On the weak invariance principle for non adapted sequences under projective criteria.

Transcription

1 O the weak ivariace priciple for o adapted sequeces uder projective criteria. Jérôme Dedecker, Florece Merlevède, Dalibor Voly To cite this versio: Jérôme Dedecker, Florece Merlevède, Dalibor Voly. O the weak ivariace priciple for o adapted sequeces uder projective criteria.. 37 pages <hal-0005> HAL Id: hal Submitted o 3 Apr 006 HAL is a multi-discipliary ope access archive for the deposit ad dissemiatio of scietific research documets, whether they are published or ot. The documets may come from teachig ad research istitutios i Frace or abroad, or from public or private research ceters. L archive ouverte pluridiscipliaire HAL, est destiée au dépôt et à la diffusio de documets scietifiques de iveau recherche, publiés ou o, émaat des établissemets d eseigemet et de recherche fraçais ou étragers, des laboratoires publics ou privés.

2 O the weak ivariace priciple for o adapted sequeces uder projective criteria. Jérôme Dedecker, Florece Merlevède ad Dalibor Volý April 3, 006 Abstract I this paper we study the cetral limit theorem ad its weak ivariace priciple for sums of o adapted statioary sequeces, uder differet ormalizatios. Our coditios ivolve the coditioal expectatio of the variables with respect to a give σ-algebra, as doe i Gordi (969 ad Heyde (974. These coditios are well adapted to a large variety of examples, icludig liear processes with depedet iovatios or regular fuctios of liear processes. Mathematics Subject Classificatios (000: 60 F 05, 60 F 7. Key words: Cetral limit theorem, weak ivariace priciple, projective criteria, martigale approximatio, fuctios of liear processes. Short Title: Limit theorems for o adapted sequeces. Jérôme Dedecker, Uiversité Paris VI, Laboratoire de Statistique Théorique et Appliquée, Boîte 58, Plateau A, 8 ème étage, 75 rue du Chevaleret, 7503 Paris, Frace. dedecker@ccr.jussieu.fr Florece Merlevède, Laboratoire de Probabilités et Modèles Aléatoires, Uiversité Paris VI, et C.N.R.S UMR 7599, Boîte 88, 75 rue du Chevaleret, 7503 Paris, Frace. merleve@ccr.jussieu.fr. Dalibor Volý, Uiversité de Roue, LMRS, Aveue de l Uiversité, 7680 Sait Etiee du Rouvray, Frace. dalibor.voly@uiv-roue.fr

3 Itroductio ad otatios Let (Ω, A, P be a probability space, ad T : Ω Ω be a bijective bimeasurable trasformatio preservig the probability P. Let X 0 be a square itegrable radom variable with mea 0. Defie the the statioary sequece (X i i Z by X i = X T i, ad let S = X + + X ad σ = S. I this paper, we shall address the cetral limit questio ad its ivariace priciple; amely we wat to fid a sequece of positive umbers with, ad coditios esurig that s S coverges i distributio to a mixture of ormal distributios (CLT, or more precisely that {s S [t], t [0, ]} coverges i distributio i the Skorohod space to a mixture of Wieer distributios (WIP. We shall provide sufficiet coditios ivolvig quatities of the type E(X k M 0, where M 0 is a σ-algebra of A satisfyig M 0 T (M 0. We do ot assume here that X 0 is M 0 - measurable, sice i may cases the atural filtratio M i = T i (M 0 is geerated by some auxiliary sequece, typically the iovatios (ε i i Z of a liear process X k = i Z a iε k i. The first result to metio i this cotext was obtaied by Gordi (969, for statioary ad ergodic sequeces. As a cosequece of a geeral result ivolvig martigale approximatios, he proved that the CLT holds with = uder the coditios (. E(X k M 0 < ad X k E(X k M 0 <. k Followig Gordi s approach, Heyde obtaied the two followig results for statioary ad ergodic sequeces. For regular sequeces (i.e. E(X 0 M 0 ad E(X 0 M X 0, he proved i 974 that S / coverges to N (0, σ uder the coditios (. (E(X k M 0 E(X k M coverges i L S to m, ad lim = m = σ, k Z which is close to optimality i the case where = (see our propositio. Next, Heyde proved i 975 that the WIP holds for = provided the two series (.3 E(X k M 0 ad (X k E(X k M 0 coverge i L, k k which clearly improves o (.. Notice that (.3 is a ecessary ad sufficiet coditio i order to get the represetatio, X 0 = m + g g T, where (m T i i Z is a martigale k

