THE FUNCTIONAL CENTRAL LIMIT THEOREM AND WEAK CONVERGENCE TO STOCHASTIC INTEGRALS I

Transcription

1 Ecoomeric Theory, 6, 000, ried i he Uied Saes of America+ THE FUNCTIONAL CENTAL LIMIT THEOEM AND WEAK CONVEGENCE TO STOCHASTIC INTEGALS I Weaky Depede rocesses OBET M. DE JONG Michiga Sae Uiversiy JAMES DAVIDSON Cardiff Uiversiy This paper gives ew codiios for he fucioa cera imi heorem, ad weak covergece of sochasic iegras, for ear-epoch-depede fucios of mixig processes+ These resus have fudamea appicaios i he heory of ui roo esig ad coiegraig regressios+ The codiios give improve o exisig resus i he ieraure i erms of he amou of depedece ad heerogeeiy permied, ad i paricuar, hese appear o be he firs such heorems i which viruay he same assumpios are sufficie for boh modes of covergece+. INTODUCTION Asympoic heory for iegraed processes is a area of research where resus from fucioa imi heory are crucia+ These resus are he mai uderpiig of he ecoomeric aaysis of modes wih iegraed ad coiegraed variabes+ hiips ~986, 987!, hiips ad Durauf ~986!, ark ad hiips ~988, 989!, ad Johase ~988, 99! are he we-kow semia coribuios o wha is ow a very exesive ieraure+ I his heory, he sampe saisics whose disribuios are sough are ypicay fucios of sampe momes i which he daa may be ~a! saioary or ~b! iegraed or ~c! a mixure of boh+ The asympoic aaysis of each of hese cases requires a differe echique+ Case ~a! is he sadard oe eadig o Gaussia imi disribuios+ I case ~b!, weak covergece o fucioas of Browia moio or reaed Gaussia processes mus usuay be proved, ad he echique of aaysis is o combie a muivariae fucioa cera imi heorem ~FCLT! wih he coiuous mappig heorem+ I is impora, espe- We hak Bruce Hase, eer hiips, ad wo aoymous referees for heir commes o earier versios of his paper+ Ay errors are ours aoe+ Address correspodece o James Davidso, Cardiff Busiess Schoo, Cardiff Uiversiy, Coum Drive, Cardiff CF 3EU, UK+ 000 Cambridge Uiversiy ress $9+50 6

2 6 OBET M. DE JONG AND JAMES DAVIDSON ciay for appicaios o ecoomic daa, ha a wide aiude shoud be permied i he amou ad ype of depedece ad heerogeeiy i he radom variabes uder cosideraio+ I case ~c!, he imis i quesio are sochasic iegras ~Iô iegras!, ad excep i he uivariae case, o show weak covergece cas for a differe echique of proof+ esus of ype ~b! are appied i a he previousy cied sudies, ad resus of ype ~c! are aso crucia i a bu he firs wo+ I is oeworhy i view of he ow rouie use of ess based o hese asympoics ~wih criica vaues obaied by simuaio! ha i he aer case, he avaiabe proofs of weak covergece impose reaivey srige codiios o he amou ad form of permied depedece+ For exampe, Srasser ~986!, Cha ad Wei ~988!, ad Jegaaha ~99! impose a marigae differece assumpio, ruig ou seria correaio of he icremes, a eas of he iegraor process+ hiips ~988b! cosiders iear processes wih idepede ad ideicay disribued ~i+i+d+! iovaios, ad Hase ~99! aows srog mixig, bu a he cied codiios are sroger ha are kow o be sufficie for he FCLT for he same muivariae process+ Moreover, he resus give by hiips ~988a! ad Davidso ~994! coai errors+ I his paper, we give ew codiios for he muivariae FCLT ad sochasic iegra covergece+ The former resu domiaes he exisig oes i he ecoomerics ieraure usig comparabe assumpios, such as Woodridge ad Whie ~988! ad Davidso ~994, Theorem 9+8!+ The codiios are oy a ie sroger ha he bes comparabe oes for he ordiary cera imi heorem ~CLT!+ Moreover, our resus for sochasic iegra covergece impose viruay he same codiios as he FCLT ad so represe a subsaia improveme over previous resus+ Secio ses ou he mai assumpios; Secio 3 discusses he FCLT ad Secio 4 he correspodig sochasic iegra covergece resu+ Secio 5 cocudes he paper+ The proofs are gahered i Appedixes A C+. DEFINITIONS AND ASSUMTIONS A key issue i his heory is he mehod of characerizig weak depedece of he uderyig ime series+ We foow auhors such as Gaa ad Whie ~988! ad öscher ad rucha ~99! i workig wih he cocep of ear-epoch depedece o a mixig process+ This framework has cosiderabe geeraiy+ Whereas addiioa depedece ca be aowed i specific cases such as iear processes ~see Davidso 000; hiips ad Soo 99!, our assumpio is more ikey o be robus i cases of pariay specified modes, i which aspecs of he shor-ru daa geeraio process are ukow+ Such siuaios are edemic i ecoomeric research+ I addiio o icudig ifiie-order movig averages uder suiabe summabiiy codiios, ear-epoch depedece ca be show o be saisfied i various oiear dyamic processes+ See Davidso ~000! for exampes+ Mixig processes are aso aowed+

