4Convergence of the p-Series for stationary sequen

NewYorkJournalofMathematicsNewYorkJ.Math.3A(1997)15{30.Convergenceofthep-SeriesforStationarySequencesI.AssaniAbstract.Let(Xn)beastationarysequence.Weprovethefollowing(i)Ifthevariables(Xn)areiidandE(jX1j)1thenlimp!1+(p−1)1Xn=1jXn(x)jpnp1=p=E(jX1j);a:e:(ii)IfXn(x)=f(Tnx)where(X;F;;T)isanergodicdynamicalsystem,thenlimp!1+(p−1)1Xn=1f(Tnx)np1=p=Zfda.e.forf0;f2LlogL:Furthermorethemaximalfunction,sup1p1(p−1)1=p1Xn=1f(Tnx)np1=pisintegrableforfunctions,f0;f2LlogL:TheselimitsarelinkedtothemaximalfunctionN(x)=k(Xn(x)n)k1;1.Contents1.Introduction152.Convergenceofthep-seriesforiidsequences183.Convergenceofthep-seriesforergodicstationarysequences23References301.IntroductionLetZnbeasequenceofindependent,identicallydistributedrandomvariablesand(an)asequenceofpositiverealnumbers.Thea.e.convergenceoftheweightedaverages()PNn=1anZnAn;ReceivedJune9,1997.MathematicsSubjectClassication.28D05,60F15,60G50.Keywordsandphrases.pSeries,maximalfunction,iidrandomvariablesandstationarysequences.c1997StateUniversityofNewYorkISSN1076-9803/971516I.AssaniwhereAn=PNn=1an,hasbeencharacterizedbyB.Jamison,S.OreyandW.Pruitt([JOP]).Theyprovedthatthecondition(0)supn~Nnn1where~Nn=#fk:akAk1ngisnecessaryandsucientforthea.e.convergenceoftheweightedaverages()toE(Z1).In[A1],interestedbythea.e.convergence(y)ofaveragesoftheformPNn=1Xn(x)g(Sny)N;weconsideredthemaximalfunctionN(x)=supnNn(x)nwhereNn(x)=#fk:Xk(x)k1ng,(Xk0).Weprovedthefollowing:(1)IfXnareiidrandomvariablesandE(jX1j)1thenN(x)isnitea.e.(2)IftheXnaregivenbyanergodicdynamicalsystem(i.e.,Xn(x)=f(Tnx)where(X;F;;T)isanergodicdynamicalsystemandfameasurablenon-negativefunction)thenforallp;1p1thereexistsaniteconstantCpsuchthat()fx:N(x)gCppZjfjpdforall0:Furthermoreforallp,1p1,forallf2Lp+wehavelimn!1Nn(x)n=Rfda.e.(Acloserinspectionoftheproofof()showsthattheconstantCpisoftheformCp−1whereCisanabsoluteconstantindependentofp.)If0p1,and(xi)i1isasequenceofnonnegativerealnumbers,setk(xi)kp;1=sup0p#fi1;jxijg1=p:Itiseasilyseenthatforrpk(xi)kp;1Xijxijp1=ppp−r1=pk(xi)kr;1(cf.[SW]).Inparticular,forallp,1p2wehave(3)(p−1)1=pXijxijp1=pp1=pk(xi)k1;1:Ask(xi)k1;1supn#fk:xk1=ngn,forboundedsequencesthepreviousinequalityappliedpointwisetoastationarysequence(Xn)ofintegrablefunctionsgivesusnotonlytheexistenceofthep-series(p−1)1=pXijXi(x)ijp1=pforallp;1p1;Convergenceofthep-SeriesforStationarySequences17butalsotheinequality(4)sup1p1(p−1)1=pXijXi(x)ijp1=p2k(Xi(x)i)k1;1ifk(Xi(x)i)k1;11.Theinequality(4)andsomeofourpreviousresultssuggestthestudyofthelimitwhenptendsto1+oftheseries(p−1)1=p1Xi=1jXi(x)ijp1=p:Denition.Let(Xn)beastationarysequenceofintegrablefunctions.Thepseriesassociatedtothissequenceisthea.e.series(whenitexists):(p−1)1=p1Xi=1jXi(x)ijp1=p:Inthisnote,usinganelementarylemmaonsequenceofrealnumbers,wewillshowthatfor(Xn)iidwithE(jX1j)1thep-series(p−1)1=p1Xi=1jXi(x)ijp1=pconvergesa.e.toE(jX1j)whenptendsto1+.Thesameargumentshowsthatthepseries(p−1)1=p1Xi=1QHj=1Xj;i(xj)ip1=pconvergesa.e.toHYj=1E(jXj;1j)where(Xj;n)nareiidrandomvariablessatisfyingtheconditionE(jXj;1j)1,andthevariablesxjareselectedinauniversalwayspeciedin[A1].WecanremarkthatforeachpthefunctionGp(x)=(p−1)1=pP1i=1jXi(x)ijp1=pisnotintegrable,asGp(x)(p−1)1=psupijXi(x)ij,andfor(Xi)iidwithE(jX1jlogjX1j)=1,thefunctionsupijXi(x)ijisnotintegrable,asshownbyD.Burkholderin[B].SoF(x)=sup1p1Gp(x)isasupremumofnonintegrablefunc-tions.ThismakesthehandlingofthefunctionF(x)somewhatdelicate.Inthesecondpartofthisnotewewillfocusontheergodicstationarycase.Wewillconsideranergodicdynamicalsystem(X;F;;T)andanonnegativemeasur-ablefunctionf.Using(2)wewillshowrstthatNn(f)(x)n=#fk:f(Tkx)k1=ngn18I.AssaniconvergesinL1normtoRfd.Thenusingextrapolationmethodswewillshowthat(5)f(Tkx)k1;111forf2L(LogL):Oneofourinterestsin(5)liesinthefollowingobservation:Ifwedenotebyf(Tnx)nadecreasingrearrangementofthesequencef(Tnx)n,thenwehave(6)f(Tkx)k1;1=supnnf(Tnx)n:Henceforf2L(LogL),(6)providesuswithsomeinformationonthedecreasingrateofthesequencef(Tnx)n.Using(5),wewillprovethatforf2LlogL,f0,(60)M1(x)=sup1p1(p−1)1=p1Xn=1f(Tnx)np1=p;and(7)limp!1+(p−1)1=p1Xn=1f(Tnx)np1=p=Zfda.e.,():TheintegrabilityofM1(x)forfinLLogLextendstheresultsontheintegrabilityofthesupnf(Tnx)nintheergodiccase.Wedonotknowatthepresenttimeif(7)holdsforf2L1.Finally,inthethirdpartofthispaperwewillstudytheconnectionbetweenthemaximaloperatorsM1(f)(x)=sup1p1(p−1)1=p1Xn=1f(Tnx)np1=p;M2(f)(x)=supN1NNXn=1f(Tnx)andf(Tnx)n1;1=N(f)(x):IfthereisnoambiguitywewillsimplydenotethesemaximalfunctionsbyM1(x),M2(x)andN(x).2.Convergenceofthep-seriesforiidsequences2.1.Theonedimensionalcase.Thenextelementarylemmawillbeusefulfortheconvergencewearelookingfor.Convergenceofthep-SeriesforStationarySequences19Lemma1.Let(xn)nbeasequenceofnonnegativenumberssuchthatxkk!k0and#fk:xkk1=ngn7!x,then(a)limp!1+(p−1)1=pP1n=1(xnn)p
1=p=x:(b)Ifxkkisadecreasingrearrangementofthesequence(xkk)kthenk
xkkconvergestox.Proof.WedenotebyR