ˆn i 1 ni ni Ž. Ý Ý CENTRAL LIMIT THEOREM FOR LINEAR PROCESSES

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1 The Aals of Proalty 997, Vol. 5, No., CENTRAL LIMIT THEOREM FOR LINEAR PROCESSES BY MAGDA PELIGRAD AND SERGEY UTEV Uversty of Ccat ad Isttute of MathematcsNovosrs I ths paper we study the CLT for partal sums of a geeralzed lear process X a, where sup a, max a 0asad s are tur, parwse mxg martgale dffer- eces, mxg sequeces or assocated sequeces. The results are mportat aalyzg the asymptotcal propertes of some estmators as well as of lear processes.. Itroducto. Let 4 e a cetered sequece of radom varales ad let a,4 e a tragular array of umers. May statstcal procedures produce estmators of the type Ž.. S a. To gve a example let us cosder the olear regresso model yž x. gž x. Ž x., where gž x. s a uow fucto ad Ž x. s the ose. Now we fx the desg pots x ad we get y gž x. Ž x. gž x., where 4 s a cetered sequece of radom varales. The oparametrc estmator of g x s defed to e g x w Ž x. y where ˆ x x x x h / ž h / wž x. Kž K, where K s a erel fucto. It s ovous that gˆ Ž x. Eg Ž x. ˆ s of the type Ž... We shall see later o that the asymptotc ehavor of the sums of varales of the form. X aj j, where aj j j ca e otaed y the study of the sums of the form Ž.., ad our results mprove o some ow results aout CLT for sums of the form Ž... Our paper s orgazed the followg way: Secto cotas the deftos Receved Aprl 994; revsed Septemer 996. Research supported part y a NSF Grat ad Taft Travel Grat. AMS 99 suject classfcatos. 60G09, 60F05. Key words ad phrases. Lear process, depedet radom varales, cetral lmt theorem. 443

2 444 M. PELIGRAD AND S. UTEV ad the results, Secto 3 cotas the proofs ad the Appedx cotas the statemets of some ow results used the proofs.. Results. Our frst theorem solves the prolem descred the troducto for some sequeces of martgale dffereces. DEFINITION.. We call X 4 a parwse mxg sequece f for every x real, sup cov IŽ X x., IŽ X x. 0 as. THEOREM.. Let 4 e a parwse mxg martgale dffereces se- quece of radom varales, ad let a ;4 e a tragular array of real umers such that. sup a ad max a 0 as. Assume ž / 4. s a uformly tegrale famly ad var a. The D 3. a NŽ 0,. as. As a corollary of the aove theorem we prove the followg. COROLLARY.. Let ; j Z4 j e a parwse mxg martgale dfferece sequece of radom varales whch s uformly tegrale L. Let a j; 4 j Z e a sequece of real umers such that j aj. Let X a ad S X. Assume varž S. j j j as ad f varž. 0. The j j S NŽ 0,. as. D Ths result s a exteso of Theorem Iragmov ad L Ž 97. from..d. to e the parwse mxg martgale case. It should e oted that the ergodcty caot replace the codto of parwse mxg Theorem.. We have the followg example. EXAMPLE.. There s a sequece, 4 of martgale dffereces whch s strctly statoary ad ergodc, havg fte secod momets ad there are 4 umers a, satsfyg. ad. ad such that a does ot coverge to a ormal dstruto.