4 differece sequece i L ad g is i L (see Volý (993. Also (.3 is a sufficiet coditio for the fuctioal law of the iterated logarithm (see Heyde (975. Followig Heyde s approach (974, our aim is to provide sufficiet coditios based o P 0 (X k = E(X k M 0 E(X k M, for the CLT (cf. Theorem, Sectio ad for the WIP (cf. Theorem, Sectio 3 uder geeral ormalizatios. For istace, as a cosequece of Theorem, we obtai that if X 0 is M 0 -measurable ad / is a slowly varyig fuctio at ifiity, the the CLT holds uder the coditios E(S M 0 = o (, ad P 0 (X i m i L. Now, as a cosequece of Theorem, we obtai that if the sequece is regular ad (.4 P 0 (X i <, i Z the the WIP holds uder the ormalizatio =. I Propositio 4, we give a couterexample showig that (.4 caot be weakeed to (. for the WIP to hold with =. Of course, such results are well adapted to liear processes with depedet iovatios (see Sectio 4, but they ca also be successfully applied to fuctios of liear processes geerated by idepedet iovatios (see Sectio 5. For istace, we obtai as a cosequece of Corollary 6 that if ( ε k i X k = f i + i 0 ( ( E f i 0 ε k i, i + where f is Lipschitz with cotiuous derivative f, ad (ε i i Z is iid with mea zero ad fiite variace, the { S[t] } ( ( E, t [0, ] coverges i distributio to ε 0 f ε i W log i + i the Skohorod space, where W is a stadard Browia motio. I Sectio 6, we go back to coditios à la Gordi. More precisely we derive from (.4 the followig improvemet of (.: the WIP holds with = provided that E(X k M 0 X (.5 k E(X k M 0 < ad <. k k k k Most of the results of this paper are ew (except Corollary. However parts of them were kow i the particularly cases where X 0 is M 0 -measurable ad/or =. This is the reaso why we have made a lot of detailed remarks all alog this paper. 3 i 0

5 . Notatios We have already itroduced the map T ad the sequece (X i i Z. We ow fix the other otatios which we shall use i this paper. We deote by I the σ-algebra of all T -ivariat sets. The probability P is ergodic if each elemet of I has measure 0 or. We deote by (D([0, ], d the space of all fuctios from [0, ] to R which have lefthad limits ad cotiuous from the right, equipped with the Skorohod distace d (see Billigsley (968, Chapter 3. For a σ-algebra M 0 satisfyig M 0 T (M 0, we defie the odecreasig filtratio (M i i Z by M i = T i (M 0. Let M = k Z M k ad M = k Z M k. Let H i be the space of M i -measurable ad square itegrable radom variables, ad deote by H i H i the orthogoal of H i i H i. Let P i be the projectio operator from L to H i H i, that is P i (f = E(f M i E(f M i for ay f i L. Defiitio. We say that the radom variable X 0 is regular if E(X 0 M = 0 almost surely, ad X 0 is M -measurable. Defiitio. Followig Defiitio 0.5 i Bradley (00, a sequece (h( of positive umbers is said to be slowly varyig i the strog sese if there exists a cotiuous fuctio f : (0, (0, such that f( = h( for all N, ad f(x is slowly varyig as x teds to ifiity. sequece (h( is slowly varyig i the strog sese. I what follows, we shall say that h( is a svf if the Sufficiet coditios for the CLT. As i the itroductio, ( deotes a sequece of positive umbers such that. I the theorem below, we give a ecessary ad sufficiet coditio for the ormalized partial sum S / to be well approximated by M /, where M is a martigale with statioary icremets adapted to the filtratio M. Theorem. Let m be a elemet of H 0 H. The followig coditios are equivalet C 0 ( : S lim m T i = 0. i= 4

