofLp(R),thereexistsgf2Lq(R)where1=p+1=q=1(q=1ifp=1)suchthatf(h)=ZRh(x)gf(x)dxforallh2Lp(R).Therefore,foreachfinthedualspaceofLp(R),byrsttakingh()=V(!)()andthentakingh()=v(),wegetE[f(V)]=Zf(V(!)())dP(!)=ZZRV(!)(x)gf(x)dxdP(!)=ZRZV(!)(x)dP(!)gf(x)dx(byFubini'sTheorem)=ZRv(x)gf(x)dx=f(v). Hence,theexpectedvalueEVofVexistsandisgivenbyEV=v=RVdP.2 ThefollowingexampleshowsthattheexpectedvalueEVcanexistevenifEkVk=1. Example2.2.4(Taylor(1978,p.41)). FortherealseparableBanachspace`2,denearandomelementVsuchthatV=nv(n)withprobabilityc=n2wherev(n)istheelementof`2having1initsnthpositionand0elsewhereandcisanappropriateconstant.NotethatEkVk2=1Xn=1nc n2=c1Xn=11 n=1, 33

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However,byProposition 2.2.3 ,EV=1Xn=1nv(n)P(V=nv(n))=1Xn=1nv(n)c n2=c 1,c 2,...,c n,...2`2. Letf"n,n1gbeasymmetricBernoullisequence;i.e.,f"n,n1gisasequenceofindependentandidenticallydistributed(i.i.d.)randomvariableswithPf"n=1g=Pf"n=)]TJ /F6 11.955 Tf 9.3 0 Td[(1g=1=2,n1. AsymmetricBernoullisequenceisalsoreferredtoasaRademachersequence.LetX1=XXX,anddeneC(X)=((v1,v2,...)2X1:1Xn=1"nvnconvergesinprobability).Let1p2.ThenarealseparableBanachspaceXissaidtobeofRademachertypepifthereexistsaconstant0
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reallineRisofRademachertype2.TherealseparableBanachspacec0(Example 2.1.2 )isnotofRademachertypepforanyp2(1,2]andforq2[1,2),therealseparableBanachspacesLqand`qarenotofRademachertypepforanyp2(q,2].AdetaileddiscussionoftheabovecanbefoundinChapter9ofLedouxandTalagrand(1991).TherealBanachspace`1isnotevenseparableaswasmentionedinExample 2.1.2 2.3UsefulLemmas TheclassicalandcelebratedreallineversionofLevy'sTheorem(e.g.,ChowandTeicher(1997,p.72)),whichassertsthatthepartialsumsfromasequenceofindependentrandomvariablesconvergealmostcertainlytoarandomvariableSifandonlyiftheyconvergeinprobabilitytoS,hasbeenextendedtoarealseparableBanachspacesettingbyItoandNisio(1968)andisstatedasfollows. Lemma2.3.1(ItoandNisio(1968)). LetfVn,n1gbeasequenceofindependentrandomelementsinarealseparableBanachspaceXandsetSn=nXi=1Vi,n1.ThenSnconvergesa.c.toarandomelementSinXifandonlyifSnP!S. Remark2.3.1. ItfollowsfromLemma 2.3.1 thatinthedenitionofC(X),thecondition1Xn=1"nvnconvergesinprobabilityisequivalenttothecondition1Xn=1"nvnconvergesa.c. Nowweintroducethenotionofregularvariationwhichhasbeenprovedfruitfulinanincreasingnumberofapplicationsinprobabilitytheory(Feller(1971,VIII.8andVIII.9)foradetaileddiscussion).ApositiveBorelfunctionLdenedon[0,1)issaidtovaryslowly 35

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(atinnity)(orbeslowlyvarying(atinnity))iflimx!1L(cx) L(x)=1forallc>0.ApositiveBorelfunctionRon[0,1)issaidtovaryregularly(orberegularlyvarying)withexponent(<<1)ifitisoftheformR(x)=xL(x)withLslowlyvarying;i.e.,limx!1R(cx) R(x)=cforallc>0. Clearly,afunctionisslowlyvaryingifandonlyifitisregularlyvaryingwithexponent=0,andapositiveBorelfunctionLisslowlyvaryingifandonlyif1=Lisslowlyvarying.Forexample,allpowersofjlogxjareslowlyvarying.Similarly,afunctionapproachingapositivenitelimitisslowlyvarying. Feller(1971)introducedthefollowingtwoabbreviations: Zu(x)=Zx0yuZ(y)dy,Zu(x)=Z1xyuZ(y)dy,0varyslowly.Thentheintegralsin( 2.5 )convergeat1foru<)]TJ /F6 11.955 Tf 9.29 0 Td[(1anddivergeforu>)]TJ /F6 11.955 Tf 9.3 0 Td[(1.Ifu)]TJ /F6 11.955 Tf 23.47 0 Td[(1,thenZuvariesregularlywithexponentu+1.Ifu<)]TJ /F6 11.955 Tf 9.3 0 Td[(1,thenZuvariesregularlywithexponentu+1,andthisremainstrueforu=)]TJ /F6 11.955 Tf 9.29 0 Td[(1ifZ)]TJ /F11 7.97 Tf 6.58 0 Td[(1<1. Lemma2.3.3(Feller(1971,p.281)). (i) IfZvariesregularlywithexponentandZu<1,thenu++10andlimx!1xu+1Z(x) Zu(x)= where=)]TJ /F6 11.955 Tf 9.3 0 Td[((u++1)0. 36

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(ii) IfZvariesregularlywithexponentandifu)]TJ /F6 11.955 Tf 21.92 0 Td[((+1)thenlimx!1xu+1Z(x) Zu(x)= where=u++10. TheBorel-Cantellilemma(e.g.,ChowandTeicher(1997,p.42))playsanindispensableroleinprobabilitytheoryforestablishinga.c.convergenceresultsandisstatedasfollows.ForasequenceofeventsfAn,n1gwerecallthatlimsupn!1An=1Tn=11Sk=nAkandthatlimsupn!1Anisalsoconvenientlydenotedby[Ani.o.(n)]wherei.o.(n)signiesinnitelyofteninn. Lemma2.3.4(Borel-CantelliLemma). IffAn,n1gisasequenceofeventsforwhich1Pn=1PfAng<1,thenPlimsupn!1An=0or,equivalently,Pnliminfn!1Acno=1. TheclassicalreallineversionoftheKronecker'slemma(e.g.,ChowandTeicher(1997,p.114))carriesovertoaBanachspace(e.g.,Taylor(1978,p.101))andthisBanachspaceversionisstatedasfollows. Lemma2.3.5. Letfvn,n1gbeasequenceofelementsinarealBanachspaceandletfbn,n1gbeasequenceofrealpositivenumberstendingtoinnity.If1Xn=1vn bnconverges,then1 bnnXi=1vi!0. Thefollowinglemma(Lemma 2.3.6 ),theBeppo-LeviTheorem(e.g.,ChowandTeicher(1997,p.90,Corollary2)),isadirectcorollaryoftheMonotoneConvergenceTheorem. Lemma2.3.6(Beppo-Levi). LetfXn,n1gbeasequenceofnonnegativerandomvariableson(,F,P).ThenE 1Xn=1Xn!=1Xn=1EXn. 37

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ForarealseparableBanachspaceXandp2[1,1),itiswellknownthattheclassofrandomelementsVforwhichEkVkp<1formsaBanachspacewithnorm(EkVkp)1=p(e.g.,HilleandPhillips(1985,p.89)).Wethushavethefollowingresult. Lemma2.3.7. LetfVn,n1gbeasequenceofrandomelementsinarealseparableBanachspaceXandletp2[1,1).IfEkVnkp<1,n1 andlimn!1supm>nEkVm)]TJ /F8 11.955 Tf 11.95 0 Td[(Vnkp=0,thenthereexistsarandomelementVinXsuchthatlimn!1EkVn)]TJ /F8 11.955 Tf 11.96 0 Td[(Vkp=0. Remark2.3.2. WhenXistherealline,Lemma 2.3.7 reducestothewell-knownCauchyconvergencecriterionforrandomvariables(e.g.,ChowandTeicher(1997,p.99)).TheproofofLemma 2.3.7 givenbyHilleandPhillips(1985,p.89)followsalongthelinesoftheproofoftheCauchyconvergencecriterionforrandomvariables,mutatismutandis. Thefollowinglemma(Lemma 2.3.8 ),theLevyCentralLimitTheorem(e.g.,ChowandTeicher(1997,p.317)),isadirectcorollaryoftheLindeberg-FellerCentralLimitTheorem(e.g.,ChowandTeicher(1997,p.314)).TogetherwiththeKolmogorovZero-OneLaw(e.g.,ChowandTeicher(1997,p.64)),weobtainLemma 2.3.9 andapplyitinExample 3.2.11 below. Lemma2.3.8(LevyCentralLimitTheorem). LetSn=nPi=1XiwherefXn,n1gisasequenceofi.i.d.randomvariableswithEX1=,Var(X1)=22(0,1).ThenSn)]TJ /F8 11.955 Tf 11.95 0 Td[(n p nd!N(0,1); 38

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i.e.,limn!1PSn)]TJ /F8 11.955 Tf 11.95 0 Td[(n p n0. BytheKolmogorovZero-OneLaw(e.g.,ChowandTeicher(1997,p.64)),Plimsupn!1Sn p nM=1. Therefore,Plimsupn!1Sn p n=1=P(1\M=1limsupn!1Sn p nM)=1.2 Remark2.3.3. Lemma 2.3.9 alsofollowsimmediatelyfromtheHartmanandWintner(1941)lawoftheiteratedlogarithm. 39