3 CLT FOR LINEAR PROCESSES 445 We shall troduce ow some measures of depedece etwee two algeras. ad DEFINITION.. Let A ad B e two -algeras of evets ad defe Ž A, B. sup PŽ B A. PŽ B., A A, B B, PŽ A. 0 Ž A, B. sup corrž f, g. fl Ž A., gl Ž B. Ž A, B. sup PŽ AB. PŽ A. PŽ B. A A, B B 4 m DEFINITION.3. Let e a stochastc sequece ad let F Ž, m.. Ž a. We call the sequece -mxg f 0 where Ž. sup F, F. Ž. We call the sequece -mxg f 0 where Ž. sup F, F. Ž. c We call the sequece strogly mxg f 0 where Ž. sup F, F. It s well ow that the -mxg codto s the most restrctve ad the strog mxg s the weaest amog all. See Bradley Ž 986. for a survey. The ext theorem solves the same prolem as Theorem. for these three classes of depedet radom varales. The codtos mposed to the momets ad mxg rates are the same suffcet codtos, some sese mmal, requred for the valdty of CLT for strctly statoary sequeces. See Pelgrad Ž 986. for a survey, ad Douha, Massart ad Ro Ž 994. for a recet result o strog mxg sequeces. Therefore, the ext theorem exteds the ow results for strctly statoary mxg sequeces from equal weghts to geeral weghts, weaeg at the same tme the assumpto of statoarty. THEOREM.. Let a 4 e a tragular array of real umers satsfyg. ad let 4 e a cetered stochastc sequece satsfyg.. Assume that oe of the followg three codtos s satsfed: Ž a. 4 s -mxg. Ž. 4 s -mxg ad Ž j. j. Ž. c For a certa 0, 4 s strogly mxg, 4 s uformly tegrale, f var 0 ad. The.3 holds.

4 446 M. PELIGRAD AND S. UTEV REMARK.. I Theorem.Ž. c the codto f var 0 ca e re- moved ut ths requres further addtoal wor ad t wll e cosdered elsewhere. Our last theorem refers to assocated sequeces of radom varales. DEFINITION.4. We call the famly Ž X,..., X. assocated f for ay coordatewse ocreasg fuctos fž x,..., x. ad gž x,..., x. we have cov fž X,..., X., gž X,..., X. 0 wheever ths s defed. A sequece X 4 s called assocated f every fte famly of varales s assocated. The ext theorem exteds the well ow CLT of Newma ad Wrght Ž 98. to geeral weghts whle eepg the same codtos as the classcal case. THEOREM.3. Assume a 4 are oegatve umers satsfyg.. Let 4 e a assocated sequece of radom varales whch s cetered, satsfes. ad.4 cov, 0 as u uformly. j j: ju The.3 holds. 3. Proofs. 3.. Proof of Theorem.. I order to prove Theorem. we shall apply Theorem 3. from Hall ad Heyde Ž 980. whch s stated the Appedx for coveece Ž Theorem A.. It s easy to see that uder. ad. for every 0, 3. a E I a 0 as. ad therefore, for every 0, E max a I a 0 as, whch proves Ž a. ad Ž c. of Theorem A. I order to verfy Ž. we shall prove the followg lemma whch wll coclude the proof of Theorem.. LEMMA 3.. Suppose that f s a cotuous fucto such that X fž.4 s a uformly tegrale famly L. Let t ;4 e a tragular array of real umers such that Ž 3.. sup t T

5 ad Ž 3.3. lm max t 0. CLT FOR LINEAR PROCESSES 447 The X EX t 0 as. P PROOF. Let M ad 0 e two real umers ad defe M M, sup fž t. fž s.. t, s M ; ts M For, deote y Y fž j. IŽj Ž j.. jm. From the defto of Y we deduce that for every, X I M M Y a.s., whece, y Jese equalty ad 3., ž Ž. / ž Ž. / E X Y t EX EY t Ž. E X Y t T sup EX Y T sup E X I M TŽ., whece the facts that X 4 s a uformly tegrale famly L, 4 s ouded proalty ad f s cotuous show that t s eough to prove the valdty of the lemma for Y wth M ad fxed., j Deote y X IŽ j.. We have M, j, j Ž Y EY. t fž j. Ž X EX. t jm, j, j Ž. X EX t ad therefore t remas to estalsh the lemma for X, j wth ad j fxed., j, j Deote y sup covž X, X. ad y the codto of parwse mxg Ž as. Wth the aove otato we have the followg estmate:,, j, j 3.5 var t X t E X t t Ovously we have, j 3.6 t E X T max t.

