On the optimality of McLeish s conditions for the central limit theorem

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1 O the optimality of McLeish s coditios for the cetral limit theorem Jérôme Dedecker a a Laboratoire MAP5, CNRS UMR 845, Uiversité Paris-Descartes, Sorboe Paris Cité, 45 rue des Saits Pères, 7570 Paris cedex 06, Frace. Abstract We costruct a family of statioary ergodic sequeces for which the cetral limit theorem (CLT) does ot hold. These examples show that McLeish s coditios for the CLT are sharp i a precise sese. Résumé Sur l optimalité des coditios de McLeish pour le théorème limite cetral Nous costruisos ue famille de suites strictemet statioaires et ergodiques pour lesquelles le théorème limite cetral a pas lieu. Ces exemples motret que les coditios de McLeish pour le théorème limite cetral sot optimales e u ses précis. Itroductio Let (Ω, A, P) be a probability space, ad let T : Ω Ω be a bijective, bi-measurable trasformatio preservig the probability P. We assume here that the couple (T, P) is ergodic, meaig that ay A A satisfyig T (A) = A has probability 0 or. Let F 0 be a σ-algebra of A satisfyig F 0 T (F 0 ). Let X 0 be a F 0 -measurable, square itegrable ad cetered radom variable, ad defie the statioary sequece (X i ) i Z by X i = X 0 T i. Let the S = X + X + + X, ad for t [0, ], W (t) = S [t] + (t [t])x [t]+. Note that W is a radom variable i the space (C([0, ]), ) of cotiuous bouded fuctios equipped with the uiform metric. The followig weak ivariace priciple (WIP) is essetially due to McLeish [0], Theorem.5. The preset form, which ca be deduced from Haa s criterio [5][6], has bee stated i []. Theorem. Assume that there exists a sequece (a ) 0 of positive umbers such that ( ) a k < ad 0 k=0 a E(X F 0 ) <. () 0 The the series σ = k Z Cov(X 0, X k ) coverges absolutely ad E(S ) coverges to σ. Moreover the process / W coverges i distributio i (C([0, ]), ) to σw, where W is a stadard Wieer process. Remark I [], the ergodicity is ot required, ad Coditio () is show to be sufficiet for the coditioal WIP (which implies the stable covergece i the sese of Réyi []). I [0] the o adapted case is also cosidered (i.e. X 0 is ot supposed to be F 0 -measurable). The o-adapted versio of Theorem. is give i [3]. As quoted i [3], Coditio () holds as soo as E(X F 0 ) <. ()

2 Note that the CLT for / S uder () was already quoted i Aexe A. of my PhD thesis [] (as a cosequece of Heyde s criterio [7]). At that time, my advisor Emmauel Rio asked me whether the CLT could be true if 0 E(X F 0 ) < ad E(S ) σ. The aswer is egative as I poited out i Aexe A.3 of my PhD thesis (this is metioed i Propositio 7 of [] without proof). Recetly, the questio was asked agai by Christophe Cuy at the coferece Des martigales aux systèmes dyamiques held i Mare-la-vallée (October 8-0, 04). O this occasio, I showed him the couterexample of my PhD thesis, ad he coviced me to write a ote o the subject, ad to give a more geeral statemet. As we shall see i Corollary.4, the CLT is ot true eve if a = l( e) i the secod term of (). I would like to thak here Christophe Cuy for his suggestio. To be complete, ote that the coditio 0 E(X F 0 ) < is sufficiet for the CLT whe X = f(y ) is a fuctio of a ormal Markov chai, ad F 0 = σ(y i, i 0): this ca be deduced from [4], as idicated to me by the referee. I would like to thak the referee, who also idicated to me Lemma. below, ad the refereces [8] ad [3]. Mai result ad discussio Let us start with a prelimiary remark. For ay sequece (ψ ) 0 of positive umbers such that 