; TeX output 1997.11.24:1651Nffkz0K`yff cmdunh10NewYorkJournalofMathematicsSffff2 .(5html:|{Ycmr8NewXYJorkJ.Math.Gw html:X8html:!2@cmbx83A G" html:(1997)>html:15{30@ html:.\qNff cmbx12Convergencef@ofthe1DFff cmmib10p-Series 0forf@StationarySequences/" jhtml:N cmbx12I.Assani3, html:UQBP%fcmcsc8Abstract.LetX(2cmmi8X;cmmi6n7)b‰EџB1=pSH=\tE(jXq1*j);ja:e:P(ii)IfXXn7(x)\t=f(T.:-:nLqx)where(X&;jF(;;T.:)isanergoBpC‰II‰OVB1=p]œ=\tlZ fda.e.Xfor-f>;\t0;@f2LXlogeL:'ՍPFJurthermoreXthemaximalfunction,.VpsupkP1;\t0;NIntroGduction6 html:x15 P2.\html:ConvergenceUUofthe b> cmmi10p-seriesforiidsequences html:_D18P3.\html:ConvergenceUUofthep-seriesforergoGdicstationarysequences4 html:23Phtml:References.; html:30!UT8html: html:1.Intro`ductionDLet虮Z0ercmmi7n ZbGeasequenceofindependent, iidenticallydistributedrandomvqariables8andO(anq~)asequenceofpGositiverealnumbGers.JThea.e.convergenceOoftheweighted8averagesUUhtml: html:8(!", cmsy10)7b u cmex10PލŝN%ŝnٓRcmr7=1Ʈanq~Zn7bfe2 (֍PAn4;8Mff< ReceivÎedXJune9,1997. ##fcmti8Mathematics~SubjectClassi cation.@28D05,X60F15,60G50. Key1wordsandphrases.cpSeries,maximalfunction,iidrandomvariablesandstationarysequences.**۞Ѵc( 01997StateUniv9ersityofNewYjorkbISSN1076-9803/9715*$816ԜF C cmbxti10I.$Assani@8wheretAn8=Pލ USN% USn=1|anq~,hasbGeencharacterizedbyB.Jamison,S.OreyandW.Pruitt 8([html:JOP html:]).qTheyUUprovedthattheconditionhtml: html:_8(0)supǍn<$x䍒~Nnwfe za (֍niE<1 8wherex䍑j~Nn=#fkh+:Í!WakJǟofe bAk= M1Jǟ&fe~n oxgisnecessaryandsucientforthea.e.convergenceof p8the4 weightedaverages(html: html:)to- msbm10E(Z1|s).fIn[html:A1  html:],:interestedbythea.e.fconvergence(y[٫)8ofUUaveragesoftheform׍ PލʀGN%ʀGn=1ۼpXnq~(x)g[٫(S^n y) feUT˟ (֍&gNz ;덑8weOyconsideredthemaximalfunctionN^O!cmsy7(x)g=supv7qƯn&hŐNnl(x)Őʉfeן-n7NwhereOyNnq~(x)=#fk:c&h933Xk_(x)933ʉfeߟkR] tѬ1K&fe~n g,UU(Xk0).qW*eprovedUUthefollowing:8html: html:58(1)'4IfnXnareiidrandomvqariablesandE(jX1|sj)<1nthenN^(x)is nitea.e.^%html: html:8(2)&zIfݽtheXnO;aregivenbyanergoGdicdynamicalsystem(i.e.,Xnq~(x)=f(Tc^n x)!wherez(X:;F9;;Tc)isanergoGdicdynamicalsystemandf ameasurablenon-!negativefunction)thenforallp;Q11No html:_O()C(fx:N(x)>g<$KCpKwfe  (֍-rp(cZyjfjpRdforUUall4">0:x!F*urthermore%forallp,Y1"e  html:)showsthattheconstantCpdisofthe form ~C&fe[Ɵp1l֫whereUUC qisanabsoluteconstantindepGendentofp.) O< IfUU00opR#fi1;UPjxij>g^\t۬1=p:ثItUUiseasilyseenthatforr5<pRCk(xiTL)kp;1O^ #X tKi?jxijpR^ Ɵ۬1=pw^<$8p Vwfe (֍p8r"^)X۬1=p6k(xi)kr;12ڍ(cf.q[SW]).UUInparticular,forallp,1 html:3)Y(p81)1=p ^X tܯi$-jxiTLjpR^ Ɵ۬1=pwp1=p k(xi)k1;1X: rAsqk(xiTL)k1;1랱Fsup$qƯn&hsڬ#fk+B:xk_1=ngsڟʉfe1XnNH,PforbGoundedsequencesthepreviousinequalityappliedpGointwisetoastationarysequence(Xnq~)ofintegrablefunctionsgivesusnotonlyUUtheexistenceofthep-series>G(p81)1=p ^X tܯi$-j<$33XiTL(x)33wfej (֍ ԻiбjpR^ Ɵ۬1=p,qforUUallIgence$ofthep-SeriesforStationarySei>quences#ث17@8butUUalsotheinequality!48(html: html:4)DsupIxܬ14 html:)andsomeofourpreviousresultssuggest i"8theUUstudyofthelimitwhenptendsto1^+ oftheseriesp(p81)1=p ^M1 X tگi=1$-j<$33XiTL(x)33wfej (֍ ԻiбjpR^ Ɵ۬1=po:"w8 "V cmbx10De nition.Letd(Xnq~)bGeastationarysequenceofintegrablefunctions.![Thepseries 8assoGciatedUUtothissequenceisthea.e.qseries(whenitexists):!mp(p81)1=p ^M1 X tگi=1$-j<$33XiTL(x)33wfej (֍ ԻiбjpR^ Ɵ۬1=po:(~̍DInthisnote,JusinganelementarylemmaonsequenceofrealnumbGers,Jwewill8showUUthatfor(Xnq~)iidwithE(jX1|sj)<1UUthep-seriesk0(p81)1=p ^M1 X tگi=1$-j<$33XiTL(x)33wfej (֍ ԻiбjpR^ Ɵ۬1=p,qconvergesUUa.e.qto|~E(jX1|sj)"8whenUUptendsto1^+. JÍDTheUUsameargumentshowsthatthepseries"N(p81)1=p ^M1 X tگi=1$- $- $- $- *(Qލ2'߯H%2'jg=1B)6XjT;i(xj6)(feA1  html:].nVDW*eRcanremarkthatforeachpthefunctionGpR(x)=(p31)^1=p_Q`X5Pލp1%pi=1.gj&h33Xi*(x)33ʉfe6ij^p`Uݬ1=p8isZnot': cmti10inte}'grable,asZGpR(x)(pBW html:].LSoFc^s(x))= bsupI1second html:partofthisnotewewillfoGcusontheergodicstationarycase.oKW*e8willUconsideranergoGdicdynamicalsystem(X:;F9;;Tc)Uandanonnegativemeasur-8ableUUfunctionf.qUsing(html:2 html:)wewillshow rstthat#~<$îNnq~(f)(x)ßwfe( (֍\nϰ=ӍK#fk:&hf(Trkrx)ʉfe۟ &k!q1=ngK _feY' (֍)Ǯn"$818ԜI.$Assani@8convergesinL^1MNnormtoĴR ֮fd.ZThenusingextrapGolationmethodswewillshow 8thatU8(html: html:5)- - - - -   J^<$îf(Tc^kOx)ßwfe? (֍ uk5^    ̕ 1;1k; k; k; k; k; u 1=<1forfڧ2L(LogUULY):"DOneyofourinterestsin(html:5 html:)liesinthefollowingobservqation:Ifwedenoteby?&h933f(Trnt0ncmsy5 )5x)933ʉfeZc Un\aUUdecreasingrearrangementofthesequence&hf(Trnx)ʉfe,( Un,thenwehave!<8(html: html:6)  o2^<$ٮf(Tc^kOx)ٟwfe? (֍ ukK^, , , , ۺ 1;1Wi=supǍwnn<$33f(Tc^nr Hx)33wfe$Rh (֍ unr&ή: Í8HenceforfB2/L(LogUULY),W(html:6 html:)providesuswithsomeinformationonthedecreasing 8rateUUofthesequence&hf(Trnx)ʉfe,( Un. YnDUsingUU(html:5 html:),wewillprovethatforfڧ2Llog?L,f0,html: html:!p8(6^09)NM፬1(x)= QsupI1 html:7)u rlimq"p!1+Y(p81)1=p ^k1 `X n=1#"^<$+Ef(Tc^n x)+Ewfe $- (֍ ʮnM ^TfۯpYk^`aߟ۬1=poݠ=cZifdUUa.e.,q():"9э8TheintegrabilityofM^l1(x)forfinLLogLextendstheresultsontheintegrabilityof8thesupn&hf(Trnx)ʉfe,( Un$ŦintheergoGdiccase.@W*edonotknowatthepresenttimeif(html:7 html:)holds 8forǮf2L^1B:.Finally*,inthehtml:third html:partofthispapGerwewillstudytheconnection8bGetweenUUthemaximaloperators%G8M፬1(f)(x)= QsupI1 html:2.Convergenceofthe2DF cmmib10p-seriesforiidsequencesYn82.1.Theonedimensionalcase.TheSnextelementarylemmawillbGeusefulfor8theUUconvergenceweareloGokingfor.4$eConveri>gence$ofthep-SeriesforStationarySei>quences#ث19@8LemmaThtml: html:1.L}'et-(xnq~)nbeasequenceofnonnegativenumberssuchthat<$`Qxk`Qwfe (֍kH!ǁka0and`HӍ933#fk:ÍKxkKofe>:kܙ1=ng933 _feHO (֍!iۮnͱ7!iëx~4,thenjЍG8(a)Yphtml: html:limwx㐯p!1+'(p81)^1=p_QbPލ1%n=1-d(h33xn33ʉfe pP?in ֶ)^pRbI߬1=pYb=ix~4:F(b)Yphtml: html:If<$Ѯxk+Bџwfe (֍YWkPrJisade}'creasingrearrangementofthesequence(<$33xk33wfe (֍k )k u.thenk;<$ xk+B wfe (֍YWkPrYpc}'onvergesto6x K.8ProQof.W*edenotebyRn=O fk:Í:k{1=ngandNn=O #fk:Í:k{1=ng=#Rnq~. 