3 Our defiiio of ear-epoch depedece is as foows+ Le X deoe a riaguar array of radom variabes defied o he probabiiy space ~V, F,! ad e 7X7 p deoe ~E6X6 p! 0p for p + DEFINITION + X is caed L -NED o radom variabes V if for m 0, 7X E~X 6F m, m!7 d ~m!,.) where F s s~v s,+++,v! F for s, d is a array of posiive cosas, ad ~m! r 0 as m r + We refer o he d as he ear-epoch-depedece ~NED! magiude idices ad o he ~m! as he NED umbers+ A sequece such as ~m! is said o be of size if ~m! O~m «! for some «0, ad we aso say ha X is NED of size o he process V + I he appicaio, V ca be a mixig process+ Of he differe mixig coceps ha have bee defied, he ecoomerics ieraure usuay adops eiher uiform ~f-! or srog ~a-! mixig, ad we cosider boh of hese cases, wih simiar size ermioogy for he a- ad f-mixig umbers+ For defiiios ad deais see he precedig refereces ad aso Davidso ~994!+ De Jog ~997! appears o provide he mos geera CLT for NED fucios of mixig processes currey avaiabe+ Leig $X % deoe a riaguar sochasic array, i is show i ha paper ha K d X & N~0,!, where K is a ieger-vaued icreasig sequece, if he foowig assumpio hods+ Assumpio + K ~a! X has mea zero ad 7 X 7 + ~b! There exiss a posiive cosa array c such ha $X 0c % is L r -bouded for r uiformy i ad + ~c! X is L -NED of size _ o V, where V is a a-mixig array of size r0~r!, or X is L -NED of size _ o V, where V is a f-mixig array of size r0~~r!!, ad d 0c is bouded uiformy i ad + ~d! For some sequece b such ha b o~k! ad b o~!, eig b #, M i max ~i!b ib c ad M, r max r b K c, max M i o~b 0!,.) i r ad r i WEAKLY DEENDENT OCESSES 63 M i O~b!+.3) I he case of f-mixig, r is aowed aso if he assumpio of uiform iegrabiiy of X 0c is added o Assumpio ~b!+

4 64 OBET M. DE JONG AND JAMES DAVIDSON 3. A FUNCTIONAL CENTAL LIMIT THEOEM Le K ~j! X ~j! X for 3.) where $K ~j!, % is a sequece of ieger-vaued, righ-coiuous, icreasig fucios of j, wih K ~0! 0 for a ; K ~j! is odecreasig i for a ad K ~j! K ~j '! r as r if j j ' + The referece case obviousy is X 0 X, for some sequece of radom variabes X, wih K This framework is basicay he same as ha of Woodridge ad Whie ~988! ad Davidso ~994, Ch+ 9!+ THEOEM 3++ Le Assumpio hod for X ad assume ha h~j! im EX ~j! 3.) r exiss for a ad ha 3 im dr0 sup im sup d# K ~j d! r K ~j! c ) d The X ~j! & X~j!, where X~j! is a Gaussia process havig amos surey ~a+s+! coiuous sampe pahs ad idepede icremes+ The ie of argume we adop o prove Theorem 3+ corass wih ha of Woodridge ad Whie ~988! ad Davidso ~994!+ They obai he FCLT by a direc proof ha geeraes he cera imi heorem as a coroary, whereas we sar wih he fiie dimesioa disribuios+ Uder Assumpio, ~X ~j!,+++,x ~j k!! d & ~X~j!,+++,X~j k!! 3.4) for ay fiie coecio of coordiaes j,+++,j where he imi disribuios are a+s+ coiuous ad Gaussia+ This foows from Theorem of De Jog ~997! ad he Cramér Wod heorem ~Davidso, 994, Theorem 5+5!+ d Accordig o Theorems 5+4 ad 5+5 of Biigsey ~968!, X & X, where X is coiuous wih probabiiy oe, if ~3+4! hods ad X is sochasicay equicoiuous+ This is he propery ha for a «0, im dr0 im sup ~ sup r sup 6X ~j! X ~j '!6 «! ) $j ' :6j j ' 6 d% Therefore, o compee he proof i suffices o show ha ~3+5! hods for he paria sum process ad ha he icremes are idepede i he imi+ These argumes are se ou i Appedix B+