0 ( k=0 ψ k) =, oe ca fid a sequece (u ) 0 of positive umbers such that ( ) u = ad ψ <. (3) 0 0 To see this, ote that the secod coditio i (3) writes also 0 ( k=0 ψ k) u <, ad that the followig lemma holds. Lemma. Let (v ) 0 be a o-decreasig sequece of positive umbers. The 0 v = if ad oly if there exists a sequece (u ) 0 of positive umbers such that 0 u = ad 0 v u <. Proof of Lemma.. If such a (u ) 0 exists, the, writig u = u v / v / ad applyig Cauchy-Schwarz s iequality, we see that 0 v =. For the other implicatio, assume that 0 v = ad let y = k=0 v k ad u = (y v ). Sice (y ) 0 ad (v ) 0 are o-decreasig, oe ca see that u (y v + ) y + y x dx ad v + u + y + y x dx, ad the results follows. k= We are ow i positio to state the mai result of this ote. Theorem. Let (ψ ) 0 be a sequece of positive umbers such that: ψ for ay oegative iteger, ad 0 ( k=0 ψ k) =. For ay sequece (u ) 0 of positive umbers satisfyig (3), there exists a statioary ergodic sequece (X i ) i Z of square itegrable ad cetered radom variables, such that E(X M 0 ) u k for M 0 = σ(x i, i 0), ad lim E(S ) =, (4) k= but the sequece / S does ot coverge i distributio. Remark Note that, by (3) ad the first part of (4), 0 ψ E(X M 0 ) <. Hece, Theorem. shows that the coditios of Theorem. o the sequece (a ) 0 caot be relaxed, if we assume moreover that a for ay oegative iteger. Note that, by defiitio, M 0 = σ(x i, i 0) is the smallest σ-algebra such that X 0 is M 0 -measurable ad M 0 T (M 0 ). u k

3 Remark 3 Oe ca modify the statemet of Theorem. as follows: there exists a statioary ergodic sequece (X i ) i Z of square itegrable ad cetered radom variables, such that the first coditio of (4) holds, but the sequece / S is ot stochastically bouded. It suffices to choose the sequece α i = u i ( [0,/] ]/,] ) i the costructio of Sectio 3. We ow give some examples of weights satisfyig the assumptios of Theorem. or Theorem.. Defiitio.3 For x > 0, let l (x) = l(x e), ad for ay iteger k, defie l k by iductio as follows: for x > 0, l k (x) = l l k (x). For ay positive iteger k ad ay positive umber a, defie the ( k ) L,a (x) = (l (x)) a ad for k >, L k,a (x) = l i (x) (l k (x)) a. The followig corollary is a direct cosequece of Theorem. ad Theorem.. Corollary.4 For ay positive iteger k, the followig statemets hold:. If for some k ad a >, 0 L k,a() E(X F 0 ) <, the Coditio () is satisfied, ad the coclusio of Theorem. holds.. For ay k, there exists a statioary ergodic sequece (X i ) i Z of square itegrable ad cetered radom variables, such that 0 L k,() E(X M 0 ) < for M 0 = σ(x i, i 0) ad E(S ), but the sequece / S does ot coverge i distributio. Let us metio that the couterexample give i Theorem. is differet from the couterexample of Peligrad ad Utev []. I Theorem. of their paper, they show that, for ay sequece c 0, there exists a statioary ergodic sequece (X i ) i Z of square itegrable ad cetered radom variables (i their example X i = g(y i ) where Y i is a coutable Markov chai ad F 0 = σ(y i, i 0)) such that E(S F 0 ) c <, (5) 3/ = but / S is ot stochastically bouded. This proves that the coditio of Maxwell ad Woodroofe [9] (Coditio (5) with c ) for the CLT ad the WIP (see agai [] for the WIP) is sharp (ote that () also implies Maxwell-Woodroofe s coditio). The couterexample of Peligrad ad Utev is differet from ours because firstly we deal with the quatity E(X F 0 ) istead of E(S F 0 ), ad secodly i our case E(S ) which implies the stochastic boudedess of / S. I the paper [3], there is a example of a statioary ergodic sequece such that E(S F 0 ) = o( / l()) ad the CLT fails, but agai the variace does ot grow liearly. I [8] there is a example for which E(S F 0 ) = o( / l()) ad E(S ) ad the CLT fails. I Remark of [8] the authors ask the followig questio: does E(S F 0 ) = o( / l()) ad E(S ) imply the CLT? Thaks to Theorem. we are able to give a egative aswer to this questio: it suffices to take ψ ad u = (L, ()), which implies that E(X F 0 ) = o( / / l()). Note that this also proves that coditio () is sharp. 