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ArandomelementV0issaidtobestochasticallydominatedbyarandomelementVifforsomeconstantD<1,PfkV0k>tgDPfkDVk>tg,t0. (2.6) AsequenceofrandomelementsfVn,n1gissaidtobestochasticallydominatedbyarandomelementVifforsomeconstantD<1,PfkVnk>tgDPfkDVk>tg,t0,n1. (2.7) StochasticdominationoffVn,n1gisofcourseautomaticwithV=V1andD=1iftherandomelementsfVn,n1gareidenticallydistributed.ItfollowsfromLemma5.2.2ofTaylor(1978,p.123)(orLemma3ofWeiandTaylor(1978))thatstochasticdominanceofasequenceofrandomelementsfVn,n1gcanbeaccomplishedbytherandomelementsinthesequencehavingaboundedabsoluterthmoment(r>0).Specically,ifsupn1EkVnkr<1forsomer>0,thenthereexistsarandomelementVwithEkVks<1forall01inLemma5.2.2ofTaylor(1978,p.123)(orLemma3ofWeiandTaylor(1978))isnotneededaswaspointedoutbyAdler,Rosalsky,andTaylor(1992).) Lemma2.3.10(Adler,Rosalsky,andTaylor(1989)). LetV0andVberandomelementsinarealseparableBanachspacesuchthatV0isstochasticallydominatedbyVinthesensethat( 2.6 )holdsforsomeconstantD<1.ThenE[kV0kI(kV0k>t)]DE[kDVkI(kDVk>t)],t0. Lemma2.3.11(AdlerandRosalsky(1987a)). LetV0andVberandomelementsinarealseparableBanachspacesuchthatV0isstochasticallydominatedbyVinthesensethat( 2.6 )holdsforsomeconstantD<1.Thenforallq>0andt0,E[kV0kqI(kV0kt)]DtqPfkDVk>tg+DE[kDVkqI(kDVkt)]. 40

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Lemma2.3.12(AdlerandRosalsky(1987a)). LetfVn,n1gbeasequenceofrandomelementsinarealseparableBanachspaceX.SupposethatfVn,n1gisstochasticallydominatedbyarandomelementVinXinthesensethat( 2.7 )holdsforsomeconstantD<1.Letfcn,n1gbeasequenceofpositiveconstantssuchthatmax1jncpj1Xj=n1 cpj=O(n)forsomep>0 and1Xn=1PfkVk>Dcng<1. (2.8) Thenforall00wherelogedenotesthelogarithmtothebasee. 41

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CHAPTER3STRONGLAWSOFLARGENUMBERSINRADEMACHERTYPEp(1p2)BANACHSPACESFORINDEPENDENTSUMMANDS 3.1Objective WiththepreliminariesaccountedforinChapter2,ourobjectiveinthischapteristoestablishverygeneralSLLNsfornormedweightedsumsofindependentBanachspacevaluedrandomelementswhicharenotnecessarilyidenticallydistributed.TheunderlyingBanachspaceisassumedtobeofRademachertypep(1p2)andthesequenceofrandomelementsisassumedtobestochasticallydominatedbyarandomelement.ThemainresultsthatwillbeestablishedareTheorems 3.2.1 and 3.2.2 ,whicharenewevenwhentheunderlyingBanachspaceistherealline.SpecialcasesofthemainresultsincluderesultsofWoyczynski(1980),Teicher(1985),Adler,Rosalsky,andTaylor(1989),andSung(1997). 3.2MainResults Therstmainresult,Theorem 3.2.1 ,maybepresented.ItsproofwillbegivenafterRemark 3.2.1 andExample 3.2.1 Wenowpresenttherstmainresult,Theorem 3.2.1 ,whichisanewresultwhentheunderlyingBanachspaceisthereallineR.ItsproofwillbegivenafterRemark 3.2.1 andExample 3.2.1 Theorem3.2.1. Let1p2andletfVn,n1gasequenceofindependentrandomelementsinarealseparableRademachertypepBanachspaceX.SupposethatfVn,n1gisstochasticallydominatedbyarandomelementVinthesensethat( 2.7 )holdsforsomeconstantD<1.Letfan,n1gandfbn,n1gbesequencesofconstantssatisfying0
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wherefcn,n1gisasequenceofconstantssatisfying0Dcng<1, (3.4) thenfVn,n1gobeystheSLLNPni=1ai(Vi)]TJ /F8 11.955 Tf 11.95 0 Td[(EVi) bn!0a.c. (3.5) Remark3.2.1. Thefollowingexampleshowsthatconditions( 3.2 )and( 3.3 )areindependentinthesensethattheydonotimplyeachother. Example3.2.1. Let1p2and>0.Letcn=n,n1.Then,forn1,wehavethefollowinginequalities. Ifp>1,thencpn1Xj=nc)]TJ /F5 7.97 Tf 6.59 0 Td[(pj=np1Xj=nj)]TJ /F14 7.97 Tf 6.59 0 Td[(pnpZ1n)]TJ /F11 7.97 Tf 6.59 0 Td[(1x)]TJ /F14 7.97 Tf 6.58 0 Td[(pdx=npx1)]TJ /F14 7.97 Tf 6.58 0 Td[(p 1)]TJ /F9 11.955 Tf 11.95 0 Td[(p1x=n)]TJ /F11 7.97 Tf 6.58 0 Td[(1Cnpn1)]TJ /F14 7.97 Tf 6.59 0 Td[(p=O(n). 43

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Ifp1,thencpn1Xj=nc)]TJ /F5 7.97 Tf 6.58 0 Td[(pj=np1Xj=nj)]TJ /F14 7.97 Tf 6.59 0 Td[(p=16=O(n). If0<<1,thenforn2,cnnXj=1c)]TJ /F11 7.97 Tf 6.58 0 Td[(1j=n+nnXj=2j)]TJ /F14 7.97 Tf 6.59 0 Td[(n+nZn1x)]TJ /F14 7.97 Tf 6.59 0 Td[(dx=n+nx1)]TJ /F14 7.97 Tf 6.58 0 Td[( 1)]TJ /F9 11.955 Tf 11.95 0 Td[(nx=1=n+n 1)]TJ /F9 11.955 Tf 11.96 0 Td[()]TJ /F8 11.955 Tf 21.15 8.08 Td[(n 1)]TJ /F9 11.955 Tf 11.95 0 Td[(=O(n). If=1,thencnnXj=1c)]TJ /F11 7.97 Tf 6.59 0 Td[(1j=nnXj=1j)]TJ /F11 7.97 Tf 6.59 0 Td[(1nZn+111 xdx=nlogxn+1x=1=nlog(n+1) andsocnnXj=1c)]TJ /F11 7.97 Tf 6.58 0 Td[(1j6=O(n). If>1,thencnnXj=1c)]TJ /F11 7.97 Tf 6.58 0 Td[(1j=nnXj=1j)]TJ /F14 7.97 Tf 6.59 0 Td[(nZn+11x)]TJ /F14 7.97 Tf 6.59 0 Td[(dx=nx1)]TJ /F14 7.97 Tf 6.58 0 Td[( 1)]TJ /F9 11.955 Tf 11.95 0 Td[(n+1x=1=n (1)]TJ /F9 11.955 Tf 11.96 0 Td[()(n+1))]TJ /F11 7.97 Tf 6.59 0 Td[(1+n )]TJ /F6 11.955 Tf 11.96 0 Td[(1 andsocnnXj=1c)]TJ /F11 7.97 Tf 6.58 0 Td[(1j6=O(n). Insummary,wehavethefollowingfourcases: 44

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(i) Bothconditions( 3.2 )and( 3.3 )holdifp>1and0<<1;i.e.,1=p<<1and11and1;i.e.,>1ifp=1or1if10andforalln1,EkVnk CD21Xi=0PVn CD2>i1Xi=0PfkVnk>D2cig(sincecn=O(n))1Xi=0DPfkVk>Dcig(by( 2.7 ))<1(by( 3.4 )), whichimpliesby3.ofProposition 2.2.2 thatthefVn,n1gallhaveexpectedvalues. DeneWn=VnI(kVnkD2cn),n1.Weshallprovethefollowingthreestatements: (i) P1n=1PfVn6=Wng<1. (ii) Pni=1ai(Wi)]TJ /F8 11.955 Tf 11.96 0 Td[(EWi) bn!0a.c. 45

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(iii) Pni=1ai(EWi)]TJ /F8 11.955 Tf 11.96 0 Td[(EVi) bn!0. Weprove(i)asfollows.Notethat1Xn=1PfVn6=Wng=1Xn=1PfkVnk>D2cngD1Xn=1PfkVk>Dcng(by( 2.7 ))<1(by( 3.4 )). Weprove(ii)asfollows.SincefVn,n1gandfcn,n1gsatisfytheconditionsofLemma 2.3.12 ,1Xn=1EkWnkp cpn=1Xn=11 cpnE[kVnkpI(kVnkD2cn)]<1. (3.6) Thus,forn1,supm>nEmXi=1ai(Wi)]TJ /F8 11.955 Tf 11.95 0 Td[(EWi) bi)]TJ /F5 7.97 Tf 18.31 14.94 Td[(nXi=1ai(Wi)]TJ /F8 11.955 Tf 11.95 0 Td[(EWi) bip=supm>nEmXi=n+1ai(Wi)]TJ /F8 11.955 Tf 11.96 0 Td[(EWi) bipCsupm>nmXi=n+1ai bipEk(Wi)]TJ /F8 11.955 Tf 11.96 0 Td[(EWi)kp(sinceXisofRademachertypep)C2psupm>nmXi=n+1EkWikp cpi(by( 3.1 ))=C2p1Xi=n+1EkWikp cpi=o(1)(by( 3.6 )). Therefore,byLemma 2.3.7 ,EnXi=1ai(Wi)]TJ /F8 11.955 Tf 11.95 0 Td[(EWi) bi)]TJ /F8 11.955 Tf 11.96 0 Td[(Sp!0 46

