Conditional central limit theorem via martingale approximation

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1 Coditioal cetral it theorem via martigale approximatio Dedicated to the memory of Walter Philipp Magda Peligrad Departmet of Mathematical Scieces, Uiversity of Ciciati, PO Box 20025, Ciciati, Oh , USA. address: AMS 2000 subject classi catios. Primary 60F7; secodary 60J0. Key words ad phrases. Martigale approximatio, cetral it theorem Abstract I this paper we survey ad further study partial sums of a statioary process via approximatio with a martigale with statioary di ereces. Such a approximatio is useful for trasferrig from the martigale to the origial process the coditioal cetral it theorem. We study both approximatios i L ad i L 2 : The results complemet the work of Dedecker Merlevède ad Volý (2007), Zhao ad Woodroofe (2008), Gordi ad Peligrad (2009). The method provides a uitary treatmet of may itig results for depedet radom variables. Itroductio ad Results This paper has double scope. First, to survey some recet results o martigale approximatio ad the, to poit out additioal classes of statioary stochastic processes that ca be studied via a martigale approximatio. The error term will be well adapted to derive the coditioal cetral it theorem (CLT) ad also its fuctioal form for processes associated to partial sums. I this sectio we give the de itios ad state the results. The, i Sectio 2, we prove the ew results ad i Sectio 3 we give examples. The statioary processes ca be itroduced i several equivalet ways. We assume that ( ) 2Z deotes a statioary Markov chai de ed o a probability space (; F; P) with values i a measurable space. The margial distributio ad the trasitio kerel are deoted by (A) = P( 0 2 A) ad Q( 0 ; A) = P( 2 Aj 0 ). I additio Q deotes the operator Qf() = R f(z)q(; dz): Deote by F k the eld geerated by i with i k, ad for a measurable Supported i part by a Charles Phelps Taft Memorial Fud grat ad NSA grat H

2 fuctio f de e X i = f( i ), S = P X i (i.e. S = X 0, S 2 = X 0 + X ;...). i=0 For ay itegrable variable X we deote E k (X) = E(XjF k ): I our otatio E 0 (X ) = Qf( 0 ) = E(X j 0 ): Notice that ay statioary sequece (X k ) k2z ca be viewed as a fuctio of a Markov process k = (X i ; i k); for the fuctio g( k ) = X k. The statioary stochastic processes may also be itroduced i the followig alterative way. Let T : 7! be a bijective bi-measurable trasformatio preservig the probability. Let F 0 be a -algebra of F satisfyig F 0 T (F 0 ). We the de e the odecreasig ltratio (F i ) i2z by F i = T i (F 0 ). Let X 0 be a radom variable which is F 0 -measurable. We de e the statioary sequece (X i ) i2z by X i = X 0 T i. I this paper we shall use both frameworks. I additio we assume that ( ) 2Z is ergodic. The variable X 0 will be assumed cetered at its mea, i.e. E(X 0 ) = 0: The martigale approximatio as a tool i studyig the asymptotic behavior of the partial sums S = P i=0 X i; is goig back to Gordi (969) who proposed to decompose the origial statioary sequece ito a square itegrable statioary ad ergodic martigale M = P i= D i adapted to F, S = M + R, where R is a coboudary, i.e., a telescopig sum of radom variables, with the basic property that su jjr jj p < for some p. More precisely, X = D + Z Z ; where Z is aother statioary sequece i L 2 or i L (here ad everywhere i the paper we deote by jj:jj p the orm i L p ). For provig CLT for statioary sequeces, a weaker form of martigale approximatio was poited out by may authors (see for istace the survey by Merlevède-Peligrad-Utev, 2006). A importat step forward was the result by Heyde (974). I the cotext of statioary ad ergodic sequeces of radom variables with ite secod momet, Heyde obtaied the decompositio S = M + R with E(R 2 =)! 0 as! () with (M ) a martigale adapted to (F ) with statioary square itegrable di ereces, uder the coditio E 0 (S ) E (S )! D 0 i L 2 ad E(S 2 =)! E(D 2 0). Recetly, two iterestig papers, oe by Dedecker-Merlevède-Volý (2007) ad the other by Zhao-Woodroofe (2008), deal with ecessary ad su ciet coditios for martigale approximatio with a error term as i (), showig the uiversality of averagig. Dedecker-Merlevède-Volý (2007) proved amog other thigs that () holds if ad oly if! jje 0(S )jj 2 = = 0 ad! X jje 0 (S l ) E (S l ) D 0 jj 2 = 0 for some D 0 2 L 2. l= 2