8T*oUUprove(html:a html:)itisenoughtoshowthat ڂ@limV3p!1+q(p81)^ ²1 PX \tn=1m(<$33xn33wfe ( (֍n )pR^ޫ=ix~4:"'8W*eUUcanwritetheseries(p81)(Pލ ;1% ;n=1(h33xn33ʉfe pP?in ֶ)^pR)UUinthefollowingway;!Mk'f(p81)^ ²1 PX \tn=1m(<$33xn33wfe ( (֍n )pR^öT=(p81)^/kX 7Gn2R1(<$33xn33wfe ( (֍n )p2+ ןX jn2.qymsbm7N.;nR1#З(<$33xn33wfe ( (֍n)pR^+эöT=Ap2+8BpR:8As-lim-p!1)Apfj=0wejustneedtoconsiderBp=(p)1)P 8n2N.;nR1.P(h33Xn33ʉfe vFBdn ܬ)^pR.\SButwehave&us (p81)61 X n=1<$Nn+1aNnwfe1M (֍ݫ(n+1)rpHBpfj(p81)61 X n=1<$Nn+1aNnwfe1M (֍VܮnrpFڮ:!v>8ItisthenenoughtoprovethatBp ӫissqueezedintotwotermstendingtothe 荑8sameclimit2x . \W*ewillonlyprovecthattheterm(p1)Pލ 81% 8n=1?Nn+1 r Nn?!fe%D#npJcon-8vergesyto$x .^2Thesameargumentshowsthesameconclusionforthesecondterm ?8(p81)Pލ 81% 8n=1?Nn+1 r Nn?!fe%D (n+1)pE . l8DW*eUUhaveZX w(p81)61 X n=1<$Nn+1aNnwfe1M (֍Vܮnrpf=(p81)^\t<$33N133wfe V (֍s1rp$+.1 X n=2<$(;nZ)^pR))(< _feN (֍|g(n1)rpc&^!2:f(p81)^G<$33N133wfe V (֍s1rp$+p61 X n=2<$Nnwfe za (֍n$x<$1lwfe# (֍(n1)rp(?/^-M:#f8AshNnʉfe <%nNconvergesUUtox aƫandPލ 㐲1% 㐯n=2 +%1 R&feȟ(n1)p= )1K&fe[Ɵp1ޙweconcludethat#M lim(p!1+(p81)61 X n=1<$Nn+1aNnwfe1M (֍VܮnrpH=ix~4:F$820ԜI.$Assani@D(html:b: html:)UUT*oobtaintheconvergenceUUofthesequencek5ʯxc3kʟ/fe >>:k+B"tox aƫwecanobservethat~zSlimt!1Ӎꆫ#f`:ÍKx`KofeWV9` K1K&fes{t )gꆟ _fe<6 (֍bt =ix~4:ō8Ifwetaketheincreasingsequencetk B= sk+Br9&fe >xc3kwhere5xc3k/fe >>:k+B:pisthek^th 铫termofthe ɍ8decreasingUUrearrangementofthesequenceÍxkofe>:kwecanseethatĕ<${߮xk+B{ߟwfe (֍YWkPr#ٱ8#f`:<$Kx`Kwfe  (֍`<$Kxk+BKwfe (֍YWkPreg=kw<$lxk+Blwfe (֍YWkPr'p/convergesUUtoa`wPxf.l: 8ThisUUendstheproGofofthislemma.ȣz* msam10DInU2thispartweonlyconsidersequencesXnưofiidnonnegativerandomvqariables 8suchz3thatE(X1|s)<1.aThisz3assumptioncanbGemadeinviewofthenatureofour8pUUseries.E8Theorem*html: html:2.L}'et L(Xnq~)beasequenceofiidnonnegativerandomvariablessuchthat8E(X1|s)<1.Thenwehave @alim^}Cn!1<$v>fNnq~(x)v>fwfe (֍ |n1L=E(X1|s)a.e.^\iwith4CNnq~(x)=#^k:<$KXk됫(x)Kwfe (֍ k"D<$z1Kwfe (֍n .^^ ;8(html: html:a)P[limp!1+B(p81)1=p ^k1 `X n=1$F^<$-\Xnq~(x)-\wfe8 (֍ nIȼ^NyۯpS}˟^X۬1=phI=E(X1|s),a.e.8(html: html:b)&B8ProQof.By=Wthepreviouslemma,B#(html:b: html:)isanimmediateconsequenceof(html:a html:),soweare8leftUUwithproving(html:a html:).DInUUourproGofofLemma1in[html:A1  html:],weshowedthatwehave 2{S S S S <$ZBXnq~(x)ZBwfe8 (֍ nvT vT vT vT |< 1;1<1a.e.,UUbGecauseDv fe 㑟$limRn!1Ӎh#fk:&hKXk_(x)Kʉfeߟku tѬ1K&fe~n gh _feLs (֍#|n@=E(X1|s):8W*eUUprovedthisbynotingthats-Nnq~(x)=#fk:<$KXk됫(x)Kwfe (֍ k"D<$z1Kwfe (֍n .g=Sf1 X n=1A1C?^ @x:<$KXk됫(x)Kwfe (֍ k<$z1Kwfe (֍n .^:8ThenUUweconsideredGsOVnq~(x)=Sf1 X n=1A1C?^ @x:<$KXk됫(x)Kwfe (֍ k"D<$z1Kwfe (֍n .^8^x:<$KXk됫(x)Kwfe (֍ k<$z1Kwfe (֍n .^:8Kolmogorov'sinequalityforsumsofindepGendentrandomvqariablesleadstothe 8followingUUinequalityforeach>0.l1 /X ¯n=1f  <$Nn2 X(x)8E(Nn2)wfeJ (֍ 7lnr2P P P P V g<1:\$eConveri>gence$ofthep-SeriesforStationarySei>quences#ث21@8AnUUapplicationoftheBorel-Cantellilemmagaveus2ş fe 㑟$lim<$INn2 X(x)Iwfe߻ (֍ 1Wnr2#=lim@"n<$E(Nn2 X)wfeK (֍ nr27V=E(X1|s):~@8ThenUUasimpleinterpGolationallowedustoclaimthat8(html: html:8)2 fe 㑟$limßn<$҄tNnq~(x)҄twfe (֍ |nwZ=E(X1|s):肍8ButUUalsoin[html:A1  html:],Theorem3showsthatforeachp,18(html: html:9)limn!1ӍZ<#fk:&hKYk_(x)Kʉfehk1=ngZ< _feSǴ (֍&㍮n;=E(Y1|s) y8for,(Ynq~)sequenceofiidrandomvqariableswhereE(jY1|sj^pR)<1,forsome18 html:)and(html:9 html:)wegetcE(X1S^8M)=lim@"nӍ#fk:&hKXk_(x)^MKʉfe#zʟk+o`1=ng _fecg (֍.