On the strong law of large numbers for sums of random elements in Banach spaces

112 Pages·2001·4.1 MB·English

by  Cantrell, Amy M., 1971-

ONTHESTRONGLAWOFLARGENUMBERSFOR
SUMSOFRANDOMELEMENTSINBANACHSPACES
By
AMYM.CANTRELL
ADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOL
OFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENT
OFTHEREQUIREMENTSFORTHEDEGREEOF
DOCTOROFPHILOSOPHY
UNIVERSITYOFFLORIDA
2001
ACKNOWLEDGMENTS
Iwouldliketothankmyadvisor,AndrewRosalsky,forhisinspirationalatti-
tudeandunderstanding. Hisdirectionandmotivationhavemadethisworkpossible.
Iwouldalso liketothankthemembersofmy dissertationcommittee, Dr. Malay
Ghosh,Dr.RamonLittell,Dr.JamesHobert,andDr.IreneHueter. IthankDr.Ron
Randlesforhissupportandencouragementbeforeandthroughoutmystudyatthe
UniversityofFlorida.
IthankmyparentsforalwaysbelievingthatIcouldaccomplishmygoals,and
fortheirconstantloveandsupport. Ithankmyhusband,Emory,forhislove,en-
couragementandunderstanding.
Igivemythankstoalloftheteachersthatinspiredandmotivatedmydesireto
learn,withspecialthankstoRitaCater,RonGoolsby,JamesBentley,andThomas
Polaski. I wishtogivespecialthankstoMrs. AnneClosefor hermanyyears of
support,encouragementandfriendship.
11
TABLEOFCONTENTS
ACKNOWLEDGMENTS
ii
ABSTRACT iv
CHAPTERS
1 INTRODUCTION 1
2 STRONGLAWSOFLARGENUMBERSFORSUMSOFRANDOM
ELEMENTS 11
2.1 Introduction 11
2.2 NecessaryConditionsforaSLLN 14
2.3 SufficientConditionsforaSLLN 21
2.3.1 SLLNsfor SumsofIndependent RandomElementsinRade-
macherTypepBanachSpaces 21
2.3.2 SLLNsforCompactlyUniformlyIntegrableSequencesofInde-
pendentRandomElements 36
2.3.3 SLLNsforBanachSpaceValuedSummandsIrrespectiveof
TheirJointDistributions 44
2.4 SomeInterestingExamples/Counterexamples 51
3 STRONGLAWSOFLARGENUMBERSFORSUMSOFRANDOM
VARIABLES 65
3.1 Introduction 65
3.2 NecessaryConditionsforaSLLN 65
3.3 SufficientConditionsforaSLLN 66
3.3.1 SLLNsforSumsofIndependentRandomVariables 66
3.3.2 SLLNsforRandomVariableSummandsIrrespectiveofTheir
JointDistributions 77
3.4 SomeInterestingExamples/Counterexamples 80
4 SUMMARYANDIDEASFORFUTURERESEARCH 97
4.1 Summary 97
4.2 IdeasForFutureResearch 98
REFERENCES 101
BIOGRAPHICALSKETCH 105
iii
AbstractofDissertationPresentedtotheGraduateSchool
oftheUniversityofFloridainPartialFulfillmentofthe
RequirementsfortheDegreeofDoctorofPhilosophy
ONTHESTRONGLAWOFLARGENUMBERSFOR
SUMSOFRANDOMELEMENTSINBANACHSPACES
By
AmyM.Cantrell
May2001
Chairman: AndrewRosalsky
MajorDepartment:Statistics
Forasequenceofrandomelements{Vn, n>1}inarealseparableBanachspace
A,sufficientconditionsareprovidedforthestronglawoflargenumbersX^Li(Vj~
Ci)/bn —> 0 almostcertainlytoholdwhere {c„,ra > 1} and {bn > 0,n > 1} are
suitablesequencesofcenteringelementsinXandnormingconstants,respectively.
Inthecaseofindependentrandomelements,separatenecessaryconditionsare
alsoprovidedforastronglawoflargenumbers. Thenecessityresultextendsareal
lineresultofMartikainentoaBanachspacesetting.Thesufficiencyresultsinthecase
ofindependentsummandsarenewevenwhenXisthereallineandaredividedinto
twocategories. ThefirstassumesthatXisofRademachertypep(1<p<2). The
resultisgeneralenoughtoincludeasspecialcasesastronglawofAdler,Rosalsky,
andTaylorforsumsofindependentandidenticallydistributedrandomelementsand
astronglawofHeydeforsumsofindependent(real-valued)randomvariables. The
secondimposesnoconditionsontheunderlyingBanachspace; instead,itassumes
IV
thatthesequenceofrandomelementsiscompactlyuniformlyintegrable. Theresult
includesasspecialcasesastronglawofAdler,Rosalsky,andTaylorforcompactly
uniformlyintegrablesequencesandastronglawofTaylorandWeiforuniformlytight
sequences.
Strong laws are also provided where no conditions are imposedon thejoint
distributionsoftherandomelementsorontheunderlyingBanachspace,andthese
resultsarenewevenwhentheBanachspaceistherealline.