3 The, Zhao-Woodroofe (2008) showed that () is equivalet to jje 0(S )jj 2 = = 0 ad (2)! X! [E 0(S l ) E (S l )] = D 0 i L 2. l= The approximatio of type () is importat sice it makes possible to trasfer from martigale the coditioal CLT i the followig form. For ay g cotiuous ad bouded E 0 (g(s = ))! E(g(jjD 0 jj 2 N)) i probability. (here N deotes a stadard ormal variable). This coditioal form of the CLT is a stable type of covergece that makes possible the chage of measure with a majoratig measure, as discussed i Billigsley (968), Rootzé (976), ad Hall ad Heyde (980).. Coditioal CLT via martigale approximatio The rst result represets a combiatio of ideas from the papers of Heyde (974) ad Zhao ad Woodroofe (2008). Theorem Assume (X i ) i2z is a statioary sequece of radom variables with ite secod momet. The the martigale approximatio () holds if ad oly if X l= [E 0 (S l ) E (S l )]! L2 D 0 ad! Moreover the martigale is uique. E(S 2 ) = E(D 2 0). (3) I order to state the other martigale approximatio results it is coveiet to itroduce a semi-orm associated to a statioary sequece (X j ) j2z. For p > 0 de e the plus orm i L p : jjx 0 jj +p = sup! X p jj X j jj p. This otatio was used i the space L 2 by Zhao ad Woodroofe (2008). For m xed we cosider the statioary sequece I Markov operators laguage j=0 Y0 m = m E 0(X + ::: + X m ); Yk m = Y0 m T k. (4) Y m 0 = m (Q + ::: + Qm )(f)( 0 ). We give ext several equivalet coditios i terms of plus orm i L 2. Next theorem exteds Theorem 2 i Gordi ad Peligrad (2009) ad simpli es a coditio i Theorem 2 i Zhao ad Woodroofe (2008). 3

4 Theorem 2 Assume (X i ) i2z is a statioary sequece with ite secod momet. The the followig four statemets are equivalet (a) (b) (c) jjy 0 m jj +2 = 0 m! m! m! Moreover, the martigale is uique. m m mx jje i (X 0 )jj +2 = 0 i= mx jje i (X 0 )jj 2 +2 = 0 i= (M) martigale approximatio () holds. A iterestig problem is to d characterizatios for a martigale approximatio of the type S = M + R with E(jR j= )! 0, (5) where (M ) a martigale adapted to (F ) with statioary square itegrable di ereces. This decompositio is still strog eough to let us trasport the coditioal CLT from the martigale to the origial stochastic process. I this cotext we shall establish several results. Next theorem deals with su ciet coditios for a martigale approximatio of the type (5). Deote D m 0 = m mx [E 0 (S l ) E (S l )]. l= Theorem 3 Assume that (X i ) i2z is a statioary sequece with ite rst momets. The (d) ad (e) below are equivalet ad ay oe implies there is a uique martigale such that (5) holds. (d) sup jj S p jj < C ad jjy 0 m jj + = 0 m! (e) jj E 0 (S )jj p = 0 ad D m! 0 coverges i L 2 as m!. I the followig remark we commet o relatio (d) i Theorem 3. Remark 4 Uder coditios of Theorem 3 we have implies (d 0 ) m! jjy m 0 jj + = 0 (e 0 jj E 0 (S )jj ) p = 0 ad D m! 0 coverges i L as m!. However (e 0 ) does ot imply (d 0 ). 4