>ns荍limfe 㑎Ӎ#fk:&hKXk_(x)Kʉfeߟku1=ngܟ _feU+ (֍'InO񍍍= fe 㑟$limӍ#fk:&hKXk_(x)Kʉfeߟku1=ngܟ _feU+ (֍'In],=E(X1|s): y8As]3lim@ğME(X1^>M)4=E(X1|s)]3wehaveobtainedaproGofof(html:a html:)fromwhich(html:b: html:)now8followsUUeasily*.%8č82.2.=zTheWm9ultidimensionalcase.The^previoussituationcanbGeextendedtoa8moreUUgeneralsituation.qIn[html:A1  html:]weprovedthefollowing:=S| html:10)Z fe 㑟$limnӍ:#fk:eK#cmex7Qr HD i=1DXi;k Tì(xi*)K ͉fe2utk:j  tѬ1K&fe~n g: _fei (֍17n=gH ݱY ti=1fE(Xi;1 ):y8Thedicultyresidesinthewaythosesetsoffullmeasurex䍑e ivareobtained;Dthey8areUUindepGendentoftheincomingvqariables(XjT;n )forjY>i. DW*e!~wanttoprovethatin(html:10  html:)weactuallyhaveconvergencetoQލ GH% Gi=1>E(Xi;1 ).8MoreUUpreciselywehave:o$822ԜI.$Assani@8Theoremohtml: html:3.Given|Hzap}'ositiveintegerandanonnegativesequenceofiidvari- 8ablesq(X1n m)n 1onthepr}'obabilityqmeasurespace( 1|s;F1;1)qsatisfyingthec}'ondition @8E(X1;1 )<1,bitV]isp}'ossibleto ndasetoffullmeasurex䍑+e 1bsuchthatifx1C2x䍑me 1̺the8followingholds:DF;or\allpr}'obability\measurespaces( 2|s;F2;2)\andallnonne}'gativeiidsequences8(X2;n )n suchthatE(X2;1 )ޮ<1,itisp}'ossibleto ndasetoffullmeasurex䍑qde 2usuch p8thatifx2C2x䍑me 2Vthefollowingholds:DF;orallpr}'obabilitymeasurespaces( H;FH;;H)andalliidse}'quences(XH;nx)n8of~nonne}'gativerandomvariablessatisfyingE(XH;1 )Lh<1~wecan ndasetoffull8me}'asurex䍑i html:11)"e'8and xs9limo2p!1+ ^G(p81)^ ²1 PX \tn=1\ !6|Qލ*EH%*Ei=19 html:12)%M8ProQof.As_Tpreviouslywejustneedtoprove(html:11  html:)toget(html:12 html:).W*euseinductionto8proveUU(html:11  html:).qTheresultistrueforH=1,asshownintheprevioustheorem. ODLet\usassumethattheresultistrueforH|791.xHenceifck=wQލ Y@H1% Y@i=1 Xikܫ(xiTL) 8whereUUxid2x䍑me iwehavenT8(html: html:13)lim(n!1Ӎ۫#fk:ÍKckKofe8k斱 tѬ1K&fe~n g۟ _fe?5 (֍}n=H1  Y ti=1E(Xi;1 ):"2=8TheideaoftheproGofisthesameasinLemma1in[html:A1  html:].W*ehaveforxiб2x䍑ٴe i ,81iH+[1,(XH;nx)asequenceofnonnegativeiidrandomvqariablesandforall8>01 wGX 'kn=1㔮^xH \ɫ:    <$ONn2 X(xH)8E(Nn2)OwfeR (֍$Dnr2[h [h [h [h a ձ^ G<18(html: html:14)"F98where b9UNn2 X(xH)=ӍK#fk:ck_XH8;k =\(xmHlu)Rfe+k3'1=n^2|sgK _fep\P (֍2nr2uή:ҍ8Theinequality(html:14  html:)isobtainedbyapplyingKolmogorov'sinequalitytotheseries8ofUUindepGendentrandomvqariables!m1 jX jk+B=1~G1E\F AxmHlu:33ck_XH8;k =\(xH)33fW )8Pɰk+1=n^8^xH \ɫ:<$Kck됮XH;k (xH)Kwfe4B (֍k<ر1=n^:|$eConveri>gence$ofthep-SeriesforStationarySei>quences#ث23@8TheUUBorel-Cantellilemmaappliedto(html:14  html:)givesuslim3n!1<$VNn2 X(xH)8E(Nn2)VwfeR (֍$Dnr2?6=0a.e.&UY(xH):8AsUUlim8n!1ߍ)z E(NHn2Ό))z Sfen2G=Qލ 8H% 8i=1WE(Xi;1 )UUwehave#5limn!1<$9Nn2 X(xH)9wfe&ul (֍ 0nr23=gH ݱY ti=1fE(Xi;1 )a.e.