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forsomeX-valuedrandomelementS.Thus,nXi=1ai(Wi)]TJ /F8 11.955 Tf 11.96 0 Td[(EWi) biP!Swhich,byLemma 2.3.1 ,impliesnXi=1ai(Wi)]TJ /F8 11.955 Tf 11.96 0 Td[(EWi) bi!Sa.c.Henceweobtain(ii)viaLemma 2.3.5 Weprove(iii)asfollows.Notethat1Xn=1kan(EWn)]TJ /F8 11.955 Tf 11.96 0 Td[(EVn)k bn1Xn=1janj bnE[kVnkI(kVnk>D2cn)]D21Xn=1janj bnE[kVkI(kVk>Dcn)](byLemma 2.3.10 )C1Xn=11 cnE[kVkI(kVk>Dcn)](by( 3.1 ))=C1Xn=11 cn1Xi=nE[kVkI(DciDcng<1(by( 3.4 )). 47

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Then,byLemma 2.3.5 intherandomvariablecase(thereallineversionoftheKroneckerlemma),kPni=1aiE(Vi)]TJ /F8 11.955 Tf 11.95 0 Td[(Wi)k bnPni=1kaiE(Vi)]TJ /F8 11.955 Tf 11.95 0 Td[(Wi)k bn!0proving(iii). Since(i)ensuresthatPfliminfn!1[Vn=Wn]g=1byLemma 2.3.4 ,thenwehavefrom(ii)thatPni=1ai(Vi)]TJ /F8 11.955 Tf 11.95 0 Td[(EWi) bn!0a.c.Combiningthiswith(iii)yieldstheSLLN( 3.5 ).2 Remark3.2.2. InTheorem 3.2.1 ,thereisatrade-offbetweentheRademachertypeandcondition( 3.2 );thelargerpis,amorestringentconditionisimposedontheBanachspaceXwhereascondition( 3.2 )becomeslessstringent.Toseethis,supposethat( 3.2 )holdsforsomep2[1,2)andletp02(p,2].Since0rforsomeintegerr2. Notethatstrictinequalityappearsinthiscondition. 48

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TherstcorollaryofTheorem 3.2.1 isthefollowingproposition,Proposition 3.2.1 ,whichisTheorem1ofAdler,Rosalsky,andTaylor(1989)andisoneofthemainresultsofthatarticle. Proposition3.2.1(Adler,Rosalsky,andTaylor(1989,Theorem1)). Let1p2andletfVn,n1gasequenceofindependentrandomelementsinarealseparableRademachertypepBanachspaceX.SupposethatfVn,n1gisstochasticallydominatedbyarandomelementVinthesensethat( 2.7 )holdsforsomeconstantD<1.Letfan,n1gandfbn,n1gbesequencesofconstantssatisfying0Dbng<1.ThentheSLLN( 3.5 )holds. Proof.Takecn=bn=an,n1inTheorem 3.2.1 .Thecorollaryfollowsimmediately.2 Wenowpresenttwoillustrativeexamples,Examples 3.2.2 and 3.2.3 ,toshowthatTheorem 3.2.1 canfailiftheRademachertypephypothesisisdispensedwith.RecallthattherealseparableBanachspacesc0and`1arenotofRademachertypepforany1
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Denen=n(1)]TJ /F5 7.97 Tf 6.59 0 Td[(q)=q,n1.Thenn#0.Letfk,k1gbeasequenceofindependentrandomvariableswithdistributionsgivenby1=...=7=0a.c.andPfk=1g=Pfk=)]TJ /F6 11.955 Tf 9.3 0 Td[(1g=1 2(1)]TJ /F8 11.955 Tf 11.95 0 Td[(Pfk=0g)=1 logk,k8. Fork1,denek=p nwherenissuchthat2n)]TJ /F11 7.97 Tf 6.58 0 Td[(1k<2nandtakeVtobetherandomelementinc0withcoordinates(kk)k1.ThenVisclearlyalmostcertainlyboundedandhaszeroexpectation.LetfVn,n1gbeasequenceofi.i.d.copiesofV.Thenconditions( 3.1 )and( 3.4 )hold.Bothconditions( 3.2 )and( 3.3 )holdbyExample 3.2.1 (i)taking=1=q.Insummary,alloftheconditionsofTheorem 3.2.1 aresatisedexceptforXbeingofRademachertypep.However,inExample7.11ofLedouxandTalagrand(1991,p.190),itisshownthatPni=1Vi=nnP90,whichimpliesPni=1(Vi)]TJ /F8 11.955 Tf 11.95 0 Td[(EVi) n1=q90a.c. (3.8) Hence,theSLLN( 3.5 )fails. Remark3.2.4. Notethatitfollowsimmediatelyfrom( 3.8 )andq
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Forn1,letv(n)betheelementof`1having1initsnthpositionand0elsewhere.DeneasequenceofindependentrandomelementsfVn,n1gin`1byrequiringthefVn,n1gtobeindependentwithPfVn=v(n)g=PfVn=)]TJ /F8 11.955 Tf 9.29 0 Td[(v(n)g=1=2,n1. ThenfVn,n1gisstochasticallydominatedinthesensethat( 2.7 )holdswithV=V1andD=1,butisnotcomprisedofidenticallydistributedrandomelements.ClearlyV1satisescondition( 3.4 ).AsinExample 3.2.2 ,alloftheconditionsofTheorem 3.2.1 aresatisedexceptforXbeingofRademachertypep.Furthermore,sinceq>1,kPni=1Vik1 n1=q=n1)]TJ /F19 5.978 Tf 7.91 3.25 Td[(1 q!16=0a.c.Hence,theSLLN( 3.5 )fails. Remark3.2.5. IfinExample 3.2.3 weinsteadtaketherealseparableBanachspaceXtobe`q(whichisnotofRademachertypep),thenarguingasinExample 3.2.3 weobtainkPni=1Vikq n1=q=n1=q n1=q=190a.c. Hence,theSLLN( 3.5 )fails. Thefollowingexample,Example 3.2.4 ,showsthatTheorem 3.2.1 canfailifcondition( 3.4 )doesnothold. Example3.2.4. LetfXn,n1gbeasequenceofi.i.d.randomvariableswithX12L1butX162Lqforsomeq2(1,2).Letp2(q,2]anddeneasequenceofindependentrandomelementsfVn,n1gin`p(whichisofRademachertypep)byVn=(Xn,0,0,...),n1.ThenfVn,n1gisstochasticallydominatedinthesensethat( 2.7 )holdswithV=V1andD=1.Nowforeachn1,theexpectedvalueofVnexistssinceEkVnkp= 51

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E(jXnjp)1=p=EjXnj<1.Letan=1,bn=cn=n1=q,n1.Thencondition( 3.1 )clearlyholdsandconditions( 3.2 )and( 3.3 )holdbyExample 3.2.1 (i).NotethatEkV1kq=EjX1jq=1andso1Xn=1PfkV1k>n1=qg=1;thatis,condition( 3.4 )fails.Thus,allofthehypothesesofTheorem 3.2.1 aresatisedexceptfor( 3.4 ).Notethatforalln1,Pni=1(Vi)]TJ /F8 11.955 Tf 11.96 0 Td[(EVi) n1=q=(Pni=1(Xi)]TJ /F8 11.955 Tf 11.95 0 Td[(EXi),0,0,...) n1=qandsoPni=1(Vi)]TJ /F8 11.955 Tf 11.95 0 Td[(EVi) n1=qp=(jPni=1(Xi)]TJ /F8 11.955 Tf 11.95 0 Td[(EXi)jp)1=p n1=q=jPni=1(Xi)]TJ /F8 11.955 Tf 11.95 0 Td[(EXi)j n1=q90a.c. bythereallineversionoftheMarcinkiewicz-ZygmundSLLNrecallingthatEjX1jq=1.Hence,theSLLN( 3.5 )fails. Thefollowingexample,Example 3.2.5 ,showsthatthehypothesesofTheorem 3.2.1 aresatisedbutthoseofProposition 3.2.1 (Theorem1ofAdler,Rosalsky,andTaylor(1989))arenotsatised.Consequently,Theorem 3.2.1 isabonadeimprovementofProposition 3.2.1 (Theorem1ofAdler,Rosalsky,andTaylor(1989)). Example3.2.5. Let1
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elementVinthesensethat( 2.7 )holdsandEkVk1=<1.Then,1Xn=1PfkVk>Dcng=1Xn=1PV D>cn=1Xn=1PV D>n=1Xn=1P(V D1=>n)EV D1=<1. Thus,condition( 3.4 )holds.Since)]TJ /F9 11.955 Tf 11.96 0 Td[(<0,itfollowsthatcnjanj bn=n)]TJ /F14 7.97 Tf 6.59 0 Td[(=o(1),whichimpliesthatcondition( 3.1 )holds.Furthermore,byExample 3.2.1 (i),bothconditions( 3.2 )and( 3.3 )hold.Insummary,alloftheconditionsofTheorem 3.2.1 aresatised. Ontheotherhand,if>1,thenbn janjnXj=1jajj bj=nnXj=1j)]TJ /F14 7.97 Tf 6.59 0 Td[(6=O(n).whereasif=1,thenbn janjnXj=1jajj bj=nnXj=1j)]TJ /F11 7.97 Tf 6.59 0 Td[(1=(1+o(1))nlogn6=O(n).Therefore,condition( 3.7 )ofProposition 3.2.1 fails.Consequently,Theorem 3.2.1 ensuresthatfVn,n1gobeystheSLLN( 3.5 )whereasProposition 3.2.1 (Theorem1ofAdler,Rosalsky,andTaylor(1989))isnotapplicableforthisexample. ThesecondcorollaryofTheorem 3.2.1 isthefollowingtheorem,Theorem 3.2.2 ,whichisanimprovedversionofTheorem6ofAdler,Rosalsky,andTaylor(1989)aswillbediscussedindetailinRemark 3.2.6 below.Moreover,Theorem 3.2.2 isanewresultwhentheunderlyingBanachspaceisthereallineR. 53