5 If the variables are assumed to have ite secod momets coditio (d) of Theorem 3 simpli es. Theorem 5 Assume that (X i ) i2z is a statioary sequece of square itegrable radom variables. The (d 0 ) ad (e) are equivalet ad ay oe implies there is a uique martigale such that (5) holds. 2 Fuctioal coditioal CLT via martigale approximatio The theory ca be exteded to cosider the coditioal cetral it theorem i its fuctioal form. For t 2 [0; ] de e W (t) = S [t]. (6) =2 where [x] deotes the iteger part of x: We are iterested i the followig more geeral result: for all real valued fuctios g cotiuous ad bouded o D[0; ], E 0 [g(w )]! E[g(jjD 0 jj 2 W )] i probability. where W is the stadard Browia motio o [0; ]. It is well kow that a martigale with statioary ad ergodic di ereces i L 2 satis es this type of behavior, that is at the heart of may statistical procedures. The the questio becomes to d ecessary ad su ciet coditios for a martigale decompositio with the error term satisfyig for some p > 0 jj max j js j M j j jj p =! 0. (7) with (M ) a martigale adapted to (F ) with statioary square itegrable di ereces. The semi-orm associated to a statioary sequece (X j ) j2z ; relevat for this case is kx jjx 0 jj M + ;p = sup p jj max j X j j jj p :! k The followig theorem was established i Gordi ad Peligrad (2009). Below Y m 0 is de ed by (4). Theorem 6 Assume (X k ) k2z is a statioary sequece of cetered square itegrable radom variables. The, j= jjy m 0 jj M + ;2! 0 as m! (8) if ad oly if there exists a martigale with statioary di ereces satisfyig (7) with p = 2. Such a martigale is uique. 5

6 For the L case we shall establish : Theorem 7 Assume that (X i ) i2z is a statioary sequece with ite secod momets ad cetered. The (f) ad (g) below are equivalet (f) jjy 0 m jj M + ; = 0 m! (g) jj max k j E 0 (S k )jjj p = 0 ad D m! 0 coverges i L 2 as m! ad ay oe implies that (7) holds with p = : The martigale is uique. 3 Proofs Costructio of the approximatig martigale. The costructio of the martigale decompositio is based o averages. It was used i papers by Wu ad Woodroofe (2004) ad further developed i Zhao ad Woodroofe (2008), ad also used i Gordi ad Peligrad (2009). We itroduce a parameter m (kept xed for the momet), ad de e the statioary sequece of radom variables: m 0 = m mx i= E 0 (S i ) = m mx i=0 [ i m E 0(X i )]; m k = m 0 T k. Deote by D m k = m k E k ( m k ) ; M m = X Dk m. (9) The, (Dk m) k2z is a martigale di erece sequece which is statioary ad ergodic if (X i ) i2z is ad (M m ) 0 is a martigale adapted to (F ) 0. So we have the decompositio of each idividual term ad therefore k= X k = D m k+ + m k m k+ + m E k(s k+m+ S k+ ) S k = M m k + m 0 m k + X k j= = M m k + m 0 m k + R m k, where we implemeted the otatio m E j (S j+m S j ) R m k = X k j= m m E X j (S j+m S j ) = Yj m. j=0 6

7 The we have the martigale decompositio Proof of Theorem 3. S k = M m k + R m k where R m k = m 0 m k + R m k. (0) Let us show rst that (d) implies (e): First step is to otice that the martigale (M m ) 0 de ed by (9) has di ereces i L 2 : This is so sice by both parts of (d) sup! EjM m j sup! EjS R m j p K m. For each m xed the martigale (M m ) 0 has statioary ad ergodic di ereces ad by Theorem of Essee ad Jaso (985) we deduce for all m, D 0;m is i L 2. If we impose (d) the, for every two itegers m 0 ad m" xed, we have jjm m0 sup! M m" jj 2jj0 m0 jj + 2jj m" sup! K m0 ;m ". 0 jj + jjr m0 jj + jjr m" jj Agai by Theorem i Essee ad Jaso (985) we have E(D 0;m 0 D 0;m ") 2 <. Therefore, by the CLT for martigales with statioary ad ergodic di ereces, we obtai for every positive itegers m 0 ad m" M m0 M m" =) d N(0; jjd m0 0 D m" 0 jj 2 2): The, by the covergece of momets i the CLT ad (0) we obtai 0 jjd0 m0 D0 m" m jjm jj 2 = sup p! 2= M m" jj sup! jjr m0 jj + jjr m" jj p. 2= Now, by takig ito accout relatio (d); we have m 0 ;m"! jjdm0 0 D0 m" jj 2 = 0 () showig that D m 0 is coverget i L 2 : Deote its it by D 0 : Moreover, by relatio (0), for every m we deduce that jj E 0 (S )jj jjr sup p = sup m jj p!! 7