&UY(xH):"+8TheUUmonotonicityofNnӫgivesusforp^2፯n8n(pn+1)^2[R_kNp2n (xH)_k`fe& (֍pr2\nntB<$KNnq~(xH)Kwfe"L (֍ gpr2\nn*v KN:(pn+1 r )28(xH)K=%fe9Ub (֍mpr2\nnAI=v KN:(pn+1 r )28(xH)K=%fe9Ub (֍ <((pn+1)r2@<$l(pn+1)^2lwfe  (֍(pnq~)r2%|X::8ThisUUlastchainofinequalitiesimpliesthat"'1jlimgZn!1<$8Nnq~(xH)8wfe"L (֍Fٮnϫ=Flimn!1R;Np2n (xH);`fe& (֍pr2\nnE&=gH ݱY ti=1fE(Xi;1 )as<$!zRp^2፯n!zRwfe y~ (֍ html:3.Convergenceofthep-seriesforergo`dicstationarysequencesNDInMthispartthesequenceXn G˫willbGegivenbyanergoGdicdynamicalsystem 8(X:;F9;;Tc)onaprobabilitymeasurespace(X;F9;).Thesequenceisde nedby8theUUrelationXnq~(x)=f(Tc^n x)UUwherefhisanonnegativeintegrablefunction.ލ8PropQositionhtml: html:4.L}'etzͫ(X:;F9;;Tc)b}'eanergodicdynamicalsystemandf\anonneg-8ativeinte}'grablefunction.Wehave!_Ѝʫlim;n!1޴  <$KNnq~(f)Kwfe:e (֍ n1ñ8cZ1fd  I* 1 =0;8غwher}'e `<$ĮNnq~(f)(x)ğwfe( (֍\nϰ=ӍK#fk:&hf(Trkrx)ʉfe۟ &k!q1=ngK _feY' (֍)Ǯn%]8ProQof.W*eknowthatlimcn!1&h*@Nnl(f)*@ʉfe,KnE=ĴRfda.e.?forf%v2L1ɍpK[+ bforsomep,Mq1<8p婱1g(seeTheorem3in[html:A1  html:]).Thedicultyatthislevelcomesfromthenature8ofVthefunctionoff,VENnq~(f);VutheVmapNnǓisnotlinearnorpGositivelyhomogeneous.8ButUUwehavethefollowingpropGerties:ZTD(A)Yhtml: html:k&h33Nnl(f)33ʉfe,Knyk1 ?kfk1x,E#(B)Yhtml: html:IfUUfV;g.arenonnegativefunctionswithdisjointsuppGortthenwehave &hZ33Nnl(f+g@L)Z33ʉfe v –n~(=&hKNnl(f)Kʉfe,Knq+&hlNnl(g@L)lʉfeY>nforUUalln1.cD(C)Yhtml: html:F*orUUallfڧ0integrablefunctionswehavek&h33Nnl(f)33ʉfe,Knyk1Ckfk1|s:$824ԜI.$Assani@8(html:A html:)UUand(html:BW html:)areeasytocheck. DT*oestablish(html:C8 html:)wetakefɱ2:L^1gCforwhichwecan ndforeachnonnegative8numbGers ( iTL)i6Uandsets(Ai)i6Usuchthatf(P| i1 SAiۮ;Ai\AjE=ifi6=jtand8ĴRAPN8ޮ iTL1 JAiҮd(18+)ĴR fd.qW*eUUhaveȮ<$=Nnq~(f)=wfe:e (֍ n&KNnq~(Pލ ;1% ;i=12 iTL1 JAiҫ)KfeH& (֍! FnaDbyUUmonotonicity*.,8Thusq@P @P @P @P <$Nnq~(f)wfe:e (֍ noU oU oU oU 1A     Nnq~(Pލ ;1% ;i=12 iTL1 JAiҫ) feH& (֍! FnR޴ R R R Xa 14A=      IJ1 URX t i=1<$Nnq~( iTL1 JAiҫ)wfe.i (֍4nIB IB IB IB IB NW Ls1htbyUU(html:BW html:)!'A=1 X tկi=1㉴    <$Nnq~( iTL1 JAiҫ)wfe.i (֍4nIB IB IB IB NW 1XLǮ:k$8As W0<$Nnq~( iTL1 JAiҫ)wfe.i (֍4n 8=ӍK#fk:Hʍ f$cmbx71mA1i[S(trk_x)jhfec jk' 1&fe OXn i|ֱgK _fe_/џ (֍,n9 8=_KP:[n i*]6k+B=1"1(IAi1*(Tc^kOx)KӉfeG8 (֍ On`иweUUhaveysߴ s s s <$zLNnq~( iTL1 JAiҫ)zLwfe.i (֍4n9s 9s 9s 9s ǭ 1 8=򣊍[n i*] \vyoX lk+B=1<${i(AiTL){iwfe (֍ Pn4<$K(n iTL)(Ai)Kwfe0# (֍n8.= iTL(Ai):"򍍑8SoN  <$HNNnq~(f)HNwfe:e (֍ n  81 X tկi=1㉮ iTL(Ai)(18+)cZ UQfd: Ѝ8AsUUisarbitrarywehavereachedaproGofof(html:C8 html:).DW*eUUarenowinapGositiontoprovePropGositionhtml:4 html:.DF*or$eachpGositiverealnumbGerM;wecanwritefڧ=fͱ^>MY+gM ߫with$f^M;and8gM 5nonnegativeUUfunctionswithdisjointsuppGort.DW*eUUhave 8<$W`Nnq~(f)W`wfe:e (֍ nv)8cZ ㉮fd=<$KNnq~(fLo^8M)Kwfe1!{ (֍qn8ٱ8cZfLo^8Md+<$lNnq~(gM\)lwfe" (֍s&n) cZ ㉮gM\d:;*8HenceIdB fe 㑟$limy nQ: Q: Q: Q: <$XONnq~(f)XOwfe:e (֍ nv8cZ ㉮fd  bޟ 1i fe 㑟$limy nUQ UQ UQ UQ <$Nnq~(fLo^8M)wfe1!{ (֍qnML8cZ 8(fLo^8M)d   12+8 fe 㑟$limy n    <$Nnq~(gM\)wfe" (֍s&n< < < < B0؟ 