6 448 M. PELIGRAD AND S. UTEV To estmate the secod term 3.5 we splt the sum two, oe up to h ad aother after h, where h s a teger: Ž 3.7. By 3.5, 3.6 ad 3.7, t t h max t t,, h max t t h, ht max t T max. h, j h var t X T h max t T max ad the result follows y 3.3 ad 3.4 y lettg frst,, ad after, h. 3.. Proof of Corollary.. Wthout restrctg the geeralty, we ca assume sup E. We have S X a. j j j I order to apply Theorem., we fx W such that j W a j 3 ad tae W. The ž j/ j ž j/ j j j S a a T U ad we have the followg estmate Ž 3.8. ž j / j aj aj j j var U a a 0 as. j j W Therefore we have oly to prove that T NŽ 0,. D as. Accordg to Theorem. t s suffcet to show that sup a j j 0 as. Let us suppose o the cotrary that for some 0 there exst a susequece j,, such that j a.

7 CLT FOR LINEAR PROCESSES 449 Deote y A sup a ad otce that for r j, Hece r a AŽ r j.. jw jw j j a W 4 A Ž j. W 4 A W. Tag W to e the least teger greater tha or equal to 3 ad ecause, we ota for suffcetly large, whch s a cotradcto A, Costructo of Example.. Let Z 4 ad Y 4 e two depedet sequeces of radom varales such that PŽ Z 0. PŽ Z., Z Z for all ad Y 4 s a..d. sequece of stadard ormal varales. Defe ZY for. It s easy to verfy that 4 s a strctly statoary sequece of martgale dffereces wth varž.. We shall prove that 4 s ergodc y verfyg cf. Shryayev Ž 984., chapter V that for every measurale ouded fucto f ad every postve teger we have f Ž,...,. Ef Ž,...,. a.s. as. Deote y X fž,...,.. If we fx Z, say Z, the the se- quece 4 ecomes Y,0,Y,0,.... If Z 0, the sequece 4 3 cossts of 0, Y,0,Y 4,.... Therefore for Z fxed, the sequece X 4 ecomes statoary ad - depedet, therefore ergodc, ad we have IŽ Z. X IŽ Z. Ef Ž Y,0,Y 3,... a.s.,, : eve IŽ Z. X IŽ Z. Ef Ž 0, Y 3,0,.... a.s.,, : odd IŽ Z 0. X IŽ Z 0. Ef Ž 0, Y,0,.... a.s.,, : eve IŽ Z 0. X IŽ Z 0. Ef Ž Y,0,Y 4,... a.s., : odd

8 450 M. PELIGRAD AND S. UTEV Now we add all these relatos, ad y statoarty ad costructo we have: 3 4 X Ef Ž Y,0,Y,... Ef Ž 0, Y,0,Y,... Ef Ž X. a.s. Let us ow cosder the umers a 4 satsfyg.,. ad addto, :odd a 3 for all. Set S a ad let Y e a sta- dard ormal radom varale depedet of Z 4. By the costructo S has the same dstruto as Y' az. Because Z Z for every we have , : odd, : eve a Z Z a Z a Z Z Z. Therefore S has a fxed ogaussa dstruto for all Proof of Theorem.Ž a,. Ž.. I order to prove Theorem.3 uder the assumptos Ž a. ad Ž., accordg to Theorem. ad Theorem 4., respectvely, Utev Ž 990. we have oly to verfy the Ldeerg s codto Ea I a 0 as for every 0 ad ths follows exactly as 3. y. ad.. Ž Proof of Theorem. c. I order to prove ths part of Theorem., we shall frst use a trucato argumet ad after that we shall apply Theorem B from the Appedx. The proof requres the followg auxlary lemma. 4 Ž. LEMMA 3.. Assume X satsfes the codtos of Theorem. c the a a var a X C a Ž where C sup EX c. sup X, ad cž. s a umercal costat depedg o. PROOF. We have j j a a ja j var a X a var X a a, cov X, X j a a ja, j a var X a cov X, X whece the result follows y Lemma B from the Appedx.