8 ad lettig ow m! ; by (d) we obtai jj E 0 (S )jj p = 0:! This completes the proof of (d) implies (e). We prove ow that (e) implies (5). Deote as before by D 0 the it i L 2 of D0 m : Costruct the martigale M = P i= D i ad also M = P i= D i; where (D i ) i are statioary martigale di ereces distributed as D 0 ad the otice that jjm M jj p 2 = jjd 0 D 0 jj 2! 0 as!. So we have the double represetatio S = M + R = M + R ; ad by substractig them ad usig the above computatio we have jjr R jj 2 = jjm M jj 2! 0 as!. So, the proof of jjr jj =! 0 is reduced to showig that jjr jj =! 0: It remais to otice that jj E 0 (S )jj =! 0 implies both jj R jj =! 0 ad jj 0 jj =! 0 ad therefore by the de itio of R we have jjr jj =! 0: Uiqueess follows by usig the followig argumet. If there are two approximatig martigales satisfyig jjs M jj jjs N jj = p = 0,!! the the statioary martigale M N has the di ereces (D k;n D k;m ) k i L 2 ad o oe had ad o the other had we have N jjn M jj p = 0 ;! M =) N(0; E(D 0;N D 0;M ) 2 ): By the covergece of momets i CLT we get E(D 0;N D 0;M ) 2 = 0. P We show ow that (e) implies (d). We costruct the martigale M m = i= Dm i ad so Whece, by triagle iequality jjr m R jj 2 = jjm m M jj 2 = jjd m 0 D 0 jj 2 : jjr m jj jjrm R jj + jjr jj jjd m 0 D 0 jj 2 + jjr jj 8

9 Sice (e) implies (5), we kow that jjr jj =! 0, ad by usig the de itio of R m we obtai jjr sup m jj p jjd m! 0 D 0 jj 2. The the secod part of (d) follows by lettig m!. Moreover, because jj S p jj jj R p jj + jj M p jj it follows that sup jj S p jj jjd 0 jj 2.! Proof of the Remark 4 We argue rst that (d 0 ) implies (e 0 ): By aalyzig the proof of Theorem 3 we otice that eve without coditio su jj p S jj < C; we still have that the Cauchy covergece () holds. But without kowig that the variables are i L 2 this does ot imply that the sequece is coverget i L 2. However the sequece D0 m X 0 coverges i L 2 ad sice X 0 is i L the the sequece coverges i L : We provide ow a simple example showig that (e 0 ) does ot imply (d 0 ). Let us cosider X k = d k + " k " k where (d k ) is a statioary sequece of martigale di erece i L 2 ad (" k ) is a i.i.d. sequece of cetered variables i L with E(" 2 0) = ; which is idepedet of (d k ). We take ow the ltratio F k = ((d j ; " j ); j k) : I our otatio we have that The, m 0 = m mx i= E 0 (X 0 + ::: + X i ) = d 0 + " m " 0 D0 m = m E 0 ( m ) = d m ". It follows that D0 m coverges i L to d which is i L 2. I additio we have that jje 0 (S )jj = = o(). It follows that (e 0 ) is satis ed. O the other had, we have for m, Y m 0 = m E 0(X + ::: + X m ) = m " 0 : It follows that jjy0 m jj + = sup p Ej m! X " i j = : j= 9

