﻿ On the Strong Law of Large Numbers for Weighted Sums of Random Elements in Banach Spaces
 UFDC Home  |  Search all Groups  |  UF Institutional Repository  |  UF Institutional Repository  |  UF Theses & Dissertations |   Help

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

## Material Information

Title:
On the Strong Law of Large Numbers for Weighted Sums of Random Elements in Banach Spaces
Physical Description:
1 online resource (100 p.)
Language:
english
Creator:
Liao, Yuan
Publisher:
University of Florida
Place of Publication:
Gainesville, Fla.
Publication Date:

## Thesis/Dissertation Information

Degree:
Doctorate ( Ph.D.)
Degree Grantor:
University of Florida
Degree Disciplines:
Statistics
Committee Chair:
Rosalsky, Andrew J
Committee Members:
Ghosh, Malay
Khare, Kshitij
Cantrell, Amy

## Subjects

Subjects / Keywords:
probability -- slln
Statistics -- Dissertations, Academic -- UF
Genre:
Statistics thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

## Notes

Abstract:
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.
General Note:
In the series University of Florida Digital Collections.
General Note:
Includes vita.
Bibliography:
Includes bibliographical references.
Source of Description:
Description based on online resource; title from PDF title page.
Source of Description:
This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility:
by Yuan Liao.
Thesis:
Thesis (Ph.D.)--University of Florida, 2012.
Local:
Electronic Access:
RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2014-05-31

## Record Information

Source Institution:
UFRGP
Rights Management:
Applicable rights reserved.
Classification:
lcc - LD1780 2012
System ID:
UFE0043915:00001

Full Text

PAGE 1

PAGE 2

c2012YUANLIAO 2

PAGE 3

Tomyteacherswithgratitude Tomyparentswithlove 3

PAGE 4

PAGE 5

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

PAGE 6

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

PAGE 7

PAGE 8

PAGE 9

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

PAGE 10

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

PAGE 11

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

PAGE 12

PAGE 13

PAGE 14

PAGE 15

PAGE 16

PAGE 17

PAGE 18

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
PAGE 19

PAGE 20

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
PAGE 21

PAGE 22

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

PAGE 23

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

PAGE 24

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

PAGE 25

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

PAGE 26

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

PAGE 27

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

PAGE 28

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

PAGE 29

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

PAGE 30

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

PAGE 31

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

PAGE 32

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

PAGE 33

=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

PAGE 34

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
PAGE 35

PAGE 36

(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

PAGE 37

(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

PAGE 38

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

PAGE 39

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

PAGE 40

PAGE 41

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

PAGE 42

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
PAGE 43

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

PAGE 44

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

PAGE 45

(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

PAGE 46

(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

PAGE 47

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

PAGE 48

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

PAGE 49

PAGE 50

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
PAGE 51

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

PAGE 52

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
PAGE 53

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

PAGE 54

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
PAGE 55

PAGE 56

(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

PAGE 57

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

PAGE 58

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
PAGE 59

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

PAGE 60

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

PAGE 61

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

PAGE 62

=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
PAGE 63

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
PAGE 64

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
PAGE 65

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

PAGE 66

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<
PAGE 67

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

PAGE 68

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
PAGE 69

If1Xn=1PfkanVk>Dbng<1, (4.2) thentheSLLNPni=1aiVi bn!0a.c. (4.3) holdsirrespectiveofthejointdistributionsofthefVn,n1g. TherstmainresultofChapter4,isTheorem 4.2.1 ,whichisa0
PAGE 70

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

PAGE 71

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
PAGE 72

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
PAGE 73

(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

PAGE 74

(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

PAGE 75

( 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

PAGE 76

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

PAGE 77

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

PAGE 78

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

PAGE 79

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
PAGE 80

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
PAGE 81

ThentheSLLN( 4.3 )holdsirrespectiveofthejointdistributionsofthefVn,n1g. Proof.Wersttreatthecasewhere0
PAGE 82

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
PAGE 83

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
PAGE 84

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
PAGE 85

( 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

PAGE 86

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
PAGE 87

0bngEkVk1Xn=1janj bn<1. 87

PAGE 88

(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
PAGE 89

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
PAGE 90

Proof.Notethat( 4.28 )ensuresthat0bng<1. ThentheSLLN( 4.26 )holdsirrespectiveofthejointdistributionsofthefVn,n1g. Proof.Takean1,n1inTheorem 4.2.4 .Thentheconclusion( 4.26 )follows.2 90