3 The couterexamples For i Z, let F i = T i (F 0 ), F = i Z F i, ad for i, k Z, let P i (X k ) = E(X k F i ) E(X k F i ). Clearly, if E(X 0 F ) = 0 almost surely, the E(X k F 0 ) = i 0 P i(x k ). The radom variables P i (X k ), P j (X k ) beig orthogoal if i j, Pythagoras s theorem ad the statioarity imply that E(X F 0 ) = P 0 (X i ). (6) 3 k= i=

4 The costructio of the sequeces (X i ) i Z is based o the followig lemmas (proofs i Sectio 4). Lemma 3. Let T p, = p+ i=p P p(x i ). If 0 E(X F 0 ) <, the E(S F 0 ) lim = 0 ad lim S T p, = 0. (7) p= Lemma 3. Let (H, H ) be a ifiite dimesioal Hilbert space. Let (u ) 0 be a sequece of positive umbers such that 0 u = ad u 0. There exists a icreasig sequece of positive itegers (t i ) i 0 such that: for ay orthoormal family (e i ) i 0 i H, there exists a sequece (h i ) i 0 satisfyig h ti = e i, h j h j H u j for ay j, (8) ad for t i < j < t i+, h j = b j e i + c j e i+ with b j + c j =. Let us ow costruct the sequece (X i ) i Z of Theorem.. Let λ be the Lebesgue measure over [0, ], ad let B be the σ-algebra of Borel sets of [0, ]. We deote by X the probability space X = ([0, ], B, λ). Let (ψ ) 0 ad (u ) 0 be two sequeces as i Theorem.. Let (α i ) i 0 be a sequece of fuctios i H = L (X ) (to be chose latter) such that λ(α i ) = 0, α i = ad α i u i. (9) i=0 We cosider the space Ω = X Z ad the probabilty P = λ Z. The trasformatio T is the shift o Ω defied by (T (ω)) i = ω i+. Clearly P is ivariat by T ad the couple (T, P) is ergodic. Startig from the sequece (α i ) i 0 ad from the projectios π j (ω) = ω j, we defie the sequece (A i ) i 0 of fuctios of L (P) by: A i = α i π 0. The sequece (X i ) i Z is the defied by: X 0 = A j T j = α j π j ad X i = X 0 T i = A j T i j. (0) j=0 j=0 Note that these series are well defied i L (P) because (A j T i j ) j 0 is a sequece of idepedet radom variables ad j 0 A j T i j = j 0 α j j 0 u j <. Let F i = σ(π j, j i). Clearly, X 0 is F 0 -measurable ad F is P-trivial by the 0 law. With the otatios of the begiig of this sectio, we have P 0 (X i ) = A i for ay positive iteger i. Hece, it follows from (6) ad (9) that E(X F 0 ) k u k. Note that this is also true with the σ-algebra M 0 = σ(x i, i 0), ad the first coditio of (4) is satisfied. O the other had T p, = p+ i=p P p(x i ) = i=0 A i T p. Hece, the sequece (T p, ) p is i.i.d. ad p= T p, j=0 = T 0, = α i =, () where the last equality follows from (9). Applyig Lemma 3., we ifer that lim E(S ) =, ad (4) is fully satisfied. It remais to choose α i i such a way that / S does ot coverge i distributio. To do this, we use Lemma 3. with a appropriate orthoormal family (e i ) i 0 of L (X ). Let first f = ( [0, [ [, [) ad g = i=0 ( [k/ +,(k+)/ + [ [(k+)/ +,(k+)/ + [). k=0 4