5 If h~j! j he X is Browia moio+ More geeray, X beogs o a exesio of he cass of rasformed Browia moio processes B h defied i Davidso ~994, Ch+ 9+4!+ The erm h~j! mus be odecreasig everywhere bu uder he prese geeraizaio i eed o be sricy icreasig everywhere, ad icremes of he process may equa zero a+s+ The depedece ad heerogeeiy codiios of Theorem 3+ reax hose empoyed by Woodridge ad Whie ~988! ad Davidso ~994!, whose codiios are simiar+ These aer heorems do o permi a size _ of he NED coefficies bu empoy a rade-off codiio+ The rae a which max c is required o approach zero is dicaed by his rade-off codiio, ad aso he codiio sup im sup d ~0, j! K ~j d! r K ~j! c 0d 3.6) is imposed, which is obviousy sroger ha ~3+3!+ For exampe, cosider K ad c b b 0 for b ~0, _!, appropriae o he case X b b 0 u, where u is i+i+d+ wih fiie variace+ The, K ~j d! r K ~j! im sup c 0d C~~j d! b j b!0d 3.7) for C 0, ad ceary he codiio from equaio ~3+6! does o hod, whereas he codiio from equaio ~3+3! does hod+ ausibe exampes i which codiio ~3+3! fais o hod are o easy o cosruc, bu cosider he case c I~ 405!+ Noe ha max c o~! ad ha im sup r c im sup r ) Oe ca ceary fid a sequece b such ha Assumpio ~d! hods here+ However, im dr0 sup im sup c im im sup dr0 r I~ 405! im dr0 WEAKLY DEENDENT OCESSES mi im sup r 04 03, ) The imi of a process whose icremes have variaces evovig ike c i his exampe has a discoiuiy a he origi+ I oher words, X~0! 0a+s+, bu for 0,,X~j! Y a+s+, where Y is disribued as N~0, 4 5 _!+

6 66 OBET M. DE JONG AND JAMES DAVIDSON Exedig Theorem 3+ o he muivariae case is sraighforward, ad we have he foowig coroary, which foows direcy from Theorem 3+ ad Theorem 9+6 of Davidso ~994!+ THEOEM 3++ Le X be a m-vecor-vaued array ad assume ha for every m-vecor of ui egh here exiss a array c such ha he codiios of Theorem 3+ hod for ' X, a wih respec o he same fucios K ~j!+ d The X & X, where X is a m-dimesioa Gaussia process havig a+s+ coiuous sampe pahs ad idepede icremes+ Impici i he assumpios of Theorem 3+ is he exisece of a marix of covariace fucios, say, h~j! ~m m!, havig he propery ha ' h~j! is a posiive odecreasig fucio for a of ui egh+ For exampe, such a case is give by h~j! j b V for a posiive defiie marix V ad b 0+ Havig he variaces red a differe raes is aso ceary possibe, ahough i is difficu o sae a simpe codiio o h coverig a he possibe cases+ Apar from his requireme, here shoud be o difficuy i meeig he codiios of Theorem 3+ provided Theorem 3+ hods for each eeme of he vecor+ Thus, Coroary of De Jog ~997! shows ha ay cosa array of he form c b 0 g for b g, wih o resricio o sigs, wi saisfy Assumpio ~d!+ Ceary, i his case, Assumpios ~b! ad ~c! are saisfied for a choices of by he maximum of he m array cosas specified i he eemewise covergece+ Codiio ~3+3! wi hod ikewise i his case, accordig o he earier discussio+ 4. WEAK CONVEGENCE TO STOCHASTIC INTEGALS Give vecor-vaued arrays U s ~ p! ad W ~q!, we ex cosider he covergece i disribuio of sums of he ype G s ' U s W, ~ p q!+ 4.) The case W U, or more geeray of he vecors havig eemes i commo, is permied i our approach+ Leig U U ad W d W, ~U,W! & ~U,W! uder he codiios of Theorem 3++ Noe ha wih arge eough, U ~j! ad W ~j! are arbirariy we approximaed by -measurabe radom variabes uder he NED assumpio+ Hece U ad W, havig idepede icremes, are marigaes wih respec o he same firaio+ We seek codiios uder which he weak imi of G, afer ceerig, is he Iô iegra * 0 UdW ' + The oy exisig resus of his ype aowig seria correaio appear o be hose of hiips ~988b! ad Hase ~99!+ hiips ~988b! assumes iear processes wih i+i+d+ iovaios+ Hase ~99! assumes adaped srog-mixig processes+ Our resu coais hese forms of depedece as specia cases ad,

7 i geera, domiaes hem i erms of size codiios+ 4 The assumpios aso do o require U ad W o be adaped sequeces+ These ca deped o eves of he ifiiey far fuure, provided he depedece is damped a such a rae ha he imi processes are marigaes wih respec o he same firaio+ This resu hods by he appicaio of he same ype of bockig argume ha aows he CLT o be proved uder depedece+ Our heorem hods uder esseiay he same codiios as specified previousy for he FCLT+ As before, i eed o be he case ha U ad W are Browia moio, bu his propery wi hod for he eadig case of X 0 X where im r E~ X!~ X! ' V ~fiie, posiive semidefiie!+ I he heorem i is usefu o specify he joi covergece of he ripe ~U,W,G L UW!, so ha he resu may be used subsequey o cosruc he imiig disribuios of he saisics famiiar i ui roo esig ad coiegraio heory, ivovig Browia fucioas, by appicaios of he coiuous mappig heorem+ THEOEM 4++ Le he codiios of Theorem 3+ hod for X ~U ',W '! ' ad K + The ~U,W,G L UW! & U,W, d UdW ', 4.) 0 where U ad W are a+s+ coiuous Gaussia processes havig idepede icremes ad L UW s WEAKLY DEENDENT OCESSES 67 EU W ' s + 4.3) To esabish his resu we adop he approach of Cha ad Wei ~988, Theorem +4~ii!!+ Uder he saed codiios, ~U,W! d & ~U,W!+ Therefore, by he Skorokhod represeaio heorem ~Skorokhod, 956! here exis radom processes ~U,W! havig he same disribuio as ~U,W! bu ha coverge amos surey o ~U,W!, ad hese processes are used o cosruc a approximaio o he iegra+ Leig G deoe he couerpar of G for hese variabes, he joi disribuio of ~U,W,G L UW! is he same as ha specified i ~4+!+ To prove he heorem i is sufficie o show G L UW p & 0 UdW ', 4.4) because he joi covergece foows as a case of Theorem 9+6 of Davidso ~994! i which G L UW is mapped io a ~a+s+ cosa! eeme of C@0,# pq + Moreover, we ca cosider he case of scaar U ad W wihou oss of geeraiy, because he geera case he foows by appyig Theorem 30+4 of Davidso ~994!+