1H++8cZ ㉮gM\d:$eConveri>gence$ofthep-SeriesforStationarySei>quences#ث25@8By6AthetheoremmentionedatthebGeginningofthisproof,C8 html:).C8AsUUĴR gM\d!s3ʯM0,UUtheproGofofthispropositioniscomplete.\(8Theorem1html: html:5.L}'et(X:;F9;;Tc)b}'eanergodicdynamicalsystemandfڧ2Llog?L;yf 80.Thenwehavehtml: html:ݍ8(a)d d d d d   ^<$f(Tc^kOx)wfe? (֍ uk^> > > > ş 1;1Ң Ң Ң Ң Ң 0W 1s=    URsupǍ cn5n8<$lf(Tc^nr Hx)lwfe$Rh (֍ unr( ( ( ( R 1Y?<1$m8wher}'e&hf(Trnt )5x)ʉfeZc Un'藺isfora.e.xadecreasingrearrangementofthesequence&hf(Trnx)ʉfe,( Un &u.html: html:XlimS|n!1<$Nnq~(f)(x)wfe( (֍\nlj=cZqfd;>a.e.8(b)~Rhtml: html:*Ulim-p!1+8:^(p81)61 X n=1)^<$KЮf(Tc^n xKПwfe@ (֍ n6^>ۯpBg^J۟۬1=pY=cZqfd;a.e.8(c)$p8ProQof.FirstUUwecanmakethefollowingobservqations:8F*orUUallmeasurablesetsAwehave E)f E)f E)f E)f J^<$SGG1YEA_ի(Tc^kOx)SGGwfe% (֍1ɮkzca^  N 1;1=supIQt>0Ӎ.#fk:&hK1mA0[(Trkrx)Kʉfe*˟ _k'a1=tg. _fe\ (֍,?tu`h=supǍwnӍ.#fk:&hK1mA0[(Trkrx)Kʉfe*˟ _k'a1=ng. _fe_ (֍,?nw=supǍwn<$.Nnq~(1A C)(x).wfe/G (֍Wn8(html: html:15)rꍍ=N(1A C)(x):6Ӎ8BecauseUUofthemaximalinequalityfortheergoGdicaverageswehavehtml: html:^=8(16)|®fx:N(1A C)(x)>g<$d1KwfeW (֍ ;8(A)forUUall11ͮ>0:܍8(NoteUUthatN^(1A C)(x)1,UUhenceforallp1UUwealsohavehtml: html:v8(17)fx:N(1A C)(x)>g<$1Kwfe t (֍rp8(A)):8F*orUUallpGositiverealnumbGersy.wehave:qhtml: html:w8(18)|;yRfi xW 5Pi+1?=yRfi xW 5Pi+1˱<$l(i8+1)^1=i+1lwfe1 V(i8+1)r1=i+19a<$Ky[٫(i8+1)^1=iKwfe- (֍VN(i8+1)2L i8+<$cE1lwfe d (֍(i+1)r2ϟ$826ԜI.$Assani@8(applyEtheinequalityabW گarpڟ&feD4p j+  Lbrq L&fe}Ƀq ,fora=y[ٟ^i=i+1q(i+1)^1=i+1,b= M1ڟ&fe%X'H(i+1)1=i+1,e,,58p= Ki+1K&fe Oiw"andUUq"= pK);fe[Ɵp1P\=i8+1). O5 html:(html:a html:). DW*eUUtakefڧ2Llog?LanddenotebyAitheset^єAid=f2ifڧ<2i+1 tOg:8W*eUUhaveSN(f)N^H1  sX t 0i=1(2i+1 tO14MAݫ)^1 X tկi=1㉮N(2i+1 tO14MAݫ))!Q(=21 X tiei=12i,8N(1A C):8ByUUtakingtheintegralwithrespGecttothemeasurewegetE{kN(f)k1C21 X tiei=12iTLkN(1Ain)k1ҍ8UsingUU(html:17  html:)wegetbrikN(1Ain)k1"<$zJpKwfe (֍(p81)$#supI%\t>03^[t8fx:n(1Ain)(x)>tg]"<$zJpKwfe (֍(p81)%n[8((AiTL))1=p forUUall=vp;1p<1: h8((html:17  html:)y~iscombinedwiththeinequalitykg[ٱkL1  t˯p6);feȟ(p1)3sup*nqƯt>088[tfx]:jg(x)j>tg^1=p ].) ˍ8GoingUUbacktotheevqaluationofkN^(f)k1ȫweget0kN(f)k1H21 X tiei=12i<$(i8+1=i)wfe$ (֍ 1=i*((AiTL))1=i+1H=21 X tiei=12iTL(i8+1)((Ai))i=i+1H=21 X tiei=1q((2iTL(i8+1))i+1=ia(Ai))i=i+1:}Ӎ8ApplyingUU(html:18  html:)toeachterm((2^iTL(i8+1))^i+1=ia(Ai))^i=i+1ԫweUUget`'kN(f)k1ؚ21 X tiei=1q[(2iTL(i8+1))i+1=i_((Ai))<$33(i+1)^1=i33wfe'۔ (֍ ai+1*Ai+<$cE1lwfe d (֍(i+1)r2%]ؚ41 X tiei=1q[2iTLi(Ai)8(1+i)2=i +<$cE1lwfe d (֍(i+1)r2%]ؚ128uR1 X ti=1[2iTLi(Ai)8+<$cE1lwfe d (֍(i+1)r2%]jؚ<$zJ12Kwfe (֍ln 2-~[cZ f7logSfd8+1]:$eConveri>gence$ofthep-SeriesforStationarySei>quences#ث27@8ThusUUwehaveprovedthefollowinginequalityhtml: html:m8(19)`kN(f)k1C<$zJ12Kwfe (֍ln 2-~[cZ f7logSfd8+1]forUUall11ͮfڧ0;f2Llog?L:8ThisUUclearlyendstheproGofofTheoremhtml:5 html:(html:a html:).sGDItUUremainstoshow(html:b: html:).qOurgoalistoprovethatforfڧ0;f2Llog?LUUhtml: html:Ѝ8(20)<$8lim8wfe 㑟 (֍|nOcN(fLo8f^n)=00a.e.68UsingUU(html:19  html:)wehaveforallt>0,BkN(t(fLo8f^n))k1C<$zJ12Kwfe (֍ln 2-~[cZ (t(ff^n))logT(t(ff^n))d+1]%forUUallCafڧ0;f2Llog?L:o-8ThisUUlastinequalitygivesus_BkN(fLo8f^n)k1C<$zJ12Kwfe (֍ln 2-~[cZ (ff^n)logT(t(ff^n)]d+<$l1lwfe (֍Ȯt G]:n8AtUUtheexpGenseoftakingasubsequence,wederivefromitlimk8kN(fLo8f^nk됫)k1C<$zJ12Kwfe (֍ln 2f^<$l1lwfe (֍Ȯt G:­8ThenUUweeasilyget lim&fe nԢN^(fLo8f^n)=0UUa.e.qThisprovesUU(html:20  html:).rDAsUUlim8n&h?Nk_(f^n)?