PAGE 91

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),
PAGE 92

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
PAGE 93

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

PAGE 94

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
PAGE 95

(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
PAGE 96

PAGE 97

[14] W.FELLER,Alimittheoremforrandomvariableswithinnitemoments,Amer.J.Math.,68(1946),257. [15] W.FELLER,AnIntroductiontoProbabilityTheoryandItsApplications,Vol.II,2ndEdition,JohnWiley,NewYork,(1971). [16] L.T.FERNHOLZandH.TEICHER,Stabilityofrandomvariablesanditeratedlogarithmlawsformartingalesandquadratcforms,Ann.Probab.,8(1980),765. [17] R.FORTETandE.MOURIER,LoisdesgrandsnombrespourdeselementsaleatoiresprenantleursvaleursdansunespacedeBanach,inFrench,C.R.Acad.Sci.Paris.,273(1953),18. [18] P.HARTMANandA.WINTNER,Onthelawoftheiteratedlogarithm,Amer.J.Math.,63(1941),169. [19] C.C.HEYDE,Onalmostsureconvergenceforsumsofindependentrandomvariables,SankhyaSer.A,30(1968),353. [20] E.HILLEandR.S.PHILLIPS,FunctionalAnalysisandSemi-Groups,RevisedEdition,AmericanMathematicalSociety,Providence,RhodeIsland,(1985). [21] J.HOFFMANN-JRGENSENandG.PISIER,ThelawoflargenumbersandthecentrallimittheoreminBanachspaces,Ann.Probab.,4(1976),587. [22] P.L.HSUandH.ROBBINS,Completeconvergenceandthelawoflargenumbers,Proc.Nat.Acad.Sci.U.S.A.,33(1947),25. [23] K.ITOandM.NISIO,OntheconvergenceofsumsofindependentBanachspacevaluedrandomvariables,OsakaJ.Math.,5(1968),35. [24] M.J.KLASS,Propertiesofoptimalextended-valuedstoppingrulesforSn=n,Ann.Probab.,1(1973),719. [25] K.KNOPP,TheoryandApplicationofInniteSeries,2ndEnglishEdition,BlackieandSon,London,(1951). [26] A.KORZENIOWSKI,OnMarcinkiewiczSLLNinBanachspaces,Ann.Probab.,12(1984),279. [27] M.LEDOUXandM.TALAGRAND,ProbabilityinBanachSpaces:IsoperimetryandProcesses,Springer-Verlag,NewYork,(1991). [28] J.MARCINKIEWICZandA.ZYGMUND,Surlesfonctionsindependantes,inFrench,Fund.Math.,29(1937),60. [29] A.I.MARTIKAINEN,Onnecessaryandsufcientconditionsforthestronglawoflargenumbers,Teor.Veroyatnost.iPrimenen.,24(1979),814,inRussian;Englishtranslation:TheoryProbab.Appl.,24(1979),820. 97

PAGE 98

[30] A.I.MARTIKAINENandV.V.PETROV,OnatheoremofFeller,Teor.Veroyatnost.iPrimenen.,25(1980),191,inRussian;Englishtranslation:TheoryProbab.Appl.,25(1981),191. [31] E.MOURIER,ElementsaleatoiresdansunespacedeBanach,inFrench,Ann.Inst.HenriPoincare.,13(1953),161. [32] V.V.PETROV,Theorderofgrowthofsumsofdependentrandomvariables,Teor.Veroyatnost.iPrimenen.,18(1973),358,inRussian;Englishtranslation:TheoryProbab.Appl.,18(1974),348. [33] B.J.PETTIS,Onintegrationinvectorspaces,Trans.Amer.Math.Soc.,44(1938),277. [34] A.ROSALSKYandG.STOICA,Onthestronglawoflargenumbersforidenticallydistributedrandomvariablesirrespectiveoftheirjointdistributions,Statist.Probab.Lett.,80(2010),1265. [35] H.L.ROYDEN,RealAnalysis,3rdEdition,Macmillan,NewYork,(1988). [36] S.SAWYER,Maximalinequalitiesofweaktype,Ann.ofMath.,84(1966),157. [37] A.N.SHIRYAEV,Probability,2ndEdition,Springer-Verlag,NewYork,(1996). [38] W.F.STOUT,AlmostSureConvergence,AcademicPress,NewYork,(1974). [39] S.H.SUNG,Almostsureconvergenceforweightedsumsofi.i.d.randomvariables,J.KoreanMath.Soc.,34(1997),57. [40] R.L.TAYLOR,Weaklawsoflargenumbersinnormedlinearspaces,Ann.Math.Statist.,43(1972),1267. [41] R.L.TAYLOR,StochasticConvergenceofWeightedSumsofRandomElementsinLinearSpaces,LectureNotesinMathematicsNo.672.Springer-Verlag,Berlin,(1978). [42] R.L.TAYLORandD.WEI,Lawsoflargenumbersfortightrandomelementsinnormedlinearspaces,Ann.Probab.,7(1979),150. [43] H.TEICHER,Almostcertainconvergenceindoublearrays,Z.Wahrsch.Verw.Gebiete,69(1985),331. [44] D.WEIandR.L.TAYLOR,Convergenceofweightedsumsoftightrandomelements,J.MultivariateAnal.,8(1978),282. [45] A.WILANSKY,FunctionalAnalysis,BlaisdellPublishingCo.GinnandCo.,NewYork,(1964). 98

PAGE 99

[46] W.A.WOYCZYNSKI,OnMarcinkiewicz-ZygmundlawsoflargenumbersinBanachspacesandrelatedratesofconvergence,Probab.Math.Statist.,1(1980),117. 99

PAGE 100