10 sice otherwise, by Theorem 2 i Essee ad Jaso (985), " 0 would have ite secod momet. Proof of Theorem 5 To show that (d 0 ) implies (e) we have to make a few small chages to the proof of Theorem 3. Because the variables are i L 2 it is easy to see that we do ot eed the coditio su jj p S jj < C: This was used oly to assure that D0 m is i L 2 : Proof of Theorem We already metioed that Zhao ad Woodroofe (2008) proved that () is equivalet to (2). The, () clearly implies! E (S 2 )= = E(D 2 0): So oe of the implicatios holds. Now we assume (3) ad costruct the martigale as i (9). It remais to estimate E(S X D i ) 2 = X [E(S2 ) + E(M) 2 2E(S D i )]. (2) i=0 i=0 By statioarity ad orthogoality of the martigale di ereces E(S X D i ) = i=0 X i= + ( = E(D 0 D 0 ) jij )E(X X id 0 ) = E[D 0 ( i=0 i )(E 0(X i ) E (X i ))] It is clear ow that coditio (3) implies E(D 0 D 0 )! E(D 2 0), whece, by (2), E(S X D i ) 2! 0 i=0 ad the result follows. Proof of Theorem 2 The fact that (a) is equivalet to (M) was established i Gordi ad Peligrad (2009). The, (M) implies (c) was prove by Zhao-Woodroofe (2008) ad the (c) implies (b) follows by Cauchy Schwarz iequality. Obviously (b) implies (a) by triagle iequality. Proof of Theorem 7 0

11 Notice that (f) implies that (d 0 ) i Theorem 5 holds. As a cosequece, by Theorem 5, we coclude that D0 m coverges i L 2 as m!. Now, because E 0 (S k ) = E 0 (S k M k ); by (0) we obtai p max j E 0(S k )j p E 0 [j m k 0 j + max k jm k j + max j R k m j ]. k Clearly the rst term i the right had side is covergig to 0 i L 2 sice m is xed; the secod term is coverget to 0 i L 2 because the variables are square itegrable (by stadard argumets); the last term is coverget to 0 i L by (f). This completes the proof (f) implies (g): Next, i order to show that (g) implies (7) with p = ; we use the decompositio (0) to get the estimate max k js k M k j max k jm k M k j + max k j 0 k+j + max k j R k j. By Doob maximal iequality for martigales ad by statioarity we coclude that p jj max jm k M k j jj 2 jjd0 D 0 jj 2. k Moreover, by costructio The,! jj max k j 0 k+j + max j R k j jj p 3 jj max j E 0(S k )j jj k k p jj max js k M k j jj jjd0 m D 0 jj 2 + p 3 jj max j E 0(S k )j jj ; k k ad the result follows by lettig!, from the the fact that D0 m L 2. It is easy to see that the martigale is uique.! D 0 i We show ow that (g) implies (f). We costruct the martigale M m = P i= Dm i ad so for ay m p jj max k jrm k R k j jj 2 = p jj max jm k m M k j jj 2 jjd0 m D 0 jj 2 : k By triagle iequality ad Doob maximal iequality we easily get p jj max k jrm k j jj p jj max k jrm k R k j jj + p jj max jr kj jj k jjd m 0 D 0 jj 2 + jj max k jr kj jj. Sice for m xed we have max k jk mj=! 0 i L 2 as! ; by takig ito accout the de itio of R k m we let! ad therefore jjy m 0 jj M + ; jjd m 0 D 0 jj 2. The result follows by lettig m! :

12 4 Applicatios ad Examples 4. Liear Processes. These results are particularly importat for liear processes with idepedet iovatios. Cosider X = X j0 a j j where P j0 a2 j < ad ( i) i2z is a i.i.d i L 2. Deote by b = a 0 +:::+a : The, accordig to Zhao ad Woodroofe (2008) the martigale represetatio () holds if ad oly if (2) holds that is equivalet to! X (b j+ b j ) 2! 0 ad j0 X b j coverges. By usig Theorem we ca add oe more characterizatio of represetatio () for liear processes with idepedet iovatios: X b j! c ad j0 j= E(S ) 2! c 2.! 4.2 Applicatios usig "projective criteria" We provide several su ciet coditios imposed to coditioal expectatios of sums or of idividual variables, that assure that the martigale represetatio (7) holds with p = 2. Their detailed proofs are i Gordi ad Peligrad (2009).. Statioary ad ergodic sequeces satisfyig the Maxwell-Woodroofe (2000) coditio: X jje 0 (S k )jj 2 (X 0 ) = <. (3) k 3=2 k= 2. Statioary ad ergodic sequeces satisfyig X p ke 0 (X )k 2 <. 3. Mixigales: j = = X jjx k E j (X 0 )jj < ad m k=0 mx j=0 This class was also studied i Peligrad ad Utev (2006). j! 0 as m!. 4. Projective criteria: E(X 0 jf ) = 0 almost surely ad X ke i (X 0 ) E i (X 0 )k 2 <. i= 2