Illustrativeexamplesareprovidedwhichcomparetheresultsorwhichshowhow
theresultsimproveuponoraredifferentfromotherresultsintheliterature.Examples
arealsoprovidedwhichshowthattheresultsaresharp.
v
>
CHAPTER
1
INTRODUCTION
Thehistoryand literatureofinvestigationon laws oflarge numbers arevast
andrich,asthisconceptiscrucialinprobabilityandstatisticaltheoryandintheir
application. Thereisnothingmorefundamentaltotheveryfoundationofstatistical
sciencethanthelawsoflargenumbers. Itisthelawsoflargenumberswhichelucidate
thenotionthatprobabilityisalimitingrelativefrequencyandhencethelawsoflarge
numbersprovidejustificationforKolmogorov’s(1933)axiomatictheoryofprobability
beingaphysicallyrealisticsubject. Indeed,thelawsoflargenumbersprovidearig-
orousformulationandjustificationforthenotionthat“thesamplemeanapproaches
thepopulationmeanasthesamplesizeapproachesinfinity”(i.e.,thesamplemeanis
aconsistentestimatorofthepopulationmean). Itistheareaofconsistencywherein
thelawsoflargenumbershavemanyapplications.
Thefirsttheoremonthelawoflargenumbers,duetoBernoulliinthelate1600’s,
wastheweaklawoflargenumbers (WLLN)forBernoullitrialswhichstatesthatif
Snisthenumberofsuccessesobservedinnindependentidenticaltrialswithsuccess
probabilitypineachtrial, then Sn/n —»pinprobabilityasn — oo. Bernoulli’s
resultwastoutedbyKolmogorovin1986asthebeginningofprobabilityproper(see
Bingham(1989)). AfterLebesgue’sworkonmeasuretheoryintheearly1900’s,the
(first) stronglawoflargenumbers (SLLN) couldbeformulated. Thefirstofsuch
SLLNsiscreditedto Borel (1909) and isthe extension ofBernoulli’sresultfrom
convergence in probabilitytoconvergencewithprobabilityone (oralmostcertain
(a.c.) convergence). Itisinterestingtoobservethattherewasoveratwohundred
1
2
yeargapbetweenthisSLLNanditsearlierWLLNcounterpart. Cantelli(1917) is
creditedwiththefirstSLLNregardingthea.c.convergenceofthesamplemeanto
thepopulationmean(seeSeneta(1992)).
Thelawsoflargenumbersprovidetheconsistencyofmanyestimatorsincluding
numerouscommonstatistics (suchas, ofcourse, thesamplemean), aswellasfor
estimatesfoundbyMonteCarlosimulation.Manyoftheseconceptsandapplications
canbeextended. InterestingapplicationsofSLLNsalsooccurinphysicsandcom-
puterscience. Thefieldofergodictheory,whichliesattheinterfaceofprobability
theoryandstatisticalphysics,hasasitsfoundationtheSLLNtyperesultprovidedby
theBirkoff-Khintchine-vonNeumannpointwiseergodictheorem(see,e.g., Breiman
(1968,Chapter6)orStout(1974,Section3.5)). Thisresultgivesconditionsforan
“ensembleaverage” tobeestimatedbya “timeaverage.” ASLLNwasderivedby
Wehr(1997)withapplicationstorandomresistornetworksandthedurabilityofcom-
positefibers. OtherapplicationsoftheSLLNtoresistancecanbefoundinEssohand
Bellissard(1989). Anotherinterestingand“non-statistical”applicationoccursinthe
fieldofnumbertheoryintheinvestigationofnormalnumbersx€ [0,1] (seeRevesz
(1968,pp.151-157)).
The primaryobjectiveofthecurrentwork isto investigate conditions under
which the SLLN for independent Banach space valued random elements obtains,
althoughsomeresultswillbepresentedforwhichtheassumptionofindependenceis
notneeded. Beforediscussionofthisobjectivesomenotationanddefinitionsmust
beintroduced.
LetA"bealinearspaceoverM;thatis,A”isavectorspaceoverM. Let||•||:
X-*[0,oo)beafunctionsatisfyingthefollowingthreeproperties:
(i)Forv6X,||v||=0ifandonlyifv—0.
3
(ii)||av||=|a||M|forallvGAandaGM.
(Hi)||«i+W2II<IKII+11^211forallvuv2GA.
Thefunction ||•||iscalledanormonA,andXisthensaidtobearealnormed
linearspace(withnorm|| ||).
Asequence{vn,n>1}inAissaidtoconvergetovGXiflim^oo||un—v||=0.
Arealnormedlinearspaceissaidtobe complete ifeveryCauchysequencein X
convergestoamemberofX. (ACauchysequenceinXis, ofcourse,asequence
{vn,n>1}inXsuchthat
mnl——i>>m0000|K-i>m||=0.)
ArealnormedlinearspaceXwhichiscompleteissaidtobearealBanachspace. A
realBanachspaceissaidtobeseparableifitcontainsacountabledensesubset.