8 68 OBET M. DE JONG AND JAMES DAVIDSON The covergece i ~4+4! is show i wo seps, usig a bockig argume+ Le k ~he umber of bocks! be a odecreasig sequece such ha k r as r ad im r k ~0 d! 0, where d is he uiform disace bewee ~U,W! ad ~U,W! excep o a se of arbirariy sma probabiiy, ad e k G * U ~j!~w ~j j! W ~j!!, 4.5) where j j j0k + Aso, e G * deoe he couerpar of G * for ~U,W!+ The firs sep, based o Cha ad Wei ~988!, is o show ha G * UdW 0 p & ) This proof ca foow ha of Theorem 30+3 of Davidso ~994!, ie for ie up o equaio ~30+78!, wih appropriae chages of oaio+ Because he disribuios of G * ad G * are he same, i suffices for he secod sep o show ha 6G G * L UW 6 p & ) Noig ha G G * L UW A B, 4.8) where A ad k j m 0 k j ~U, m W, EU, m W,! 4.9) B EU, m W,, 4.0) m where j #, for j,+++,k, he proof of ~4+7! is compeed by showig p & 0 ad B r 0+ These argumes, which are fairy eghy, are give i A Appedix C+ 5. CONCLUSION This paper has give ew weak covergece resus ha pace he asympoics uderyig he heory of coiegraig regressios o viruay he same fooig as sadard asympoics+ We prove he FCLT uder codiios simiar o he bes oes kow o us for he ordiary CLT, from he poi of view of he amou of depedece ad heerogeeiy permied i he uderyig radom

9 WEAKLY DEENDENT OCESSES 69 processes+ We aso show ha sochasic iegra covergece hods uder effecivey he same codiios, somehig ha has o bee demosraed previousy o our kowedge+ NOTES + See Hase ~99!+ Davidso s Theorem 30+3 ~994! is correced i he 997 repri of he work by revisig he codiios+ The prese paper shows ha Davidso s origia heorem is correc as saed, eve hough he proof coais a error+ d p + I his paper, & deoes covergece i disribuio, whereas & deoes covergece i probabiiy+ 3+ Summaios over a empy idex se are defied as zero here+ 4+ I ca be difficu o deermie wheher oe se of codiios acuay coais aoher+ However, we oe ha if i he hiips ~988b! mode he iear MA coefficies are -summabe ~see hiips, 988b, p+ 530!, he process is L -NED o he i+i+d+ forcig variabes of size 3 _ + EFEENCES Adrews, D+W+K+ ~988! Laws of arge umbers for depede o-ideicay disribued radom variabes+ Ecoomeric Theory 4, Biigsey, + ~968! Covergece of robabiiy Measures. New York: Wiey+ Cha, N+H+ &C+Z+Wei ~988! Limiig disribuios of eas-squares esimaes of usabe auoregressive processes+ Aas of Saisics 6, Davidso, J+ ~99! A cera imi heorem for gobay osaioary ear-epoch depede fucios of mixig processes+ Ecoomeric Theory 8, Davidso, J+ ~993! The cera imi heorem for gobay osaioary ear-epoch depede fucios of mixig processes: The asympoicay degeerae case+ Ecoomeric Theory 9, Davidso, J+ ~994! Sochasic Limi Theory. Oxford: Oxford Uiversiy ress+ Davidso, J+ ~000! Whe Is a Time Series I~0!? Evauaig he Memory roperies of Noiear Dyamic Modes+ Workig paper ~revised!+ Davidso, J+ &+M+De Jog ~997! Srog aws of arge umbers for depede heerogeeous processes: A syhesis of ew ad rece resus+ Ecoomeric eviews 6 ~3!, De Jog, +M+ ~997! Cera imi heorems for depede heerogeeous radom variabes+ Ecoomeric Theory 3, Gaa, A++ &H+Whie ~988! A Uified Theory of Esimaio ad Iferece for Noiear Dyamic Modes. New York: Basi Backwe+ Hase, B+E+ ~99! Covergece o sochasic iegras for depede heerogeeous processes+ Ecoomeric Theory 8, Jegaaha, + ~99! O he asympoic behavior of eas-squares esimaors i A ime series modes wih roos ear he ui circe+ Ecoomeric Theory 7, Johase, S+ ~988! Saisica aaysis of coiegraed vecors+ Joura of Ecoomic Dyamics ad Coro, Johase, S+ ~99! Esimaio ad hypohesis esig of coiegraed vecors i Gaussia vecor auoregressive modes+ Ecoomerica 59, McLeish, D+L+ ~975a! A maxima iequaiy ad depede srog aws+ Aas of robabiiy 3, McLeish, D+L+ ~975b! Ivariace pricipes for depede variabes+ Zeischrif für Wahrscheiichskeisheorie ud Verwade Gebiee 33, McLeish, D+L+ ~977! O he ivariace pricipe for osaioary mixigaes+ Aas of robabiiy 5, ark, J+Y+ &+C+B+hiips ~988! Saisica iferece i regressios wih iegraed processes, par + Ecoomeric Theory 4,