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Theorem3.2.2. Let1q,and( 3.3 )holdssinceq>1.Theconclusion( 3.5 )followsimmediatelyfromTheorem 3.2.1 .2 Thefollowingproposition,Proposition 3.2.2 ,whichisTheorem2ofSung(1997)andisoneofthemainresultsofthatarticle,isadirectcorollaryofTheorem 3.2.2 Proposition3.2.2(Sung(1997,Theorem2)). Let1
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Remark3.2.6. (i) Theorem 3.2.2 wasprovedbyAdler,Rosalsky,andTaylor(1989,Theorem6)usingtheadditionalconditionnXi=1jaij=O(bn). (3.10) Theorem 3.2.2 isalsovalidforq=1providedcondition( 3.10 )holdsaswasprovedbyAdler,Rosalsky,andTaylor(1989)intheirTheorem6.ThefollowingexampleofSung(1997)showsthatcondition( 3.10 )cannotbedispensedwithwhenq=1: LettheunderlyingrealseparableBanachspaceXbeRwhichisofRademachertypep=2.LetfVn,n1gbeasequenceofi.i.d.randomvariableswithV1havingprobabilitydensityfunctionf(x)=c x2(logx)2I[2,1)(x),
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(v) Example 3.2.2 alsodemonstratesthatProposition 3.2.1 ,Theorem 3.2.2 ,andtheMarcinkiewicz-ZygmundtypeSLLNofWoyczynski(1980)((iv)above)canfailiftheRademachertypep(12since3=2<<2.Letan=logn,bn=n)]TJ /F11 7.97 Tf 6.59 0 Td[(1,andcn=n)]TJ /F11 7.97 Tf 6.59 0 Td[(1 log(n+1),n1.Thencondition( 3.1 )and03=2.Then,byLemma 2.3.3 (i),limx!1xu+1Z(x) Zu(x)=)]TJ /F6 11.955 Tf 9.3 0 Td[((u++1)=)]TJ /F8 11.955 Tf 9.3 0 Td[(u)]TJ /F6 11.955 Tf 11.96 0 Td[(1=2)]TJ /F6 11.955 Tf 11.95 0 Td[(3>0. Thus,forn2,cpn1Xj=nc)]TJ /F5 7.97 Tf 6.59 0 Td[(pj=n2()]TJ /F11 7.97 Tf 6.59 0 Td[(1) (log(n+1))21Xj=n(log(j+1))2 j2()]TJ /F11 7.97 Tf 6.59 0 Td[(1)n2()]TJ /F11 7.97 Tf 6.59 0 Td[(1) (log(n+1))2Z1n)]TJ /F11 7.97 Tf 6.58 0 Td[(1(log(y+1))2 y2()]TJ /F11 7.97 Tf 6.58 0 Td[(1)dy=Zu(n)]TJ /F6 11.955 Tf 11.95 0 Td[(1) nuZ(n)=(1+o(1))(n)]TJ /F6 11.955 Tf 11.95 0 Td[(1)u+1Z(n)]TJ /F6 11.955 Tf 11.95 0 Td[(1) (2)]TJ /F6 11.955 Tf 11.95 0 Td[(3)nuZ(n)=O(n) 56

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therebyproving( 3.2 ). Similarly,letZ(x)=logxandu=1)]TJ /F9 11.955 Tf 12.15 0 Td[(.ThenZisslowlyvarying;thatis,Zvariesregularlywithexponent=0.DeneZu(x)asin( 2.5 ).Now<2ensuresthatu>)]TJ /F6 11.955 Tf 9.29 0 Td[(1andsobyLemma 2.3.3 (ii),limx!1xu+1Z(x) Zu(x)=u++1=u+1=2)]TJ /F9 11.955 Tf 11.96 0 Td[(>0. Thus,cnnXj=1c)]TJ /F11 7.97 Tf 6.58 0 Td[(1j=n)]TJ /F11 7.97 Tf 6.59 0 Td[(1 log(n+1)nXj=1log(j+1) j)]TJ /F11 7.97 Tf 6.58 0 Td[(1n)]TJ /F11 7.97 Tf 6.59 0 Td[(1 log(n+1)Zn0log(y+1) y)]TJ /F11 7.97 Tf 6.59 0 Td[(1dy=Zu(n) nuZ(n+1)=(1+o(1))nu+1Z(n) (2)]TJ /F9 11.955 Tf 11.96 0 Td[()nuZ(n+1)=O(n) therebyproving( 3.3 ). Furthermore,weprovethatcondition( 3.4 )holdswithD=1asfollows.LetZ(x)=(logx))]TJ /F11 7.97 Tf 6.58 0 Td[(3andu=)]TJ /F9 11.955 Tf 9.3 0 Td[(=()]TJ /F6 11.955 Tf 12.92 0 Td[(1).ThenZisslowlyvarying;thatis,Zvariesregularlywithexponent=0.DeneZu(x)asin( 2.5 ).ByLemma 2.3.2 ,Zu(x)<1sinceu<)]TJ /F6 11.955 Tf 9.29 0 Td[(1.Then,byLemma 2.3.3 (i),limx!1xu+1Z(x) Zu(x)=)]TJ /F6 11.955 Tf 9.3 0 Td[((u++1)=)]TJ /F8 11.955 Tf 9.29 0 Td[(u)]TJ /F6 11.955 Tf 11.96 0 Td[(1=1 )]TJ /F6 11.955 Tf 11.95 0 Td[(1>0. Thus,PfjV1j>xg=2Z1xf(y)dy=2cZ1xyuZ(y)dy=2cZu(x)=(1+o(1))2c()]TJ /F6 11.955 Tf 11.95 0 Td[(1)xu+1Z(x) 57

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whichimpliesPfjV1j>xg=(1+o(1))2c()]TJ /F6 11.955 Tf 11.96 0 Td[(1) x1 )]TJ /F19 5.978 Tf 5.76 0 Td[(1(logx)3asx!1. (3.12) Thenby( 3.12 ),PfjV1j>cng=(1+o(1))2c()]TJ /F6 11.955 Tf 11.96 0 Td[(1)(log(n+1))1 )]TJ /F19 5.978 Tf 5.76 0 Td[(1 n[()]TJ /F6 11.955 Tf 11.96 0 Td[(1)logn)]TJ /F6 11.955 Tf 11.96 0 Td[(loglog(n+1)]3=(1+o(1))2c()]TJ /F6 11.955 Tf 11.95 0 Td[(1)(logn)1 )]TJ /F19 5.978 Tf 5.75 0 Td[(1 n()]TJ /F6 11.955 Tf 11.95 0 Td[(1)3(logn)3=(1+o(1))2c ()]TJ /F6 11.955 Tf 11.96 0 Td[(1)2n(logn)3)]TJ /F19 5.978 Tf 5.75 0 Td[(4 )]TJ /F19 5.978 Tf 5.76 0 Td[(1 andsocondition( 3.4 )holdswithV=V1andD=1since>3=2(hence(3)]TJ /F6 11.955 Tf 11.99 0 Td[(4)=()]TJ /F6 11.955 Tf -457.64 -23.91 Td[(1)>1).Then,byTheorem 3.2.1 ,theSLLNPni=1(logi)Xi n)]TJ /F11 7.97 Tf 6.59 0 Td[(1!0a.c.holds. ToshowthatthehypothesesofTheorem 3.2.2 arenotsatised,notethatcondition( 3.9 )holdsforsomeq2(1,2)ifandonlyif1=q<)]TJ /F6 11.955 Tf 12.25 0 Td[(1;thatis,q>()]TJ /F6 11.955 Tf 12.24 0 Td[(1))]TJ /F11 7.97 Tf 6.59 0 Td[(1.Ontheotherhand,by( 3.12 ),1Xn=1PfjV1j>n1=qg=1Xn=1(1+o(1))2c()]TJ /F6 11.955 Tf 11.95 0 Td[(1)q3 n1 q()]TJ /F19 5.978 Tf 5.75 0 Td[(1)(logn)3<1ifandonlyif(q()]TJ /F6 11.955 Tf 12.69 0 Td[(1)))]TJ /F11 7.97 Tf 6.58 -.01 Td[(11;thatis,q()]TJ /F6 11.955 Tf 12.69 0 Td[(1))]TJ /F11 7.97 Tf 6.59 -.01 Td[(1.Therefore,inTheorem 3.2.2 ,condition( 3.9 )andtheconditionthatEjVjq<1forsome1
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Proposition3.2.3(Adler,Rosalsky,andTaylor(1989,Theorem3)). Let1p2andletfVn,n1gasequenceofindependentrandomelementsinarealseparableRademachertypepBanachspaceX.SupposethatfVn,n1gisstochasticallydominatedbyarandomelementVinthesensethat( 2.7 )holdsforsomeconstantD<1,andthatPfkVk>xgisregularlyvaryingwithexponent<)]TJ /F6 11.955 Tf 9.3 0 Td[(1. Letfan,n1gandfbn,n1gbesequencesofconstantssatisfying0Dbng<1, (3.14) thentheSLLN( 3.5 )holds. Example3.2.7. Let1<>:bn 2n=pfornodd,bn 4n=pforneven. Then( 3.9 )holdssincean bn1 2n=p=O1 n1=q. 59

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Insummary,alloftheconditionsofTheorem 3.2.2 aresatised.However,forneven,max1jn+1bj jajj n+11Xj=n+1jajj bj4n=p n+11 2(n+1)=p=2n)]TJ /F19 5.978 Tf 5.75 0 Td[(1 p n+1!1asnevenapproaches1. Therefore,condition( 3.13 )ofProposition 3.2.3 (Theorem3ofAdler,Rosalsky,andTaylor(1989))fails. Thefollowingexample,Example 3.2.8 ,showsthattheconditionsofProposition 3.2.3 (Theorem3ofAdler,Rosalsky,andTaylor(1989))aresatisedbutthoseofTheorem 3.2.2 arenotsatised. Example3.2.8. LettheunderlyingrealseparableBanachspaceXbeRwhichisofRademachertypep=2andletfVn,n1gbeasequenceofi.i.d.randomvariableswithV1havingprobabilitydensityfunctionf(x)=c jxj1+(logjxj)3I[e,1)(jxj),xg=2Z1xf(y)dy=2cZ1xyuZ(y)dy=2cZu(x). (3.15) Therefore,PfjV1j>xgisregularlyvaryingwithexponent=)]TJ /F9 11.955 Tf 9.3 0 Td[(<)]TJ /F6 11.955 Tf 9.3 0 Td[(1.Nowletan=log(n+1),bn=n1=,n1. 60