5 Let (t i ) i 0 be the sequece of Lemma 3., ad defie e i = f ti ad e i+ = g ti+. Now, we put α 0 = h 0 ad for i > 0, α i = h i h i, where (h i ) i 0 is the sequece of Lemma 3.. Applyig Lemma 3., we see that (α i ) i 0 satisfies (9). By costructio t i=0 α i = f t ad t + i=0 α i = g t+. Let us check that the two sequeces t / S t ad t / + S t + coverge i distributio to two distict laws. By Lemma 3. it is equivalet to cosider the two sequeces t / t / + t+ p= T p,t +. Now ( ( )) ix t E exp T p,t = t p= ( ( ( ))) ( ( ( t ixft λ exp = cos x t t t p= T p,t ad )))t t. t Cosequetly lim E(exp(ixt / S t )) =, provig that the sequece t / S t coverges i distributio to the Dirac mass at 0. O the other had, the radom variables (T i,t+ ) i t+ are idepedet cetered Rademacher radom variables, so that t / + coverges i distributio to a stadard ormal. coverge i distributio. 4 Proofs of Lemmas 3. ad 3. t+ p= T p,t + ad t / + S t + As a coclusio, the sequece / S does ot Proof of Lemma 3.. We begi with the first part of (7). Sice for ay positive iteger m, the sequece / E(S m F 0 ) coverges to 0 as teds to ifiity, it suffices to prove that lim sup E(S S m F 0 ) = 0. () m >m Now E(S S m F 0 ) = i=m j=m E (E(X i F 0 )E(X j F 0 )) i=m E(X i F 0 ), where the last boud holds because E (E(X i F 0 )E(X j F 0 )) E(X i F 0 ) E(X j F 0 ) E(X i F 0 ) as soo as j i. Hece, () follows easily from the fact that 0 E(X F 0 ) <. We ow prove the secod part of (7). Let S p, = i=p P p(x i ). By orthogoality ad statioarity, oe successively derives S E(S F 0 ) T p, p= = S p, p= = p= i=+ p T p, = p= P 0 (X i ) = p= p= P p (X i ) P 0 (X i ). (3) p+ i=+ Let β p = i=p i P 0(X i ) (which is fiite because of (6) ad the fact that 0 E(X F 0 ) < ). Usig Cauchy-Schwarz s iequality i l, we get P 0 (X i ) p= i=p p= β p = i i=p i= ( i i=p ) i β p, ad the last term coverges to zero as, by usig Cesàro s lemma ad the fact that β 0. Together with (3) ad the first part of (7), this completes the proof of the secod part of (7). p= 5

6 Proof of Lemma 3.. Let (u ) 0 be as i Lemma 3. (without loss of geerality, assume that u < π/ for ay positive iteger ). Defie the the icreasig sequece (t i ) i 0 by iductio : t 0 = 0, ad t i+ is the uique > t i + such that k=t i+ u k < π k=t i+ u k. The fuctio h j is the defied by h ti = e i ad, for t i < j < t i+, ( j ) ( j ) h j = cos u k e i + si u k e i+. k=t i+ k=t i+ By costructio h j, h j cos(u j ). Hece h j h j H ( cos(u j)) u j. Refereces [] J. Dedecker, Pricipes d ivariace pour les champs aléatoires statioaires, Thèse 555, Uiversité Paris Sud, (998). [] J. Dedecker, F. Merlevède, Necessary ad sufficiet coditios for the coditioal cetral limit theorem, A. Probab. 30 (00) [3] J. Dedecker, F. Merlevède, D. Volý, O the weak ivariace priciple for o-adapted sequeces uder projective criteria, J. Theoret. Probab. 0 (007) [4] M. I. Gordi, B. A. Lifšic, A remark about a Markov process with ormal trasitio operator, Third Vilius coferece o probability ad statistics, (98) [5] E. J. Haa, Cetral limit theorems for time series regressio, Z. Wahrsch. verw. Gebiete 6 (973) [6] E. J. Haa, The cetral limit theorem for time series regressio, Stoch. Processes Appl. 9 (979) [7] C. C. Heyde, O the cetral limit theorem for statioary processes, Z. Wahrsch. verw. Gebiete. 30 (974) [8] J. Klicarová, D. Volý, O the exactess of the Wu-Woodroofe approximatio, Stoch. Processes Appl. 9 (009) [9] M. Maxwell, M. Woodroofe, Cetral limit theorem for additive fuctioals of Markov chais, A. Probab. 8 (000) [0] D. L. McLeish, Ivariace priciples for depedet variables, Z. Wahrsch. Verw. Gebiete 3 (975) [] M. Peligrad, S. Utev, A ew maximal iequality ad ivariace priciple for statioary sequeces, A. Probab. 33 (005) [] A. Réyi, O stable sequeces of evets, Sakhyā Ser. A, 5 (963) [3] D. Volý, Martigale approximatio ad optimality of some coditios for the cetral limit theorem, J. Theoret. Probab. 3 (00)