10 630 OBET M. DE JONG AND JAMES DAVIDSON ark, J+Y+ &+C+B+hiips ~989! Saisica iferece i regressios wih iegraed processes, par + Ecoomeric Theory 5, hiips, +C+B+ ~986! Udersadig spurious regressios i ecoomerics+ Joura of Ecoomerics 33, hiips, +C+B+ ~987! Time series regressio wih a ui roo+ Ecoomerica 55, hiips, +C+B+ ~988a! Weak covergece o he sochasic iegra * 0 BdB ' + Joura of Muivariae Aaysis 4, hiips, +C+B+ ~988b! Weak covergece of sampe covariace marices o sochasic iegras via marigae approximaios+ Ecoomeric Theory 4, hiips, +C+B+ &S+N+Durauf ~986! Muipe ime series regressio wih iegraed processes+ eview of Ecoomic Sudies LIII ~4!, hiips, +C+B+ &V+Soo ~99! Asympoics for iear processes+ Aas of Saisics 0, öscher, B+M+ &I++rucha ~99! Basic srucure of he asympoic heory i dyamic oiear ecoomeric modes, par : Cosisecy ad approximaio coceps+ Ecoomeric eviews 0, 5 6+ Skorokhod, A+V+ ~956! Limi heorems for sochasic processes+ Theory of robabiiy ad Is Appicaios, Srasser, H+ ~986! Marigae differece arrays ad sochasic iegras+ robabiiy Theory ad eaed Fieds 7, Woodridge, J+M+ &H+Whie ~988! Some ivariace pricipes ad cera imi heorems for depede heerogeeous processes+ Ecoomeric Theory 4, AENDIX A: TECHNICAL LEMMAS FO MIXINGALES A impora oo for obaiig our resus is he mixigae propery+ The L -mixigaes were iroduced by McLeish ~975a!, ad he exesio o L p -mixigaes, p, by Adrews ~988!+ Le G deoe a array of s-fieds, icreasig i for each + DEFINITION + $X, G % is caed a L p -mixigae if for m 0, 7X E~X 6G, m!7 p a c~m!, 7E~X 6G, m!7 p a c~m!, A.) A.) ad c~m! r 0 as m r + The oaio here ad i he res of he paper is as i Davidso ~99, 993! ad De Jog ~997!+ The a are referred o as he mixigae magiude idices, ad X is caed a mixigae of size if c~m! is of size + Uder iegrabiiy codiios, radom variabes ha are NED o a mixig process are kow o be mixigaes, ad i paricuar we have he foowig sadard resu ~see, e+g+, Davidso, 994, Coroary 7+6!+ LEMMA A++ If X saisfies pars ~a! ~c! of Assumpio, $X, F, % is a L - mixigae of size _ wih mixigae magiude idices c +

11 WEAKLY DEENDENT OCESSES 63 esus of his kid are ofe used impiciy i he seque, where we proceed by showig ha cerai fucios of he variabes are NED o V ad appyig he same ype of argume+ Ahough he mixigae assumpio is o srucured eough o yied weak covergece resus wihou suppemeary codiios, because for exampe i may o be preserved uder rasformaios, i is usefu a cerai sages of he proofs+ I addiio o he various kow mixigae properies documeed i sources such as Davidso ~994!, we make use here of he foowig resus+ LEMMA A++ If $Y j, F j % is a L -mixigae wih magiude idices a j ad im sup r k a j A.3) ad for a q, k he ~E~Y j6f, j q! E~Y j6f, j q!! p & 0, A.4) k Y j p & 0+ A.5) roof. Noe ha for a m, k k Y j ~Y j E~Y j6f,!! m q m k ~E ~Y j6f, j q! E~Y j6f, j q!! k E~Y j6f,!+ A.6) k The L -orm of he firs ad hird erms is bouded by a j c~m!, which ca be made arbirariy sma by seecig a arge vaue of m+ The secod erm coverges i probabiiy o zero by he requireme of equaio ~A+4!+ See aso Adrews ~988!+ LEMMA A+3+ Le $X, G % ad $Y, G % be riaguar L -mixigae arrays of size _ wih mixigae magiude idices a X ad a Y, respecivey, where ~a X! O~! ad ~a Y! O~!+ If g is a icreasig ieger-vaued fucio of wih g r as r, he im r s 6E~X Y s!6i~6 s6 g! 0+ roof. This is aaogous o Lemma 4 of De Jog ~997!+ A.7)