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Weprovethatcondition( 3.14 )holdswithD=1asfollows.ByLemma 2.3.3 (i),limx!1xu+1Z(x) Zu(x)=)]TJ /F6 11.955 Tf 9.3 0 Td[((u+1)=>0. Thenby( 3.15 ),PfjV1j>xg=2cZu(x)=(1+o(1))2cxu+1Z(x) )]TJ /F8 11.955 Tf 9.29 0 Td[(u)]TJ /F6 11.955 Tf 11.96 0 Td[(1=(1+o(1))2c x(logx)3 (3.16) andsoPfjanV1j>Dbng=PjV1j>n1= log(n+1)=(1+o(1))2c(log(n+1)) n[1 logn)]TJ /F6 11.955 Tf 11.95 0 Td[(loglog(n+1)]3=(1+o(1))2c(logn) n)]TJ /F11 7.97 Tf 6.58 0 Td[(3(logn)3=(1+o(1))2c2 n(logn)3)]TJ /F14 7.97 Tf 6.59 0 Td[(. Thuscondition( 3.14 )holdssince3)]TJ /F9 11.955 Tf 11.96 0 Td[(>1. Furthermore,weprovethatcondition( 3.13 )holdsasfollows.LetZ(x)=(log(x+1))2andu=)]TJ /F6 11.955 Tf 9.3 0 Td[(2=.ThenZisslowlyvarying;thatis,Zvariesregularlywithexponent=0.DeneZu(x)asin( 2.5 ).ByLemma 2.3.2 ,Zu(x)<1since<2.Then,byLemma 2.3.3 (i),limx!1xu+1Z(x) Zu(x)=)]TJ /F6 11.955 Tf 9.3 0 Td[((u++1)=)]TJ /F8 11.955 Tf 9.29 0 Td[(u)]TJ /F6 11.955 Tf 11.96 0 Td[(1=2)]TJ /F9 11.955 Tf 11.96 0 Td[( >0. Thus,forn2,max1jnbpj jajjp1Xj=njajjp bpj=n2= (log(n+1))21Xj=n(log(j+1))2 j2=n2= (log(n+1))2Z1n)]TJ /F11 7.97 Tf 6.59 0 Td[(1(log(y+1))2 y2=dy 61

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=Zu(n)]TJ /F6 11.955 Tf 11.95 0 Td[(1) nuZ(n)=(1+o(1))(2)]TJ /F9 11.955 Tf 11.95 0 Td[())]TJ /F11 7.97 Tf 6.59 0 Td[(1(n)]TJ /F6 11.955 Tf 11.95 0 Td[(1)u+1Z(n)]TJ /F6 11.955 Tf 11.95 0 Td[(1) nuZ(n)=O(n) therebyproving( 3.13 ).Insummary,alloftheconditionsofProposition 3.2.3 (Theorem3ofAdler,Rosalsky,andTaylor(1989))aresatised. ToshowthatthehypothesesofTheorem 3.2.2 arenotsatised,notethatcondition( 3.9 )holdsforsomeq2(1,2)ifandonlyifq>.Ontheotherhand,by( 3.16 ),1Xn=1PfjV1j>n1=qg=1Xn=1(1+o(1))2cq3 n=q(log(n+1))3<1 ifandonlyifq.Therefore,inTheorem 3.2.2 ,condition( 3.9 )andtheconditionthatEjVjq<1forsome1
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where1xgisregularlyvaryingwithexponent<)]TJ /F6 11.955 Tf 9.29 0 Td[(1.Letan=1,bn=n1=q,n1.Thencondition( 3.13 )clearlyholdsbyExample 3.2.1 (i)taking=1=qandp=2.Furthermore,by( 3.16 ),PfjanV1j>bng=PfjV1j>n1=qg=(1+o(1))2cq2 n(logn)3 implyingcondition( 3.14 )holdswithD=1.Insummary,alloftheconditionsofProposition 3.2.3 (Theorem3ofAdler,Rosalsky,andTaylor(1989))aresatised.Clearly,condition( 3.9 )holdsandEjV1jq<1.Hence,alloftheconditionsofTheorem 3.2.2 alsohold.Consequently,Theorem 3.2.2 andProposition 3.2.3 (Theorem3ofAdler,Rosalsky,andTaylor(1989))caneachbeemployedtoestablishtheSLLN( 3.5 ). Thefollowingtwoexamples,Examples 3.2.10 and 3.2.11 ,showthatTheorem 3.2.2 issharpinthesensethatitcanfailifcondition( 3.9 )isweakenedtoan bn=O1 n1=p. (3.17) Intherstexample,Example 3.2.10 ,therealseparableBanachspaceis`p(1
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ThenfVn,n1gisstochasticallydominatedinthesensethat( 2.7 )holdswithV=V1andD=1.Clearly,wehaveEkV1kqp=1<1foreachq2(1,p).Moreover,condition( 3.17 )holdsbutthestrongercondition( 3.9 )failsforallq2(1,p).However,kPni=1Vikp n1=p=n1=p n1=p=190a.c. (3.18) Hence,theSLLN( 3.5 )fails.Ontheotherhand,itfollowsimmediatelyfrom( 3.18 )thatforall0
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whereasinExample 3.2.11 limsupn!1jPni=1ai(Vi)]TJ /F8 11.955 Tf 11.96 0 Td[(EVi)j bn=1a.c. Thenextcorollary,Corollary 3.2.1 ,extendsbothTheorem5ofTeicher(1985)andCorollary2ofSung(1997).TheargumentispatternedafterthatofCorollary2ofSung(1997). Corollary3.2.1. LetfVn,n1gbeasequenceofindependentrandomelementstakingvaluesinarealseparableRademachertypep(11,theconclusion( 3.22 )followsfromTheorem 3.2.2 .2 ThefollowingexampleillustratesCorollary 3.2.1 65

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Example3.2.12. LetfVn,n1gbeasequenceofindependentrandomelementstakingvaluesinarealseparableRademachertypep(1)]TJ /F8 11.955 Tf 9.3 0 Td[(q)]TJ /F11 7.97 Tf 6.59 0 Td[(1,Pni=1i)]TJ /F11 7.97 Tf 6.58 0 Td[(1=q(logi)(Vi)]TJ /F8 11.955 Tf 11.95 0 Td[(EVi) (logn)+q)]TJ /F19 5.978 Tf 5.75 0 Td[(1!0a.c. (3.24) (iii) For<
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Theconclusion( 3.24 )followsfromCorollary 3.2.1 (iii) Letan=n)]TJ /F14 7.97 Tf 6.58 0 Td[((logn),n1.ThennXi=1jaijq=nXi=1i)]TJ /F5 7.97 Tf 6.59 0 Td[(q(logi)q(logn)qn1)]TJ /F5 7.97 Tf 6.58 0 Td[(q 1)]TJ /F8 11.955 Tf 11.95 0 Td[(q byLemma 2.3.3 (ii).Then( 3.21 )holdssincejanjq Pni=1jaijqn)]TJ /F5 7.97 Tf 6.59 0 Td[(q(logn)q(1)]TJ /F8 11.955 Tf 11.96 0 Td[(q) (logn)qn1)]TJ /F5 7.97 Tf 6.59 0 Td[(q=O1 n. Theconclusion( 3.25 )followsfromCorollary 3.2.1 .2 67

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CHAPTER4STRONGLAWSOFLARGENUMBERSFORRANDOMELEMENTSINGENERALBANACHSPACESIRRESPECTIVEOFTHEIRJOINTDISTRIBUTIONS 4.1Objective OurobjectiveinthischapteristoobtainSLLNsirrespectiveofthejointdistributionsoftherandomelementswhereinadditionnogeometricconditionsareimposedontheunderlyingBanachspace.Wewillestablishfourtheorems(Theorems 4.2.1 4.2.2 4.2.3 ,and 4.2.4 )allofwhichhavetheassumptionthatthesequenceofrandomelementsisstochasticallydominatedbyarandomelement.Theorem 4.2.1 isa0
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If1Xn=1PfkanVk>Dbng<1, (4.2) thentheSLLNPni=1aiVi bn!0a.c. (4.3) holdsirrespectiveofthejointdistributionsofthefVn,n1g. TherstmainresultofChapter4,isTheorem 4.2.1 ,whichisa0
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InLemma 2.3.12 ,letcn=n1=qforn1andletp=1.Notethatcpn=n=n1=q)]TJ /F11 7.97 Tf 6.59 0 Td[(1"andliminfn!1cp2n cpn=liminfn!1(2n)1=q n1=q=21=q>2since1=q>1.Thenmax1jncpj1Xj=n1 cpj=O(n)byRemark 3.2.3 takingr=2.Moreover,condition( 2.8 )holdssinceEkVkq<1.Thus,byLemma 2.3.12 ,1Xn=11 cnE[kVnkI(kVnkD2cn)]<1. Therefore,E 1Xn=1kanWnk bn!=1Xn=1EkanWnk bn(byLemma 2.3.6 )C1Xn=11 n1=qE[kVnkI(kVnkD2n1=q)](by( 4.4 ))=C1Xn=11 cnE[kVnkI(kVnkD2cn)]<1 whence1Xn=1kanWnk bn<1a.c. ThenbyLemma 2.3.5 intherandomvariablecase(thereallineversionoftheKroneckerlemma),kPni=1aiWik bnPni=1kaiWik bn!0a.c.andsoPni=1aiWi bn!0a.c. (4.5) 70