Let(f2,P,P)beaprobabilityspace,letAbearealseparableBanachspacewith
norm ||•||,andletXbeequippedwithitsBorelcr-algebraB; thatis,Bisthea-
algebrageneratedbytheclassofopensubsetsofXdeterminedby viathemetric
11 11
d(vi,v2) = |K—v2\\,Vi,v2 G X. A randomelement AinAisanJF-measurable
transformationfrom12tothemeasurablespace(A,B). Theexpectedvalueormean
ofA,denotedEA,isdefinedtobethePettisintegralprovideditexists;thatis,Ahas
expectedvalueEVinAifL(EV)=E(L(A))foreveryLinX*whereA*denotes
the(dual) spaceofallcontinuouslinearfunctionalsonA. Werecallthatalinear
functionalisafunctionL:A-4EsatisfyingL(av+bw) =aL(v)+bL(w)forall
v,wGAandalla,6GM. Ofcourse,alinearfunctionalLiscontinuousifwhenever
{vn, n>1}isasequenceinAwithvn-»vGAwehaveL(v„)-»L(v). Asufficient
conditionforEVtoexististhat£||A|| < 00 (see,e.g., Taylor(1978, p.40)). A
completecharacterizationofwhenthePettisintegralexistswasprovidedbyBrooks
)
4
(1969). SeeHilleand Phillips (1957, pp. 76-85) forfurtherdiscussionanddetails
regardingthepropertiesofthePettisintegral.
An exampleillustratingexpectationin Banach spacesfollows: Let 1 < p <
ooandX =CP(M.)where,asusual,thisdenotestheclassofLebesguemeasurable
functions/: R —>• Rsuch that fR|/(x)|pdx < oo where thenorm isdefinedby
||/|| = Ifp\f(x)\pdx) . Thisclass isaBanach spaceaccordingtothefamous
Riesz-Fischertheorem(see,forexample,Royden(1988,p. 125)). ForfixedfoEX
andintegrablerandomvariablesXandY,definetherandomelementV=A”/o+Y.
WeclaimthatEV = E(X)fo+E(Y). Toverifythis,notethatforanyL E X*,
wehaveL(E(X)f0+E(Y))=E(X)L(f0)+E(Y)L(1)=E(XL(f0))+E(YL(1))=
E(XL(f0)+YL{1))=E(L{Xf0+Y)) =E{L{V))andthusthecandidateEV=
E(X)fo+E(Y)satisfiesthedefiningpropertyL(EV)=E(L(V)).Weremarkthatthe
dualspaceX*canbecompletelycharacterizedforthisexample. ForeverygE£g(M)
whereq—^(1<q<oo),thefunctionalLdefinedby
L{f)=J[rf{x)g(x)dx (/€ Cp) (1.1)
isinX* (see,e.g.,Royden(1988,p. 131)). Conversely,thefamousRieszRepresen-
tationTheorem(see,e.g.,Royden(1988,p.132))assertsthateveryLEX*isofthe
form(1.1)forsomefunctiongE£,(M)whereq= (1<q<oo). Thefunctiong
isuniqueuptosetsofLebesguemeasure0.
Let n > 1} beasymmetric Bernoullisequence; that is, {F„, n > 1}
is asequenceofindependent andidenticallydistributed (i.i.d.) randomvariables
withP{Y{ =1}=P{YX = -1} = 1/2. LetX°° =XxXxXx •• anddefine
C(X) ={(vi,v2,... E X°° :Y^=iYnvnconvergesinprobability}.Let 1 <p< 2.
ThenXissaidtobeofRademachertypepifthereexistsaconstant0<C<oo
1
5
suchthat
OO P
E IKIIPfora11(«i,«*,-••)€ C(X).
n=l n-l
Itshouldbepointedoutthattheconditionthattheseries Y„vnconverges
inprobabilityisequivalenttotheconditionthattheseries Ynvnconvergesa.c.
Indeed, ItoandNisio (1968) provedthataseriesofindependent randomelements
convergesinprobabilityifandonlyifitconvergesa.c.therebyextendingthecele-
bratedtheoremofLevy(see,e.g.,ChowandTeicher(1997,p.72))fromtherealline
toBanachspaces.
Hoffmann-JprgensenandPisier(1976)provedfor1<p<2thatarealseparable
BanachspaceisofRademachertypepifandonlyifthereexistsaconstant0<C<oo
suchthat
E
(1-2)
«=i i-
foreveryfinitecollection{Pi,-••,Vn}ofindependentrandomelementswithEV{=
0, 1<f<n. OthercharacterizationsofaBanachspacebeingofRademachertype
pareprovidedbyWoyczyriski(1978).
IfarealseparableBanachspaceisofRademachertypepforsome1 <p<2,
thenitisofRademachertypeqforall1<q<p.EveryrealseparableBanachspaceis
ofRademachertype(atleast)1whilethe£p-spacesand^-spacesareofRademacher
typemin{2,p} forp > 1. (SeeWoyczynski (1978, pp. 343-353) fordetails and a
thoroughdiscussion.) EveryrealseparableHilbertspaceandrealseparablefinite-
dimensionalBanachspaceisofRademachertype2(seePisier(1986)). Inparticular,
thereallineIRisofRademachertype2.