12 63 OBET M. DE JONG AND JAMES DAVIDSON LEMMA A+4+ If $X, G % ad $Y, G % are L -mixigaes wih mixigae umbers c X ~ j! ad c Y ~ j! ad magiude idices a X ad a Y, he s X Y s C ~a ~a s s X! 0 ~og j! c X ~ j! Y! ~og j! c Y ~ j! 0+ A.8) for 0 C + roof. Defie X E~X 6G,! E~X 6G,! A.9) ad Y si E~Y s 6G, s i! E~Y s 6G, s i! A.0) ad oe ha s X Y s i i i i s X Y si s X Y si I~ s i! s X Y si I~ s i! s X Y si I~ s i! + A.) Cosider each of hese hree ses of erms+ Noe firs ha he sequece s X Y si I~ s i!, A.) is a marigae differece wih respec o he G, ad, herefore, for some cosas C 0 ad C 0, i C C s X Y si I~ s i! i C E Y si 0 s X max E 0 EX i 0 EX i Y si 0 s max s EY si 0, A.3)

13 WEAKLY DEENDENT OCESSES 633 where hese iequaiies are, respecivey, by he Burkhoder, Cauchy Schwarz, ad Doob iequaiies+ A simiar argume hods for he secod se of erms i ~A+!, oig ha s X Y si I~ s i!, s is a marigae differece wih respec o G, s i ad aso ha for each s, s s s X max X + A.4) Fiay, we have i C 3 C 3 s X Y si I~ s i! i 7 X 7 7Y, i, i I~ i!7 0 EX sup i EY, i, i I~ i! 0 A.5) for C 3 0+ The majora of ~A+! does o exceed he sum of ~A+5! ad wo erms of he form ~A+3!+ Now, he emma foows by combiig he mixigae assumpio wih he fac ha EX EE~X 6G,! EE~X 6G,! E~X E~X 6G,!! E~X E~X 6G,!!, A.6) wih simiar equaiies for Y si, ad he fac ha for a moooe decreasig sequece $x j, j % he reaio ~x j x! 0 C ~og j! x j 0, A.7) for C 0, hods by a argume simiar o McLeish ~975a, Theorem +6!+ AENDIX B: OOF OF THEOEM 3+ Fix d 0 ad e j dj jd0 for j 0,+++,@0d#+ For ay pair j, j ' such ha 6j j ' 6 d0, e j d ~j,j '! deoe he maxima vaue of j such ha j j dj ad j ' j dj ad oe ha 0,j j djd ~j,j '! d ad 0,j ' j djd ~j,j '! d+ I addiio, defie

14 634 OBET M. DE JONG AND JAMES DAVIDSON K ~mi~j d,!! ~j,d! c B.) K ~j! ad e X ~j! X ~! if j + The sup sup 6X ~j! X ~j '!6 $j ' :6j j ' 6 d0% sup sup 6X ~j! X ~j djd ~j,j '!! X ~j djd ~j,j '!! X ~j '!6 $j ' :6j j ' 6 d0% max j 0,+++,@0d# sup ~j! X ~j dj!6 $j:0 j j dj d%6x sup ~j! X ~j dj!6 j 0 $j:0 j j dj d%6x j 0 sup ~j! X ~j dj $j:0 j j dj d%6x!6 I sup $j:0 j j dj d%6x ~j! X ~j dj!6 «04 4«E sup ~j! X ~j dj j 0 $j:0 j j dj d%6x!6 I sup $j:0 j j dj d%6x ~j! X ~j dj!6 ~j dj,d!4«e sup ~j! X ~j dj!60 ~j dj j 0 $j:0 j j dj d%6x,d! I sup ~j! X ~j dj!6 $j:0 j j dj d%6x 0 ~j dj,d! «0~4 ~j dj,d!! c max 4«E sup 6X ~j! X ~j dj!60 ~j dj,d! 4 I sup C«j 0,+++,@0d# $j:0 j j dj d% $j:0 j j dj d%6x ~j! X ~j dj!6 0 ~j dj,d! «0~4 ~j dj,d!! max j 0,+++,@0d# E sup ~j! X ~j dj!60 ~j dj $j:0 j j dj d%6x,d! I sup 6X ~j! X ~j dj!6 0 ~j dj,d! $j:0 j j dj d% 4 max ~j dj,d! j 0,+++,@0d# «B.)