ʉfeӟ Ƣk={]=ĴR kfLo^8nd,UUbGecausef^8nisclearlybGoundedwehave%ԍ<$|QNk됫(fLo^8n)|Qwfe+} (֍&k<$KNk됫(f)Kwfew (֍ k" =<$KNk됫(fLo^8n)Kwfe+} (֍&k37۫+<$lNk됫(fLo8f^n)lwfe>A (֍>k<8andUUaftertakingthelimitsweobtainCcZOURfLo^8nd<$KlimKwfe 㑟 (֍/k<$DBNk됫(f)DBwfew (֍ k2 fe 㑟$lim<$ܮNk됫(f)ܟwfew (֍ k0 fe 㑟$lim<$ܮNk됫(f^8n)ܟwfe+} (֍&kAl+8N[ff^n]:u=cZqfLo^8nd+N[ff^n]:8Finally*,UUbytakingthelim;infwithrespGecttonwecanconcludethatflim"k<$(KNk됫(f)(Kwfew (֍ k =cZqfdUUa.e.8ThisprovesTheoremhtml:5 html:(html:b: html:).[Theoremhtml:5 html:(html:cq html:)nowfollowseasilyfromLemmahtml:1 html:.[This 8endsUUtheproGofofTheoremhtml:5 html:.a$828ԜI.$Assani@8Corollaryhtml: html:6.L}'et(X:;F9;;Tc)b}'eanergodicdynamicalsystemandfڧ2Llog?L;?f 80.Thenther}'eexistsanabsoluteconstantCsuchthat `Ѧ `Ѧ `Ѧ `Ѧ `Ѧ m#supIf_15 html:weknowthat sô  ]supǍl\nn8<$lf(Tc^nr Hx)lwfe$Rh (֍ unr( ( ( ( _ 1 <1:L8AsUUfora.e.qx,foreachp,wehave@(p81)1=p ^k1 `X n=1#"^<$+Ef(Tc^n x)+Ewfe $- (֍ ʮnM ^TfۯpYk^`aߟ۬1=poݠ^ #supǍ1non<$lf(Tc^nr Hx)lwfe$Rh (֍ unr(^2^ T(p1)61 X n=1fѫ1=np ^`۬1=p ;8theUUcorollaryfollowseasily*.) ލ8Remark.BKp1)YOnecanseethatthelimitwhenptendsto1ofthep-Seriesisequalsto Ysuph;qƯn&hp]f(Trnx)p]ʉfe,( Un3W.ThisvhisthereasonwhyweonlyfoGcusontheexistenceoftheYlimitUUwhenptendsto1+.Kp2)YW*e/provedin[html:A2  html:]thatifN^(f)(x)isa.e niteforallfunctionsfG23L^1+ rYthenUUM^l2(f)(x)isalsoa.e niteforallfunctionsfڧ2L^1|s.Kp3)YTheresultsobtainedinthisnotecanbGeextendedtoincreasingsequencesofYintegersUU(pnq~)n.qThecorrespGondingmaximalfunctiontoconsiderissimplyZرk^<$f(Tc^pn M)(x)wfe, (֍n5e^=ٱk1;1X:{DT*oUUillustratethiswehavethefollowingPropGosition.8PropQositionhtml: html:7.L}'etƫ(X:;F9;;Tc)b}'eanergodicdynamicalsystem,pa xedpositive8r}'ealLnumber1 html:̃8(a). . . . ^<$Lf(Tc^pn Mx)Lwfe$9m (֍jn8^    ̣ 1;1@V<1a.e.forallfڧ2L1ɍpK[+.,8html: html:E8(b){Rsupx⍒kn5kPp11 ^pX n=k W^<$(Jf(Tc^pn Mx)(Jwfe$9m (֍jnNS^U^ۯp]Ȯ<1a.e.forallfڧ2L1ɍpK[+.aӍ8html: html:@8(c)d8supd`}N<$`M1PNwfe  (֍NLN X N'n=1 Pf(Tcpn Mx)<1a.e.forallfڧ2L1ɍpK[+.:8html: html:8(d)supdӯN<$1wfe  (֍N}eN YX }n=1;f(Tcpn Mx)<1a.e.forallfڧ2L(p;1).:8Thenwehavethefollowingimplic}'ations:WI(html:a html:)implies(html:d: html:), w(html:b html:)isequivalentto(html:cq html:), 8(html:b: html:)implies(html:a html:)and(html:cq html:)implies(html:d html:).?