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Notethat1Xn=1PfVn6=Wng=1Xn=1PfkVnk>D2n1=qgD1Xn=1PfkVk>Dn1=qg(by( 2.7 ))<1(sinceEkVkq<1). ThenPfliminfn!1[Vn=Wn]g=1byLemma 2.3.4 .Hence,inviewof( 4.5 ),weobtaintheSLLN( 4.3 ).2 Thefollowingexample,Example 4.2.1 ,showsthatTheorem 4.2.1 issharpinthesensethatitcanfailifcondition( 4.4 )isweakenedtoan bn=O1 n1=pforsomep>q. (4.6) Incidentally,Example 4.2.1 alsohasthesamemoralasExample 3.2.11 whichconcernedasequenceofindependentrandomvariables. Example4.2.1. Let0
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Thefollowingcorollary,Corollary 4.2.1 ,isadirectcorollaryofTheorem 4.2.1 andisanewversionofTheorem5ofTeicher(1985)andCorollary2ofSung(1997)whichpertainedtosequencesofi.i.d.Lprandomvariableswhere1p<2. Corollary4.2.1. LetfVn,n1gbeasequenceofrandomelementsinarealseparableBanachspace.SupposethatfVn,n1gisstochasticallydominatedbyarandomelementVinthesensethat( 2.7 )holdsforsomeconstantD<1andthatEkVkq<1forsome0
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(ii) For>)]TJ /F6 11.955 Tf 9.3 0 Td[(1=q,Pni=1i)]TJ /F11 7.97 Tf 6.59 0 Td[(1=q(logi)Vi (logn)1=q+!0a.c. (iii) For<<1=qand<<1,Pni=1i)]TJ /F14 7.97 Tf 6.58 0 Td[((logi)Vi (logn)n1=q)]TJ /F14 7.97 Tf 8 0 Td[(!0a.c. Proof. (i) InCorollary 4.2.1 ,letdn1andan=(nlogn))]TJ /F11 7.97 Tf 6.59 0 Td[(1=q,n1.Then1Xn=1janjq=1Xn=11 nlogn=1andan )]TJ 5.47 -.71 Td[(Pni=1jaijq1=q=(nlogn))]TJ /F11 7.97 Tf 6.58 0 Td[(1=q (Pni=1(ilogi))]TJ /F11 7.97 Tf 6.59 0 Td[(1)1=q(nlogn))]TJ /F11 7.97 Tf 6.59 0 Td[(1=q (loglogn)1=q=Odn n1=q. ThustheconclusionfollowsimmediatelyfromCorollary 4.2.1 (ii) InCorollary 4.2.1 ,letdn1andan=n)]TJ /F11 7.97 Tf 6.59 0 Td[(1=q(logn),n1.Then1Xn=1janjq=1Xn=1(logn)q n=1sinceq>)]TJ /F6 11.955 Tf 9.3 0 Td[(1.Moreover,an )]TJ 5.48 -.72 Td[(Pni=1jaijq1=q=n)]TJ /F11 7.97 Tf 6.58 0 Td[(1=q(logn) )]TJ 5.48 -.72 Td[(Pni=1i)]TJ /F11 7.97 Tf 6.58 0 Td[(1(logi)q1=qn)]TJ /F11 7.97 Tf 6.59 0 Td[(1=q(logn)(q+1)1=q (logn)1=q+=Odn n1=q. ThustheconclusionagainfollowsimmediatelyfromCorollary 4.2.1 73

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(iii) InCorollary 4.2.1 ,letdn1andan=n)]TJ /F14 7.97 Tf 6.59 0 Td[((logn),n1.Then1Xn=1janjq=1Xn=1(logn)q nq=1sinceq<1.NotethatnXi=1jaijq=nXi=1(logi)q iq(logn)qn1)]TJ /F5 7.97 Tf 6.59 0 Td[(q 1)]TJ /F8 11.955 Tf 11.96 0 Td[(q byLemma 2.3.3 (ii).Thus,an )]TJ 5.48 -.71 Td[(Pni=1jaijq1=qn)]TJ /F14 7.97 Tf 6.59 0 Td[((logn)(1)]TJ /F8 11.955 Tf 11.95 0 Td[(q)1=q (logn)n1=q)]TJ /F14 7.97 Tf 6.58 0 Td[(=Odn n1=q. TheconclusionagainfollowsimmediatelyfromCorollary 4.2.1 Remark4.2.1. Wewillshowbelowthatifcondition( 4.4 )isstrengthenedtotheconditionbn janjn1=qisquasi-monotoneincreasing; thatis,bn janjn1=qCbj jajjj1=q<1forsomeC1andalljn1, (4.10) thenTheorem 4.2.1 followsreadilyfromProposition 4.2.1 .Clearly,ifan6=0,n1andbn janjn1=qismonotoneincreasing, thenitisquasi-monotoneincreasing.Ontheotherhand,ifan6=0,n1andbn janjn1=qismonotonedecreasingto0, thenitisnotquasi-monotoneincreasing.Wewillalsoprovidebelowanexample,Example 4.2.3 ,whereintheconditionsofTheorem 4.2.1 aresatisedbutthoseofProposition 4.2.1 arenotsatised.Toseethatcondition( 4.10 )indeedimpliescondition 74

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( 4.4 ),notethatforalln1,( 4.10 )yieldsb1 ja1jCinfj1bj jajjj1=qCbn janjn1=q whencejanj bnCja1j b11 n1=q,n1. (4.11) Thuscondition( 4.4 )holds. ProofofTheorem 4.2.1 withcondition( 4.4 )replacedbycondition( 4.10 ). Itfollowsfrom( 4.11 )that1Xn=1PfkanVk>Dbng1Xn=1PkVk>Db1 Cja1jn1=q<1 sinceEkVkq<1.Rewritecondition( 4.10 )asfollows:bj jajjCbnj1=q janjn1=q,1jn,n1. (4.12) Thenmax1jnbj jajj1Xj=njajj bjCbn janj1Xj=njajj bj(by( 4.12 ))=C1Xj=nbn janjjajj bjC1Xj=nCn1=q j1=q(by( 4.10 ))=C2n1=q1Xj=n1 j1=q=O(n) aswasshownintheproofofTheorem 4.2.1 .TheSLLN( 4.3 )followsfromProposition 4.2.1 .2 75

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Thefollowingexample,Example 4.2.3 ,showsthattheconditionsofTheorem 4.2.1 aresatisedbutthoseofProposition 4.2.1 arenotsatised. Example4.2.3. LetfVn,n1gbeasequenceofrandomelementsinarealseparableBanachspaceXwherefVn,n1gisstochasticallydominatedbyarandomelementVinthesensethat( 2.7 )holdsforsomeconstantD<1.SupposethatEkVkq<1forsome0<>:bn 2nfornodd,bn 4nforneven. Then( 4.4 )holdssincean bn1 2n=O1 n1=q. But( 4.10 )failssince,forneven,bn ann1=q bn+1 an+1(n+1)1=q=4n n1=q(n+1)1=q 2n+1=n+1 n1=q2n)]TJ /F11 7.97 Tf 6.58 0 Td[(1!1asnevenapproaches1. TheSLLN( 4.3 )followsfromTheorem 4.2.1 .However,forneven,max1jn+1bj jajj n+11Xj=n+1jajj bj4n n+11 2n+1=2n 2n+2!1asnevenapproaches1. Thuscondition( 4.1 )ofProposition 4.2.1 isnotsatised. 76

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Thefollowingexample,Examples 4.2.4 ,showsthattheconditionsofProposition 4.2.1 aresatisedbutthoseofTheorem 4.2.1 arenotsatised. Example4.2.4. Let>2andletfVn,n1gbeasequenceofidenticallydistributedrandomvariableswithV1havingprobabilitydensityfunctionf(x)=c jxj )]TJ /F19 5.978 Tf 5.76 0 Td[(1(logjxj)3I[e,1)(jxj),0. Thus,max1jnbj jajj1Xj=njajj bj=n()]TJ /F11 7.97 Tf 6.59 0 Td[(1) log(n+1)1Xj=nlog(j+1) j)]TJ /F11 7.97 Tf 6.59 0 Td[(1n)]TJ /F11 7.97 Tf 6.59 0 Td[(1 log(n+1)Z1n)]TJ /F11 7.97 Tf 6.58 0 Td[(1log(y+1) y)]TJ /F11 7.97 Tf 6.59 0 Td[(1dy=Zu(n)]TJ /F6 11.955 Tf 11.96 0 Td[(1) nuZ(n)=(1+o(1))(n)]TJ /F6 11.955 Tf 11.96 0 Td[(1)u+1Z(n)]TJ /F6 11.955 Tf 11.96 0 Td[(1) ()]TJ /F6 11.955 Tf 11.95 0 Td[(2)nuZ(n)=O(n). Hence,condition( 4.1 )holds. Furthermore,weprovethatcondition( 4.2 )holdswithV=V1andD=1asfollows.LetZ(x)=(logt))]TJ /F11 7.97 Tf 6.59 0 Td[(3andu=)]TJ /F9 11.955 Tf 9.29 0 Td[(=()]TJ /F6 11.955 Tf 12.52 0 Td[(1).ThenZisvaryingslowly;thatis,Zvaries 77