15 WEAKLY DEENDENT OCESSES 635 for some fiie cosa C 0+ The secod iequaiy foows from subaddiiviy, he hird iequaiy is Markov s, ad he remaiig seps foow from he assumpios+ Nex, oe ha by he mixigae propery ~Lemma A+! ad Coroary 6+4 of Davidso ~994!, he sequece Y ~d,j '! sup 6X ~j! X ~j '!60 ~j ',d! $j:0 j j ' d% B.3) is uiformy square-iegrabe+ Moreover, Assumpio ~b! impies ha his propery is idepede of he segme of he daa sequece represeed by Y + ~Compare McLeish, 975b, Lemma 6+5; ad McLeish, 977, proof of Theorem +4+! I oher words, im sup r max EY ~d,j dj! I~6Y ~d,j dj!6 K! j 0,+++,@0d# max j 0,+++,@0d# im sup EY ~d,j dj! I~6Y ~d,j dj!6 K! r f~k!, B.4) where f ~K! does o deped o d ad f ~K! r 0asKr+ Because d is arbirary, i foows by ~3+5!, ~B+!, ad ~B+4! ha X ~j! is sochasicay equicoiuous if im im sup max ~j dj,d! 0+ dr0 r j 0,+++,@0d# B.5) Because he max ad he im sup i equaio ~B+5! ca simiary be ierchaged, his hods by he assumpio of equaio ~3+3!+ Nex, we show ha X~j! has idepede icremes+ I view of he Gaussiaiy, i suffices o show ha for ay se $j,+++,j k :0 j j {{{ j k % ad a i j, X~j i! X~j i! ad X~j j! X~j! are ucorreaed+ This foows because E~X~j i! X~j i!!~x~j j! X~j!! im E~X ~j i! X ~j i!!~x ~j j! X ~j!!, r B.6) where for ay fixed d 0, 6E~X ~j i! X ~j i!!~x ~j j! X ~j!!6 K ~j i! K ~j i! K ~j i! K ~j i! K ~! K ~j d! s K ~j! K ~j i! EX X s K ~j i! K ~j d! X s K ~j! X s K ~j j! s K ~j d! EX X s K ~! 6 EX X s 6I~6s 6 K ~j d! K ~j i!!+ B.7) s

16 636 OBET M. DE JONG AND JAMES DAVIDSON Noe ha im r K ~! K ~! 6EX X s 6I~6s 6 K ~j d! K ~j i!! 0 s B.8) for d 0, by Lemma A+3 ad he requireme ha K ~j! K ~j '! r for a j j ', ad aso ha K ~j d! r s K ~j! im im dr0 X s 0 by he assumpio i equaio ~3+3!+ Because d is arbirary i ~B+8!, i foows ha E~X~j i! X~j i!!~x~j j! X~j!! 0+ This compees he proof+ AENDIX C: OOF OF THEOEM 4+ Firs, wrie B k j m EU, m W, I~m q! k j m EU, m W, I~m q!, C.) where q is a odecreasig sequece such ha q r as r + We defie c U ad c W as he cosas wih respec o which Assumpio hods for U ad W, respecivey+ Noe ha because im max max$~c U!,~c W! % 0 r C.) by Assumpio ~d!, i is possibe o choose k ad q such ha im k q max max$~c U!,~c W! % 0+ r C.3)

17 E E E WEAKLY DEENDENT OCESSES 637 The secod erm i ~C+! coverges o zero, because is absoue vaue is bouded by m 6EU, m W 6I~m q! o~! C.4) by Lemma A+3+ For he firs erm i ~C+!, oe ha j k m 6EU, m W, I~m q!6 k q m O~k q max max$~c U!,~c W! %! 6EU, m W, I~m q!6 o~! C.5) by ~C+3!, oig ha vaues of exceedig q coribue zero o he sum+ p To show ha A & 0, firs defie h~a, x! xi~6x6 a! ai~x a! ai~x a! C.6) ad g~a, x! ~x a!i~x a! ~x a!i~x a! C.7) ad oe ha x g~a, x! h~a, x!+ For some K 0 o be chose, defie UE g~kc U,U! Eg~Kc U,U! ad UE m E~ UE 6G, m! C.8) ad U h~kc U,U! Eh~Kc U,U! ad U m E~ U 6G, m!, C.9) where G s~v,v,,+++!+ Noe ha U U U + Aso oe ha g ad h are Lipschiz fucios ad herefore U ad U are L -NED o V for a K, wih NED magiude idices c U ad NED umbers ~m!+ ~See Davidso, 994, Theorem 7++! Therefore, U is aso a L -mixigae of size _ wih mixigae magiude idices c U, impyig ha 7UE m 7 Cm 0 m U c C.0) for m 0 ad C 0 ad aso, by Assumpio ~b!, ha 7UE m 7 c U sup7u 0c U I~6U 60c U K!7 c U f ~K!, C.)

18 N N G G E G G 638 OBET M. DE JONG AND JAMES DAVIDSON for some f ~K! o depedig o or, where f ~K! r 0asKr+ These iequaiies furher impy ha 7UE m 7 ~Cm 0 m c U! m ~c U f ~K!! m C ' c U f ~K! m m 0 m0 m C.) for C ' 0+ Therefore, U is a L -mixigae of size _ wih mixigae magiude idices c U f ~K! m for some sma eough m 0+ Simiary, we may decompose W io W ad WG, havig he same properies wih respec o cosas c W + Noe ha A k j p ~ UE s j W, E E U s W, E U s W, E E U s W, U s W, E U s W, U s W, E U s W,!, C.3) where for ecoomy of oaio we heceforh use he symbo p j o deoe + Cosider he four sums of erms correspodig o his decomposiio+ I foows by Lemma A+4 ha he L -orms of a hese sums excep hose ivovig U s W, are of order k O j ~c s j U! m ~c W! 0f ~K! O~ f ~K! m!, C.4) where he equaiy i ~C+4! is by assumpio+ By choosig a arge eough K, he imsups of he correspodig compoes of A ca be made as sma as desired+ Accordigy, e he remaiig compoe be defied as A k j p U s j W, E~ U s W,!, ad we compee he proof by showig ha for a K 0, N Lemma A++ Firs wrie A k Y j, A C.5) p & 0, by a appicaio of C.6) where Y j j s p ~ U s j W, E U s W,!+ C.7) Defie F j s~v,j,v,j,+++! ad H, s~v,,+++,v,! ad for breviy of oaio e E deoe E~{6H,!+ The, oe ha for m 0 here exis posiive cosas C, C, ad C 3 such ha