6 C ( : (a E(S M 0 = o ( ad S E(S M = o (, (b lim l= l i= l P 0 (X i m = 0. If oe of these coditios holds the s S coverges i distributio to E(m I N, where N is a stadard Gaussia radom variable idepedet of I. Remark. Arguig as i the proof of Propositio i Dedecker ad Merlevède (00, we ca prove that if C 0 ( holds, the s S satisfies the coditioal cetral limit theorem, that is: for ay cotiuous fuctio ϕ such that x ( + x ϕ(x is bouded, ad ay iteger k, lim E ( ϕ(s S M k ( ϕ x E(m I g(xdx = 0, where g is the distributio of a stadard ormal. covergece of s S i the sese of Réyi (963. Recall that this implies the stable Remark. If X 0 is regular, the followig orthogoal decompositio is valid: (. X k = i Z P i (X k. It follows that (. E(X k M 0 = i 0 P i (X k ad X k E(X k M = i> P i (X k. Usig the statioarity, we see that C ( (a is equivalet to +i k=i+ P 0 (X k = o ( s ad i=+ i k= i P 0 (X k = o ( s. Remark 3. If C 0 ( holds ad E(m > 0 the s σ coverges to E(m. Hece C 0 (σ holds with m = m/ m. It follows that C (σ (a holds, which implies that σ / is a svf (see Theorem 8.3 i Bradley (00, ad the same is true for /. Remark 4. The coditio C (σ (a is equivalet to the existece of a sequece m i H 0 H such that lim S σ m T i = 0. i= 5

7 This has bee proved by Wu ad Woodroofe (004 if X 0 is M 0 -measurable, ad exteded to the geeral case by Volý (005. Note also that eve i the adapted case, the coditio E(S M 0 = o (σ aloe iot sufficiet for the CLT to hold eve if σ / (see Klicarová ad Volý (006. I the followig propositio, we give a sufficiet coditio for C ( (b. Propositio. The coditio C ( (b holds as soo as (.3 P 0 (X i coverges to m i L, ad i= P 0 (X k = o ( s, ad l= k=l l= k=l P 0 (X k = o ( s. I particular if X 0 is M 0 -measurable ad / is a svf, the C 0 ( holds as soo as (.4 E(S M 0 = o (, ad P 0 (X i m i L. As a cosequece, we obtai the followig corollary. Corollary. Cosider the followig coditios C : C 3 : i Z P 0(X i coverges to m i L, ad S m, X 0 is regular ad i Z P 0(X i < +. We have the implicatios C 3 C C (. Furthermore, if C 3 holds the we have E(m I = k Z E(X 0X k I. Remark 5. The fact that C implies C 0 ( is due to Heyde (974. Note that the covergece of i Z P 0(X i aloe iot sufficiet for the CLT, as show by Theorem 4 i Volý (993. However if we assume that the series i Z P 0(X i is ucoditioally coverget, the C holds (see Theorem 5 i Volý (993. I particular, the series i Z P 0(X i coverges ucoditioally as soo as C 3 holds (see Theorem 6 i Volý (993. I Sectio 7, we shall give aother proof of the implicatios C 3 C C (, ad we shall prove the last assertio of Corollary. Note also that C doeot imply C 3 as show by Theorem 8 i Volý (993. 6

8 Remark 6. If X 0 is M 0 -measurable, Heyde s coditio C is equivalet to (.4 with =. For a cetered ad square itegrable fuctio X k = f(y k of a statioary Markov chai (Y k k 0 with trasitio Kerel K ad ivariat distributio µ, the coditio C is equivalet to the two followig items:. lim sup m>0. lim [ m K k=0 K k f µ, K k f = 0, µ, k= K + m K k f k=0 µ, ] = 0, where µ, is the L (µ-orm (the coditio. is just the Cauchy criterio for the covergece of k= P 0(X i i L, ad the coditio. meas exactly that E(S M 0 = o(. The coditios. ad. are give i Theorem C of Derrieic ad Li (00 ad are due to Gordi ad Lifshitz (see the discussio o page 5 i Derrieic ad Li. Note that, uder ergodicity ad a coditio equivalet to., Woodroofe (99 proved that / (S E(S M 0 is asymptotically ormal. The followig propositio shows that the coditio C 3 is close to optimality (a proof ca be foud i Dedecker (998, Aexe A, Sectio A.3. Propositio. Let Ω = [0, ] Z, A = B Z, where B is the Borel σ-algebra o [0, ], ad P = λ Z, where λ is the Lebesgue measure o [0, ]. Let T be the shift from Ω to Ω defied by (T (ω i = ω i+. For ay sequece (v i i 0 of positive umbers such that i 0 iv i < ad i 0 v i =, there exists a strictly statioary sequece (X i = X 0 T i i Z of square itegrable ad cetered radom variables such that, takig M i = σ(x k, k i,. P 0 (X i v i,. S =, 3. for ay k, l ad ay i j, the variables P i (X k ad P j (X l are idepedet, but / S doeot coverge i distributio. 3 Sufficiets coditios for the WIP. The first result of this sectio is a criterio for the uiform itegrability of s 7 max k S k.