$eConveri>gence$ofthep-SeriesforStationarySei>quences#ث29@8ProQof.TheAimplications(html:cq html:)implies(html:b: html:)and(html:b html:)implies(html:a html:)canbGeprovedAthesame 8waywedidin[html:A1  html:]fortheusualCesaroaverages.Q(SeetheproGofofTheorem3part8b) in[html:A1  html:]).XTheimplication(html:cq html:)implies(html:d: html:)isadirectconsequenceofthestructure8ofUUL(p;q[٫)spacesasshownin[html:SWX html:].DItZremainstoproveZtheimplications(html:a html:)implies(html:d: html:)and(html:b html:)implies(html:cq html:).F*or(html:a html:)8implies֞(html:d: html:),wecannoticethat(html:a html:)impliestheexistenceofa niteconstantCpusuch8thatUUforallfڧ2L1ɍpK[+R}8(html: html:21)c:fx:  UR^<$f(Tc^pn Mxwfe Uޟ (֍ *n2n ^9~ 9~ 9~ 9~ ?X 1;1P(>g<$KCpKwfe  (֍-rp(cZyjfjpRdforUUall4">0:\s8InUUtheparticularcaseoffڧ=1A ,(html:21  html:)willgiveusthefollowinghtml: html:H8(22)h fx:supd]N<$-1.wfe  (֍N$N !X n=11m017-.A=(Tcpn Mx)>g<$KCpKwfe  (֍-rp(A)forUUall4">0H8bGecauseH    W^<$1oA(Tc^pn Mx)wfe* (֍AngF^ú ú ú ú Q 1;1d=supd]N<$-1.wfe  (֍N$N !X n=11m017-.A=(Tcpn Mx):.8As>thisinequalityisvqalidforallmeasurablesetsA,wecanconcludethatthe8maximalUUopGerator:SM(f)(x)=supd]N<$-1.wfe  (֍N$N !X n=11m0f(Tcpn Mx)fg8isofrestrictedweaktypGe(p,p)(see[html:SWX html:])./Inotherwords,themaximalopGerator8M^ JmapsthecharacteristicfunctionofanymeasurablesetAfromL(p;1)into8L(p;1).iIt=isshownin[html:SWX html:]thatthenatureofthemaximalopGeratorM^andthe8existence7ofanequivqalentnormonL(p;1),makingitaBanachspace,M^ S6maps8continuouslyjallfunctionsf,t2L(p;1)intoL(p;1).Thismeanstheexistenceofa8 niteUUconstantCpsuchthatforallfڧ2L(p;1),html: html:8(23)dfx:supd]N<$-1.wfe  (֍N$N !X n=11m0f(Tcpn Mx)>g<$KCpKwfe  (֍-rpkfkp;1+-pDforUUalle >0:8F*romUUthiswecanclearlyderive(html:d: html:).DTheyimplication(html:b: html:)implies(html:cq html:)canbGeobtainedbysummation.JF*orf~2L1ɍpK[+ flet8usdenotebyCxothe niteconstantwhichdominatesthesuponk. jThenforeach8k,UUwehaveN;IkPp1cS2k ^pX n=k W^<$(Jf(Tc^pn Mx)(Jwfe$9m (֍ 2kNS^U^ۯp]Ȯ<Cx:1Ǎ8ThisUUimpliestheinequality61supx⍒2k<$1GwfeV (֍k̼Ƭ2k ɷX 0xn=k[f(Tcpn Mx)+p<2pRCx:xፑ8F*rom,thiswecanderivebyconvexitytheuniformbGoundednessoftheaverages Z 9j¬1933&fekk@|PލK ٬2k%K n=k\f(Tc^pn Mx).qThisUUpropGertyallowsustoobtain(html:cq html:)withoutdiculty*.Wp8⍑8Remark.ItJwouldbGeinterestingtoknowif(html:a html:)implies(html:cq html:)foranyincreasing8sequenceYpnandanyp,Z1<p<1Y.~F*orp<=1,pn?=nYwealreadymentionedthat8(html:a html:)UUimplies(html:cq html:)(see[html:A2  html:]). $830ԜI.$Assani@8References8html: html: [html: html:A1]hI.UNAssani,nStrong~lawsforweightedsumsofiidrandomvariables,nDukÎeUNMath.J.88(1997),h217{246.[html: html:A2]hI.XAssani,Return~timesandBirkho theoremsinL1ofproductmeasures,XPreprinÎt.[html: html:B]hD. Burkholder,Successiveconditionalexpectationsofanintegrablefunction,AMSt33h(1962),X887{893.[html: html:JOP]hB.aJamison,*S.OreyJ,andW.Pruitt,ConvergenceAofweightedaveragesofindependenthrandom~variables,XZ.WJahrheinlicÎhkeitstheorieXVerw.Gebiete.4(1965),40{44.[html: html:SW]hE.M$SteinandG.WJeiss,kIntroductiontoFWwourierAnalysisonEuclideanSpaces,kPrincetonhUnivÎersityXPress,Princeton,NJ,1971. html: html:DepuUartment#ofMauUthematics,VUniversity#ofNoruUthCarolinaatChapelHill,VChapelHill,#NC27599 assani@math.unc.edu[TÎyp cmmi10Zcmr5ٓRcmr7K`y cmr10!