19 WEAKLY DEENDENT OCESSES 639 7Y j E Y j 7 j p ~ U s j j s p j W, E U s ~ W, E j s p E j j s p E j j ~ W, E j W, j E j U s W,! W,! W, ~ U s E U s! W, E U s E W, 7W, E j 7U s E W,! ~ U s E U s U s! ~ U s E W, 7 j j c W 0 «j ~m0k! C j C U s W, U s! U s 7 W, U 0 «j ~m0k! c s 0 ~c U s! 0 ~c W! U s j E W, C 3 m 0 «~0k! «j ~c s j U! 0 ~c W! C.8) for some «0+ The firs iequaiy foows from rearragig he erms ad he orm iequaiy; he secod iequaiy uses ieraed expecaios; he hird is he Cauchy Schwarz iequaiy ad rearragig of erms; he fourh uses he NED defiiio, Theorem +6 of McLeish ~975a! ~see aso Davidso, 994, Theorem 6+9!, ad he size

20 640 OBET M. DE JONG AND JAMES DAVIDSON assumpios; ad he fifh is obaied usig Jese s iequaiy+ For he case m 0, a excep he wo fia seps of ~C+8! hod, bu for his case we have j 7Y j E~Y j 6H, j j!7 C 4 ~c s j U! 0 ~c W! C.9) for C 4 0, usig some of he same argumes as before+ We have herefore esabished ha Y j is L -NED of size _, o a mixig process+ Because i aso possesses a is momes, i foows by Coroary 7+6 of Davidso ~994! ha $Y j, F j % is aso a L - mixigae of size _, wih respec o cosas a j j ~c s j U! ~c W! 0+ C.0) k Noe ha im sup r a j,, because by assumpio, max im sup r ~c U!, im sup r ~c W! + C.) By Lemma A+, he proof is herefore compee if we ca show ha for a q, k ~E~Y j6f, j q! E~Y j6f, j q!! We ex wrie p & 0+ C.) U ~ U U m! U m ~ U m U m!, C.3) ad eig c~m! deoe he mixigae umbers reaig o U, oe ha $ U U m, G % ad $ U m, G % are L -mixigaes wih mixigae umbers equa o c~m! for m ad c~! for m+ Therefore, by Lemma A+4 ad he assumpios, im sup r k j p j k j p j ~E ~ U s W, 6F, j q! E~ U s ~E~~ U m s U m s! W, 6F, j q! E~~ U m s U m s! W, 6F, j q!! W, 6F, j q!! m C ~c~m! ~og! m c~! ~og~!! 0 O~m «! C.4) for some C 0 ad «0+ Therefore by choosig m arge eough, he differece bewee he expressios ca be made egigibe+ A simiar argume ca be used o repace W, by W, W m, i he as expressio, ad herefore i remais o show m ha for a q, K, ad m,

21 WEAKLY DEENDENT OCESSES 64 k j p j ~E~~ U m s U m s!~ m W, W m,!6f, j q! E~~ U m s U m s!~ m W, W m,!6f, j q!! p & 0+ C.5) Noig ha m U m U m h m ~ U h U h! C.6) ad m W, W, m m h m ~ W h h, W,!, C.7) i foows ha his resu hods if for a q, K, h, ad, k j p j k ~E~~ U h s U h s!~ W, E~~ U h s U h s!~ W, W h,!6f, j q! [ ~E~Z j6f, j q! E~Z j6f, j q!! W,!6F, j q!! p & 0+ C.8) Because he erms of ~C+8! are ucorreaed, he aer saeme is rue if for a q, K, h, ad, im k r EZ j 0+ C.9) However, oe ha k EZ j j k p j j p j s p j s p j h E~ U h s U s!~ W, W,! h ~ U h s U s!~ W, W,!+ C.30) Cosider, as represeaive, he erms for which s h s h + C.3) The oher cases are reaed ideicay+ Firs, oe ha by he marigae differece propery of he four erms i equaio ~C+30!, he erms i ha equaio are zero uess s h + Therefore, for he erms ha saisfy he precedig resricio, we have, appyig Lemma A+4 oce agai,

22 64 OBET M. DE JONG AND JAMES DAVIDSON k j p j s p j h ~ U h s U s!~ W, W,! j p j I~ h!~ k j p j j p j ~ s p j h ~ U h s U s!~ h U, h W, I ~ h!~ W, W,! h U, h W,! h U, h!~ W, h U, h! W,! k O j O j p j O max j k o~!, ~c s k j ~c s j U! 0 ~c W! I~ h!c W U c, h j j U! ~c s ~c U! W! j 0 ~c W! C.3) where he as equaiy foows from he assumpio of equaio ~3+3!+ This compees he proof+