CHAPTER 4. LIMIT THEOREMS IN STATISTICS 4.1. SEQUE

McFadden,StatisticalTools©2000Chapter4-1,Page79______________________________________________________________________________CHAPTER4.LIMITTHEOREMSINSTATISTICS4.1.SEQUENCESOFRANDOMVARIABLES4.1.1.Agreatdealofeconometricsusesrelativelylargedatasetsandmethodsofstatisticalinferencethatarejustifiedbytheirdesirablepropertiesinlargesamples.Theprobabilisticfoundationsfortheseargumentsare“lawsoflargenumbers”,sometimescalledthe“lawofaverages”,and“centrallimittheorems”.Thischapterpresentsthesefoundations.Itconcentratesonthesimplestversionsoftheseresults,butgoessomewayincoveringmorecomplicatedversionsthatareneededforsomeeconometricapplications.Forbasiceconometrics,themostcriticalmaterialsarethelimitconceptsandtheirrelationshipcoveredinthissection,andforindependentandidenticallydistributed(i.i.d.)randomvariablesthefirstWeakLawofLargeNumbersinSection4.3andthefirstCentralLimitTheoreminSection4.4.Thereadermaywanttopostponeothertopics,andreturntothemastheyareneededinlaterchapters.4.1.2.ConsiderasequenceofrandomvariablesY1,Y2,Y3,....TheserandomvariablesareallfunctionsYk(s)ofthesamestateofNatures,butmaydependondifferentpartsofs.ThereareseveralpossibleconceptsforthelimitYoofasequenceofrandomvariablesYn.SincetheYnarefunctionsofstatesofnature,theselimitconceptswillcorrespondtodifferentwaysofdefininglimitsoffunctions.Figure4.1willbeusedtodiscusslimitconcepts.Panel(a)graphsYnandYoasfunctionsofthestateofNature.AlsographedarecurvesdenotedYo±anddefinedbyYo±gwhichforeachstateofNaturesdelineateang-neighborhoodofYo(s).ThesetofstatesofNatureforwhich*Yo(s)-Yn(s)*gisdenotedWn.Panel(b)graphstheCDF'sofYoandYn.Fortechnicalcompleteness,notethatarandomvariableYisameasurablereal-valuedfunctiononaprobabilityspace(S,F,P),whereFisa-fieldofsubsetsofS,PisaprobabilityonF,and“measurable”meansthatFcontainstheinverseimageofeverysetintheBorel-fieldofsubsetsoftherealline.TheCDFofavectorofrandomvariablesisthenameasurablefunctionwiththepropertiesgivenin3.5.3.4.1.3.YnconvergesinprobabilitytoYo,ifforeachg0,limn64Prob(*Yn-Yo*g)=0.ConvergenceinprobabilityisdenotedYn6pYo,orplimn64Yn=Yo.WithWndefinedasinFigure4.1,Yn6pYoifflimn64Prob(Wn)=0foreachg0.4.1.4.YnconvergesalmostsurelytoYo,denotedYn6asYo,ifforeachg0,limn64Prob(supm$n*Ym-Yo*g)=0.ForWndefinedinFigure4.1,thesetofstatesofnatureforwhich*Ym(w)-Yo(w)*gforsomem$nisWm,andYn6asYoiffProb(WnN)60.^m$n^m$nAnimplicationofalmostsureconvergenceislimn64Yn(s)=Yo(s)a.s.(i.e.,exceptforasetofstatesofNatureofprobabilityzero);thisisnotanimplicationofYn6pYo.McFadden,StatisticalTools©2000Chapter4-2,Page80______________________________________________________________________________01StateofNature01RealizationsofRandomVariablesYoYnYo+Yo-Wn01Probability01RandomVariablesCDFofYoCDFofYnFIGURE4.1.CONVERGENCECONCEPTSFORRANDOMVARIABLESPanel(a)Panel(b)McFadden,StatisticalTools©2000Chapter4-3,Page81______________________________________________________________________________4.1.5.Ynconvergesin-mean(alsocalledconvergencein22norm,orconvergenceinLspace)toYoiflimn64E*Yn-Yo*=0.For=2,thisiscalledconvergenceinquadraticmean.Thenormisdefinedas2Y2=[*Y(s)*@@P(ds)]1/=[E*Y*]1/,andcanbeinterpretedasaprobability-mSweightedmeasureofthedistanceofYfromzero.Thenormofarandomvariableisamoment.Therearerandomvariablesforwhichthe-meanwillnotexistforany0;forexample,YwithCDFF(y)=1-1/(logy)fory$ehasthisproperty.However,inmanyapplicationsmomentssuchasvariancesexist,andthequadraticmeanisausefulmeasureofdistance.4.1.6.YnconvergesindistributiontoYo,denotedYn6dYo,iftheCDFofYnconvergestotheCDFofYoateachcontinuitypointofYo.InFigure4.1(b),thismeansthatFnconvergestothefunctionFopointbypointforeachargumentonthehorizontalaxis,exceptpossiblyforpointswhereFojumps.(Recallthatdistributionfunctionsarealwayscontinuousfromtheright,andexceptatjumpsarecontinuousfromtheleft.Sinceeachjumpcontainsadistinctrationalnumberandtherationalsarecountable,thereareatmostacountablenumberofjumps.ThenthesetofjumppointshasLebesguemeasurezero,andtherearecontinuitypointsarbitrarilyclosetoanyjumppoint.Becauseofright-continuity,distributionfunctionsareuniquelydeterminedbytheirvaluesattheircontinuitypoints.)IfAisanopenset,thenYn6dYoimpliesliminfn64Fn(A)$Fo(A);conversely,Aclosedimplieslimsupn64Fn(A)#Fo(A)seeP.Billingsley(1968),Theorem2.1.Convergenceindistributionisalsocalledweakconvergenceinthespaceofdistributionfunctions.4.1.7.TherelationshipsbetweendifferenttypesofconvergencearesummarizedinFigure4.2.Inthistable,“A||B”meansthatAimpliesB,butnotviceversa,and“A}|B”meansthatAandBareequivalent.ExplanationsandexamplesaregiveninSections4.1.8-4.1.18.Onfirstreading,skimthesesectionsandskiptheproofs.4.1.8.Yn6asYoimpliesProb(Wn)#Prob(Wm)60,andhenceYn6pYo.However,^m$nProb(Wn)60doesnotnecessarilyimplythattheprobabilityofWmissmall,soYn6pYodoes^m$nnotimplyYn6asYo.Forexample,taketheuniverseofstatesofnaturetobethepointsontheunitcirclewithuniformprobability,taketheWntobesuccessivearcsoflength2/n,andtakeYntobe1onWn,0otherwise.ThenYn6p0sincePr(Ynú0)=1/n,butYnfailstoconvergealmostsurelytozerosincethesuccessivearcswra