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regularlywithexponent=0.DeneZu(x)asin( 2.5 ).ByLemma 2.3.2 ,Zu(x)<1sinceu<)]TJ /F6 11.955 Tf 9.29 0 Td[(1.Then,byLemma 2.3.3 (i),limx!1xu+1Z(x) Zu(x)=)]TJ /F6 11.955 Tf 9.3 0 Td[((u++1)=)]TJ /F8 11.955 Tf 9.29 0 Td[(u)]TJ /F6 11.955 Tf 11.96 0 Td[(1=1 )]TJ /F6 11.955 Tf 11.95 0 Td[(1>0. Thus,wehavePfjV1j>xg=2Z1xf(y)dy=2cZ1xyuZ(y)dy=2cZu(x)=(1+o(1))2c()]TJ /F6 11.955 Tf 11.95 0 Td[(1)xu+1Z(x) whichimpliesPfjV1j>xg=(1+o(1))2c()]TJ /F6 11.955 Tf 11.96 0 Td[(1) x1 )]TJ /F19 5.978 Tf 5.76 0 Td[(1(logx)3asx!1. (4.13) Then,by( 4.13 ),PfjanV1j>bng=PjV1j>n)]TJ /F11 7.97 Tf 6.58 0 Td[(1 log(n+1)=(1+o(1))2c()]TJ /F6 11.955 Tf 11.96 0 Td[(1)(log(n+1))1 )]TJ /F19 5.978 Tf 5.76 0 Td[(1 n[()]TJ /F6 11.955 Tf 11.96 0 Td[(1)logn)]TJ /F6 11.955 Tf 11.96 0 Td[(loglog(n+1)]3=(1+o(1))2c()]TJ /F6 11.955 Tf 11.95 0 Td[(1)(logn)1 )]TJ /F19 5.978 Tf 5.75 0 Td[(1 n()]TJ /F6 11.955 Tf 11.95 0 Td[(1)3(logn)3=(1+o(1))2c ()]TJ /F6 11.955 Tf 11.96 0 Td[(1)2n(logn)3)]TJ /F19 5.978 Tf 5.75 0 Td[(4 )]TJ /F19 5.978 Tf 5.76 0 Td[(1. Thus,condition( 4.2 )holdswithV=V1andD=1since>2(hence(3)]TJ /F6 11.955 Tf 10.37 0 Td[(4)=()]TJ /F6 11.955 Tf 10.37 0 Td[(1)>1).Then,byProposition 4.2.1 ,theSLLNPni=1(log(i+1))Xi n)]TJ /F11 7.97 Tf 6.59 0 Td[(1!0a.c.holds. ToshowthatthehypothesesofTheorem 4.2.1 arenotsatised,notethatcondition( 4.4 )holdsforsomeq2(0,1)ifandonlyif1=q<)]TJ /F6 11.955 Tf 12.3 0 Td[(1;thatisq>()]TJ /F6 11.955 Tf 12.3 0 Td[(1))]TJ /F11 7.97 Tf 6.58 0 Td[(1.Onthe 78

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otherhand,by( 4.13 ),1Xn=1PfjV1j>n1=qg=1Xn=1(1+o(1))2c()]TJ /F6 11.955 Tf 11.95 0 Td[(1)q3 n1 q()]TJ /F19 5.978 Tf 5.75 0 Td[(1)(logn)3<1 ifandonlyif(q()]TJ /F6 11.955 Tf 12.69 0 Td[(1)))]TJ /F11 7.97 Tf 6.58 0 Td[(11;thatis,q()]TJ /F6 11.955 Tf 12.69 0 Td[(1))]TJ /F11 7.97 Tf 6.59 0 Td[(1.Therefore,inTheorem 4.2.1 ,condition( 4.4 )andtheconditionthatEjVjq<1forsome0
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SupposethatEkVkq<1.Letcn=n1=qforn1.Thencpn=n=n1=q)]TJ /F11 7.97 Tf 6.59 0 Td[(1"andliminfn!1cp2n cpn=liminfn!1(2n)1=q n1=q=21=q>2 since1=q>1.Therefore,condition( 4.1 )holdssincemax1jnbj jajj1Xj=njajj bj=max1jncpj1Xj=n1 cpj=O(n) byRemark 3.2.3 takingr=2.Furthermore( 4.2 )holds;i.e.,1Xn=1PfkVk>Dn1=qg<1 (4.14) sinceEkVkq<1.Insummary,alloftheconditionsofProposition 4.2.1 hold.Clearly,condition( 4.4 )holds.Hence,alloftheconditionsofTheorem 4.2.1 alsohold.Consequently,Theorem 4.2.1 andProposition 4.2.1 caneachbeemployedtoestablishtheSLLN( 4.3 ). Thefollowingtheorem,Theorem 4.2.2 ,isadirectcorollaryofTheorem 4.2.1 inthecasewhere0
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ThentheSLLN( 4.3 )holdsirrespectiveofthejointdistributionsofthefVn,n1g. Proof.Wersttreatthecasewhere0
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Example4.2.6. LetfVn,n1gbeasequenceofidenticallydistributedrandomelementsinarealseparableBanachspaceX.SupposeEkV1kq<1forsome01=q,andcondition( 4.1 )holdsifandonlyif>1.Therefore,if>1=q,thenbothconditions( 4.15 )and( 4.1 )hold.Furthermore,EkV1kq<1impliesEkV1k1=<1sinceq>1=.Hence,condition( 4.2 )holdswithV=V1andD=1.Inthiscase,theconditionsofbothTheorems 4.2.2 andProposition 4.2.1 aresatised.Ifweinsteadletbn=n1=q,n1wherenow01.Moreover,condition( 4.2 )holdswithV=V1andD=1sinceEkV1kq<1.However,condition( 4.15 )fails.Thuswhenbn=n1=q,n1,theconditionsofProposition 4.2.1 aresatisedbutthoseofTheorem 4.2.2 arenot.Nextletq=1andbn=n(log(n+1)),n1where>1.Thencondition( 4.15 )holds.However,1Xj=n1 bj=O1 (log(n+1)))]TJ /F11 7.97 Tf 6.59 0 Td[(16=On bnandsocondition( 4.1 )fails.Thuswhenbn=n(log(n+1)),n1where>1,theconditionsofTheorem 4.2.2 aresatisedbutthoseofProposition 4.2.1 arenot. Thefollowingcorollary,Corollary 4.2.2 ,followsimmediatelyfromTheorem 4.2.1 (when0
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ThentheSLLN( 4.3 )holdsirrespectiveofthejointdistributionsofthefVn,n1g. Thefollowingtheorems,Theorems 4.2.3 and 4.2.4 ,providesetsofconditionsunderwhichtheSLLN( 4.3 )holdswheretherandomelementsfVn,n1garestochasticallydominatedbutarenotnecessarilyindependent.Theorems 4.2.3 and 4.2.4 arenewresultsevenwhentheunderlyingBanachspaceisthereallineR. Theorem4.2.3. LetfVn,n1gbeasequenceofrandomelementsinarealseparableBanachspaceX.SupposethatfVn,n1gisstochasticallydominatedbyarandomelementVinthesensethat( 2.7 )holdsforsomeconstantD<1.Letfan,n1gandfbn,n1gbesequencesofconstantssuchthatan6=0,n1,0
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implyingbn janj"1 (4.19) sincebn=janj". (ii) Condition( 4.17 )impliesthat1Xn=1PfkanVk>bng<1. (4.20) Toseethis,notethat( 4.17 )ensuresthat1Xj=n1 jL(j)<1 (4.21) foreachn1unlessV=0a.c.(inwhichcase( 4.2 )and( 4.3 )areimmediate).LetZ(x)=1=L(x)andletu=)]TJ /F6 11.955 Tf 9.3 0 Td[(1.ThenZisvaryingslowly;thatis,Zvariesregularlywithexponent=0.DeneZu(x)asin( 2.5 ).ThenZu(x)<1by( 4.21 ).Hence,byLemma 2.3.3 (i),limx!1xu+1Z(x) Zu(x)=)]TJ /F6 11.955 Tf 9.3 0 Td[((u++1)=0. Thus,limn!1L(n)1Xj=n1 jL(j)=1 whence( 4.17 )impliesthat1Xn=1nPbn)]TJ /F11 7.97 Tf 6.58 0 Td[(1 jan)]TJ /F11 7.97 Tf 6.58 0 Td[(1j
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( 2.7 )holdsforsomeconstantD<1,wehave1Xn=1PfkanVnk>Dbng1Xn=1DPfkanDVk>Dbng<1, which,byLemma 2.3.4 ,ensuresthatPnliminfn!1[I(kanVnk>Dbn)=0]o=1, Therefore,1Xn=1kanVnkI(kanVnk>Dbn)<1a.c. Since0Dbi)!0a.c. Hence,toprovetheSLLN( 4.3 ),itsufcestoshowthat1 bnnXi=1kaiVikI(kaiVikDbi)!0a.c. which,byLemma 2.3.5 intherandomvariablecase(thereallineversionoftheKroneckerlemma),willholdonceweshowthat1Xn=11 bnkanVnkI(kanVnkDbn)<1a.c. Hence,itsufcestoverifythat1Xn=11 bnE[kanVnkI(kanVnkDbn)]<1. (4.22) Note,byLemma 2.3.11 takingq=1,that1Xn=11 bnE[kanVnkI(kanVnkDbn)]D21Xn=1PfkanVk>bng+D21Xn=1janj bnE[kVkI(kanVkbn)]. 85

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Thenrecalling( 4.20 ),( 4.22 )willfollowprovidedwecanshowthat1Xn=1janj bnE[kVkI(kanVkbn)]<1. (4.23) First,itfollowsfrom( 4.16 )that,forallnj1,bj jajjjanj bnCjL(j) nL(n). (4.24) Then,toverify( 4.23 ),notethat1Xn=1janj bnE[kVkI(kanVkbn)]=1Xn=1janj bnnXj=1EkVkIbj)]TJ /F11 7.97 Tf 6.59 0 Td[(1 jaj)]TJ /F11 7.97 Tf 6.59 0 Td[(1j
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0bngEkVk1Xn=1janj bn<1. 87