9 CLT FOR LINEAR PROCESSES 45 Ž. I order to prove Theorem. c, we shall trucate frst the varales at the level A 0 ad deote ad I A E I A I A E I A. By Lemma 3. there s a postve costat C such that var a C sup E I A sup E I A ad otce that y the uform tegralty of 4 A Ž 3.9. lm sup var a 0. By. ad 3.9 we ota Ž 3.0. lm var a uformly. A By 3.9, 3.0 ad Theorem 4. Bllgsley 968, order to prove the theorem t s eough to show that for every fxed postve A we have a st dev a Ž. NŽ 0,. as. D To prove ths we shall verfy the codtos of Theorem B gve the Appedx. The codto Ž a. requres a uform oud o the varace of partal sums whch follows y Lemma 3.. The Ldeerg codto Ž d. s satsfed y.,. ad Ž Codto Ž. s a cosequece of.,. ad Ž We have oly to verfy codto Ž c.. Let Ž.. Notce that y Lemma A the Appedx, we ca fd a costat K such that Ž 3.. c ½ j j5 ja u ½ 5 c ½ 5 ž cov exp t a, exp t a ½ 5 / cov exp t a, exp t a ja j j u c Ž. j j j j ja ju K t u a a.

10 45 M. PELIGRAD AND S. UTEV Notce ow that y Lemma 3. ad the Holder equalty, we fd a costat K such that ž /ž / a a Ž. ž / a a a a E a A E a a K Ž A. a a Ž a. K Ž A. a Ž a.. Therefore the rght-had sde of Ž 3.. s ouded aove y c Ž. 3 a K t u c a a, where u Ž c a. ad K 3 s a costat. Due to our codto o Ž. ad the selecto of, we have Ž. Ž. Ž. Ž Ž.. K 4 ž / Ž. K5, where K ad K are costats, whch completes the proof of Theorem. 4 5 Ž. uder c Proof of Theorem.3. Wthout loss of geeralty we assume that a 0 for all. For every a ad u a we have, after smple mapulatos, Ž 3.. u j Ž j ž./ a ju Ž j. j: j u a 0 a a cov, sup cov, a. I partcular y 4., there exsts a costat C such that for every a, a a var a C a. We shall costruct ow a tragular array of radom varales Z, 4 for whch we shall mae use of Theorem C the Appedx. Fx a small postve ad fd a postve teger u u such that u j Ž j ž./ ju 0 a a cov, for every u.

11 CLT FOR LINEAR PROCESSES 453 Ths s possle ecause of Ž 3.. ad 4.. Deote y x the teger part of x ad defe K, u j Y a, j 0,,..., j uj KjK j ½, Ž. K Kj 5 A : Kj Kj K, cov Y, Y var Y. Sce covž Y, Y. varž Y. varž Y.,,, we get that for every j the set A j s ot empty. Now we defe the tegers m, m,..., m recurretly y m0 0: ad defe m mm; m m, m A 4 j j j m j Z Y, j 0,,..., j m j 4 u m,..., u m. j j j We oserve that Z a X, j 0,,.... j j It s easy to see that every set j cotas o more tha 3Ku elemets. Hece, for every fxed postve y. ad. the array Z :,..., ; 4 satsfes the Ldeerg s codto. It remas to oserve that y Theorem C ad costructo j Ł Ž j. j j ct var Zj varž Zj ž./ j j ct covž Z, Z covž Z, Z ž, / j ž./ j u ct 4 a aj covž, j. covž Y m, Y, m ž. j j / ju j E exp t Z E exp tz

12 454 M. PELIGRAD AND S. UTEV 8 ž Ž. K / ž / ct 4 var Y c t var a cž t.. Now the proof s complete y Theorem 4. Bllgsley 968. APPENDIX Ths secto cotas some of the theorems whch were used the proofs of the results of Secto 3. The followg lemma s a varat of Theorem 7.. Iragmov ad L Ž 97.. The costat comes from Theorema. Bradley ad Bryc Ž LEMMA A. Let, e two complex-valued radom varales measurale wth respect to A Ž. ad B Ž.. Let 0 ad assume E ad E ; the E EŽ. E Ž A, B.. By the applcato of Theorem. Ro 993, followed y the CauchySchwarz equalty, we formulate the followg result. LEMMA B. Let X 4 e a strogly mxg sequece of radom varales such that EX for a certa 0 ad every. The there s a umercal costat cž. depedg oly o such that for every we have m m Ž. j j covž X, X. cž. sup X. The followg theorem s a smplfed verso of Theorem 3. from Hall ad Heyde 980. THEOREM A. Let X, 4 e a square tegrale martgale dfferece sequece uder the atural fltratos. Suppose that: Ž a. max X P 0; X P ; Ž. c E max X s ouded. The S N 0, as where S X. D