13 This coditio ad related coditios were studied i Heyde (974), Haa (979), Gordi (2004) amog others. 4.3 Applicatio to mixig sequeces The results i the previous sectio ca be immediately applied to mixig sequeces leadig to sharpest possible fuctioal CLT ad providig additioal iformatio about the structures of these processes. Examples iclude various classes of Markov chais or Gaussia processes. We shall itroduce the followig mixig coe ciets: For ay two -algebras A ad B de e the strog mixig coe ciet (A,B): (A; B) = supfjp(a \ B) P(A)P(B)j; A 2 A, B 2 Bg ad the mixig coe ciet, kow also uder the ame of maximal coe ciet of correlatio (A,B): (A; B) = supfcov(x; Y )=kxk 2 ky k 2 ; X 2 L 2 (A), Y 2 L 2 (B)g : For the statioary sequece of radom variables (X k ) k2 Z ; we also de e F m the eld geerated by X i with idices m i ; F deotes the eld geerated by X i with idices i ; ad F m deotes the eld geerated by X i with idices i m: The sequeces of coe ciets () ad () are the de ed by () = (F 0 ; F ); ad () = (F 0 ; F ) : Equivaletly, (see Bradley (2007), ch. 4) () = supfjje(y jf 0 )k 2 =ky k 2 ; Y 2 L 2 (F ) ad E(Y ) = 0g We assume the variables are cetered ad square itegrable. The, if X (2 k ) < ; k= the represetatio (7) holds with p = 2: Assume X k EX2 0 I(jX 0 j Q(2(k)) < where Q deotes the cadlag iverse of the fuctio t! P(jX 0 j > t): The the represetatio (7) holds with p = 2. Notice that the coe ciet (k) is de ed by usig oly oe variable i the future. A excellet source of iformatio for classes of mixig sequeces ad classes of Markov chais satisfyig mixig coditios is the book by Bradley (2007). Further applicatios ca be obtaied by usig the couplig coe ciets i Dedecker ad Prieur (2005). 3

14 4.4 Applicatio to additive fuctioals of reversible Markov chais For reversible Markov processes (i.e. Q = Q ) the ivariace priciple uder optimal coditio is kow sice Kipis ad Varadha (986). Here is a formulatio i terms of martigale approximatio. Let ( i ) i2z deotes a statioary ad ergodic reversible Markov chai ad f such that R f 2 d < ad R fd = 0 with the property var(s )! f 2 <.! The the martigale represetatio (7) holds i L 2 holds. A proof of this fact ca be foud i Gordi ad Peligrad (2009). Ackowledgemet. The author would like to thak to Florece Merlevède for useful suggestios ad commets ad for the proof of secod part of Remark 4. Refereces [] Billigsley, P. (968). Covergece of Probability Measures, Wiley, New York. [2] Bradley, R. C. (2007). Itroductio to strog mixig coditios. Volumes -3, Kedrick Press. [3] Dedecker, J. ad Prieur, C. (2005). New depedece coe ciets. Examples ad applicatios to statistics. Probab. Theory Relat. Fields, 32, [4] Dedecker J., Merlevède F. ad Volý, D.(2007). O the weak ivariace priciple for o-adapted statioary sequeces uder projective criteria. J. Theoret. Probab [5] Essee, C.G. ad Jaso, S. (985). O momet coditios for ormed sums of idepedet variables ad martigale di ereces. Stochastic Process. Appl. 9 (985), [6] Hall, P. ad Heyde, C. C. (980). Martigale it theory ad its applicatio. Academic Press, New York-Lodo. [7] Heyde, C. C. (974). O the cetral it theorem for statioary processes. Z. Wahrsch. verw. Gebiete. 30, [8] Haa, E. J. (979). The cetral it theorem for time series regressio. Stochastic Process. Appl. 9, [9] Gordi, M. I. (969). The cetral it theorem for statioary processes, Soviet. Math. Dokl. 0,.5,

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