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(ii) Whenq=0,condition( 4.25 )ofTheorem 4.2.4 implies( 4.1 )sincebn janj1Xj=njajj bj=O(1)=O(n). Inthiscase,theconditionEkVkq<1isautomatic,and( 4.20 )cannotbedispensedwith. (iii) Example3.1ofRosalskyandStoica(2010)satisesthehypothesesofTheorem 4.2.3 butnotthehypothesesofTheorem 4.2.1 ,Proposition 4.2.1 ,Theorem 4.2.2 ,orTheorem 4.2.4 (iv) Example3.2ofRosalskyandStoica(2010)satisesthehypothesesofTheorem 4.2.1 ,Proposition 4.2.1 ,andTheorem 4.2.4 butnotthehypothesesofTheorem 4.2.2 .If,inthisexample,X1hasprobabilitydensityfunctionf(x)=c xp+1(logx)I[e,1)(x),2andcisaconstant,thentakingL(x)=(logx))]TJ /F11 7.97 Tf 6.59 0 Td[(1,x1,thehypothesesofTheorem 4.2.3 arealsosatised. NowweconsidersetsofconditionsunderwhichtheparticularSLLNoftheformPni=1Vi bn!0a.c. (4.26) holdswherefbn,n1gisasequenceofpositiveconstantssatisfying0
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Thefollowingcorollaries,Corollaries 4.2.3 4.2.4 ,and 4.2.5 ,areobtainedfromTheorems 4.2.2 4.2.3 ,and 4.2.4 ,respectively.WhentheunderlyingBanachspaceXisthereallineR,Corollaries 4.2.3 4.2.4 ,and 4.2.5 wereestablishedbyPetrov(1973,Theorem1),RosalskyandStoica(2010,Theorem2.1),andRosalskyandStoica(2010,Theorem2.2),respectively. Corollary4.2.3. LetfVn,n1gbeasequenceofidenticallydistributedrandomelementsinarealseparableBanachspaceX.SupposeEkV1kq<1forsome0
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Proof.Notethat( 4.28 )ensuresthat0bng<1. ThentheSLLN( 4.26 )holdsirrespectiveofthejointdistributionsofthefVn,n1g. Proof.Takean1,n1inTheorem 4.2.4 .Thentheconclusion( 4.26 )follows.2 90

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CHAPTER5FUTURERESEARCHANDCONCLUSIONS 5.1FutureResearch Somethoughtsconcerningfutureresearchwillnowbediscussed. TherstideaaboutfutureresearchisthatwhetherTheorem 4.2.4 canbeimproved.NotethatinTheorem 4.2.4 ,thereareineffecttwomomentconditions,namelycondition( 4.20 )andtheconditionEkVkq<1,bothofwhichweappliedtoproveTheorem 4.2.4 .Whatistherelationshipbetweenthem?Dotheyimplyeachotheroronestrictlystrongerthantheother?Oraretheynotcomparableingeneral?Wenowpresenttwoexamples,Examples 5.1.1 and 5.1.2 ,thatsatisfyallthehypothesesofTheorem 4.2.4 .Inthesetwoexamples,thehypothesisEkVkq<1isstrictlystrongerthan( 4.20 ).Weneedadditionalexamplestoclarifytherelationshipbetweenthecondition( 4.20 )andtheconditionEkVkq<1aproposofTheorem 4.2.4 Example5.1.1. LetfVn,n1gbeasequenceofidenticallydistributedrandomvariables.Let0<<1andan=1,bn=n1=,n1. Thenbn janj1)]TJ /F5 7.97 Tf 6.59 0 Td[(q1Xj=njajj bj=n1)]TJ /F17 5.978 Tf 5.75 0 Td[(q 1Xj=n1 n1==n1)]TJ /F17 5.978 Tf 5.76 0 Td[(q O(n1)]TJ /F19 5.978 Tf 8.53 3.26 Td[(1 )=O(n1)]TJ /F17 5.978 Tf 8.4 3.53 Td[(q ). Thus,( 4.25 )holdsifandonlyifq.Moreover,( 4.20 )holdsifandonlyifEjV1j<1.Therefore,allofthehypothesesofTheorem 4.2.4 aresatisedifq.Furthermore,EjV1jq<1impliesEjV1j<1forq;i.e.,EjV1jq<1implies( 4.20 )forq.However,( 4.20 )doesnotnecessarilyimplyEjV1jq<1ifq>. Example5.1.2. LetfVn,n1gbeasequenceofidenticallydistributedrandomvariableswithV1havingprobabilitydensityfunctionf(x)=c jxj1+(logjxj)3I[e,1)(jxj),
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where0<<1andcisapositiveconstant.Let0<<1andan=log(n+1),bn=n1=,n1. ThenbyLemma 2.3.3 (i),PfjV1j>xg=2Z1xf(y)dy=2cZ1xy)]TJ /F11 7.97 Tf 6.58 0 Td[((1+)(logy))]TJ /F11 7.97 Tf 6.59 0 Td[(3dy=(1+o(1))2c x(logx)3asx!1 and1Xj=njajj bj=1Xj=nlog(j+1) j1==(1+o(1)) 1)]TJ /F9 11.955 Tf 11.95 0 Td[(log(n+1) n1)]TJ /F21 5.978 Tf 5.76 0 Td[( Thus,bn janj1)]TJ /F5 7.97 Tf 6.59 0 Td[(q1Xj=njajj bj=(1+o(1))Cn1= log(n+1)1)]TJ /F5 7.97 Tf 6.59 0 Td[(qlog(n+1) n1)]TJ /F21 5.978 Tf 5.75 0 Td[( =(1+o(1))C[log(n+1)]q nq)]TJ /F21 5.978 Tf 5.76 0 Td[( (5.1)PfjV1j>n1=qg=(1+o(1))C x=q(logn)3, (5.2) andPfjanV1j>bng=PjV1j>n1= log(n+1)=(1+o(1))C n=(logn)3)]TJ /F14 7.97 Tf 6.59 0 Td[(. (5.3) By( 5.1 ),( 4.25 )holdsifandonlyif
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thoseinourmainresults.WewouldhopethattheconditionsofthemaintheoremsinChapter3and4canbestrictlyweakenedsothatthecorrespondingSLLNfailsbuttheWLLNholds. Thethirdthoughtconcerningfutureresearchismotivatedbycompleteconvergence.AsequenceofrandomvariablesfXn,n1gissaidtoconvergecompletelyto0if1Xn=1PfjXnj"g<1forall">0. ThiskindofconvergencewasintroducedbyHsuandRobbins(1947).ItiseasilyseenbyLemma 2.3.4 (theBorel-Cantellilemma)thatcompleteconvergenceto0impliesalmostcertainconvergenceto0,andtheconverseistrueifthefXn,n1gisindependent.WewouldhopethattheassumptionsofthemaintheoremsinChapter3and4canbestrengthenedtoachievecompleteconvergenceresults. Finally,wewouldconsidertoobtainSLLNsfordoublesumsofrandomelementsfVi,j,i1,j1goftheformPni=1Pmj=1Vi,j bm,n!0a.c.asm^n!1. orPni=1Pmj=1Vi,j bm,n!0a.c.asm_n!1 respectively,wherefbm,n,m1,n1gisanarrayofpositiveconstantswithbm,n!1asm^n!1 orbm,n!1asm_n!1, respectively.WealsowouldliketoconsiderobtainingcompleteconvergenceresultsfordoublesumsofrandomelementsfVi,j,i1,j1g. 93

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5.2Conclusions InthisdissertationwehavepresentedresultspertainingtotheSLLNproblemforsumsofBanachspacevaluedrandomelementsandthemainresultsarenewevenwhentheunderlyingBanachspaceistherealline. InChapter3,weestablishTheorem 3.2.1 ,averygeneralSLLNfornormedweightedsumsofindependentBanachspacevaluedrandomelementswhicharenotnecessarilyidenticallydistributed.TheunderlyingBanachspaceisassumedtobeofRademachertypep(1p2)andthesequenceofrandomelementsisassumedtobestochasticallydominatedbyarandomelement.SpecialcasesofTheorem 3.2.1 are: (i) Proposition 3.2.1 whichisTheorem1ofAdler,Rosalsky,andTaylor(1989) (ii) Theorem 3.2.2 whichisanimprovedversionofTheorem6ofAdler,Rosalsky,andTaylor(1989). Theorem 3.2.2 containsProposition 3.2.2 whichistheresultofSung(1997).(Theorem6ofAdler,Rosalsky,andTaylor(1989)doesnotcontainSung's(1997)result.)Theorem 3.2.2 alsocontainsTheorem4.1ofWoyczynski(1980)(Remark 3.2.6 (iv))andTheorem5ofTeicher(1985)(Corollary 3.2.1 ).Theorems 3.2.1 and 3.2.2 arenewevenwhentheunderlyingBanachspaceistherealline. InChapter4,weobtainSLLNsirrespectivethejointdistributionsoftherandomelementsandnogeometricconditionsareimposedontheunderlyingBanachspace.Weestablishedfourtheorems(Theorems 4.2.1 4.2.2 4.2.3 ,and 4.2.4 )allofwhichhavetheassumptionthatthesequenceofrandomelementsisstochasticallydominatedbyarandomelement.Theorem 4.2.1 isa0
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(i) Corollary 4.2.1 whichisadirectcorollaryofTheorem 4.2.1 andisanewversionofTheorem5ofTeicher(1985)andCorollary2ofSung(1997)whichpertainedtosequencesofi.i.d.Lprandomvariableswhere1p<2 (ii) Corollary 4.2.2 whichfollowsimmediatelyfromTheorem 4.2.1 (when0
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BIOGRAPHICALSKETCH YuanLiaowasbornin1980inChangchun,China.UpongraduationfromthehighschoolafliatedwiththeNortheastUniversityinChinainJuly1999,heenrolledasanundergraduatestudentintheDepartmentofMathematicsattheUniversityofScienceandTechnologyofChinawhereheearnedaBachelorofArtsDegreeinMathematicsinJuly2004.InSeptember2004,heenteredamastersprograminstatisticsintheDepartmentofMathematicsattheGraduateUniversityoftheChineseAcademyofScienceswhereheearnedaMasterofArtsDegreeinStatisticsinJuly2007.InAugust2007,heenteredaPh.D.programintheDepartmentofStatisticsattheUniversityofFlorida.DuringhisgraduateeducationatUniversityofFlorida,hewasappointedasateachingassistantfordifferentclassesintheDepartmentofStatistics.HismainresearchinterestsareProbabilityTheoryandLimitTheoryforBanachSpaceValuedRandomElements. 100