13 CLT FOR LINEAR PROCESSES 455 A careful aalyss of the proof of Theorem 4. Utev 990 gves the followg statemet. THEOREM B. Let X, 4 e a tragular array of radom var- ales such that the followg hold. Ž. a var X C varž X. ja j ja j for every 0 a where C s a uversal costat; var Ž j Xj. Ž. lm f 0; var X j ž / c c Ž c. cov exp t X, exp t X h Ž u. var X j j j t j ja ju ja for every a c where h Ž u. 0, h Ž. t t ad u s of the form u Ž c a. for a certa 0 ; Ž. d EXI X 0 as for every 0, where Ž deotes var X.. The S NŽ 0,. as. D The ext theorem s tae from Newma ad Wrght 98. THEOREM C. Let Z,4 e a assocated famly of radom var- ales. The j Ł j j j j j j j E exp t Z E exp tz ct var Z var Z. Acowledgmets. We would le to tha Emmauel Ro for potg out to us that pot Ž. c of Theorem. holds ths gve form, whch s more geeral tha a earler verso of the paper. May thas go to Rchard Bradley who commeted aout the parwse mxg codto. He provded us wth two examples provg that ether does parwse mxg mply ergodcty or does ergodcty mply parwse mxg. Hs examples motvated ad spred Example.. We also wat to tha Q-Ma Shao who otced that uform tegralty could replace the strctly statoarty assumpto whch was used a prevous verso of ths paper. We tha also the referee who poted out a gap the proof of Theorem. ad for hs may useful suggestos whch mproved the presetato of ths paper. REFERENCES BILLINGSLEY, P. Ž Covergece of Proalty Measures. Wley, New Yor. BRADLEY, R. C. Ž Basc propertes of strog mxg codtos. Progr. Proa. Statst BRADLEY, R. C. ad BRYC, W. Ž Multlear forms ad measures of depedece etwee radom varales. J. Multvarate Aal COX, J. T. ad GRIMMET, G. Ž Cetral lmt theorems for assocated radom varales ad the percolato model. A. Proa

14 456 M. PELIGRAD AND S. UTEV DOUKHAN, P., MASSART, P. ad RIO, E. Ž The fuctoal cetral lmt theorem for strogly mxg processes. A. Ist. H. Pocare HALL, P. ad HEYDE, C. C. Ž Martgale Lmt Theory ad Its Applcatos. Academc Press, New Yor. IBRAGIMOV, I. A. ad LINNIK, YU. V. Ž 97.. Idepedet ad Statoary Sequeces of Radom Varales. Wolters, Groge. NEWMAN, C. M. ad WRIGHT, A. L. Ž 98.. A varace prcple for certa depedet sequeces. A. Proa PELIGRAD, M. Ž Recet advaces the cetral lmt theorem ad ts wea varace prcple for mxg sequeces of radom varales. Prog. Proa. Statst RIO, E. Ž Covarace equaltes for strogly mxg processes. A. Ist. H. Pocare RIO, E. Ž Aout the Ldeerg method for strogly mxg sequeces. ESIAM: Proalty ad Statstcs 356. SHIRYAYEV, A. N. Ž Proalty. Sprger, New Yor. UTEV, S. A. Ž Cetral lmt theorem for depedet radom varales. Proa. Theory Math. Statst DEPARTMENT OF MATHEMATICAL SCIENCES UNIVERSITY OF CINCINNATI P.O. BOX 005 CINCINNATI,OHIO INSTITUTE OF MATHEMATICS NOVOSIBIRSK RUSSIA AND DEPARTMENT OF MATHEMATICS AND STATISTICS LATROBE UNIVERSITY BUNDOORA, VIC 3084 AUSTRALIA