9 Propositio 3. We say that the coditio C 4 ( holds if (a sup E(S k M 0 = o (, sup S k E(S k M = o (, k k C 4 ( : (b for some positive sequece (u i i Z such that u i is bouded, i= ( P lim lim sup E 0 (X i I A u P 0 (X i >Au = 0. i i If C 4 ( holds, the i= (3. the sequece max k S k s is uiformly itegrable. Remark 7. A sufficiet coditio for C 4 ( (a is that (3. E(X k M 0 = o( ad X k E(X k M = o(. k= k= Note that (3. implies that (3.3 X 0 is regular, ad P 0 (X k = o(. Now if = h( with h( a svf, the (3. ad (3.3 are equivalet. The proof of this equivalece will be doe i Sectio 7. k Theorem. Assume that s [t] / is bouded for ay t [0, ]. If C ( (b holds ad C 4 ( holds, the {s S [t], t [0, ]} coverges i distributio i (D([0, ], d to E(m IW, where W is a stadard Browia motio idepedet of I. Remark 8. Agai, if C ( (b holds ad C 4 ( holds, the W = {s S [t], t [0, ]} satisfies the coditioal WIP, that is: for ay cotiuous fuctio ϕ from (D([0, ], d to R such that x ( + x ϕ(x is bouded, ad ay iteger k, ( lim E (ϕ(w M k ϕ x E(m I P W (dx = 0, where P W is the distributio of a stadard Wieer Process. Agai, this implies the stable covergece of the processes W. 8

10 Remark 9. I the coditio C 4 (, the fact that s i= u i is bouded esures that lim if s > 0. This excludes the geeral class of examples discussed i Herrdorf (983 for which the ormalizig sequece satisfies lim if s = 0, the cetral limit theorem holds, but the ivariace priciple fails. As a cosequece of Theorem, we obtai the followig corollary. Corollary. If the coditio C 3 holds, the { / S [t], t [0, ]} coverges i distributio i (D([0, ], d to ηw, where W is a stadard Browia motio idepedet of I, ad η = k Z E(X 0X k I. Remark 0. Let us recall a result due to Haa (979: if. X 0 is M 0 -measurable ad C 3 holds,. P is weak mixig (which implies that P is ergodic, that is 3. lim if lim /σ > 0, P(A T k B P(AP(B = 0 for ay A, B i A, k= the {σ S [t], t [0, ]} coverges i distributio i (D([0, ], d to W, where W is a stadard Browia motio. I fact, if C 3 holds the σ coverges to k Z E(X 0X k, so that the last coditio reduces to k Z E(X 0X k > 0. Applyig Corollary, we see that the coditio. of Haa ca be replaced by the weaker oe E(X 0 X k I = E(X 0 X k almost surely, for ay k Z. Fially, ote that, if X 0 is M 0 -measurable, Corollary is due to Dedecker ad Merlevède (003, Corollary 3. By comparig the corollaries ad, oe ca ask if the WIP holds uder the Heyde s coditio C. The followig propositio gives a egative aswer to this questio. Propositio 4. There exists X 0 L measurable with respect to a σ-algebra M 0, ad a bijective ad bimeasurable trasformatio T preservig the probability P such that X 0 is regular, P is ergodic ad the coditio C is satisfied, but the WIP doeot hold for =. 9

11 4 Applicatios to liear processes with depedet iovatios Let X 0 = i Z a i ε 0 T i with (a i i Z belogig to l. The followig result shows that if (ε 0 T i i Z satisfies C 3, the (X i i Z satisfies C 3 also. Corollary 3. Let (a i i Z be a sequece of real umbers i l. Let ε 0 be a regular radom variable i L ad let ε k = ε 0 T k. Defie the X 0 = i Z a iε i. If (4. P 0 (ε i <, i Z the C 3 holds ad { / S [t], t [0, ]} coverges i distributio i (D([0, ], d to ηw, where W is a stadard Browia motio idepedet of I, ad η = ( E(X 0 X k I = a i E(ε 0 ε k I. k Z i Z k Z Remark. I Theorem 5 of Dedecker ad Merlevède (003, a similar result was give, but for causal liear processes ad causal iovatios oly, that is X 0 = a i ε i, ε 0 is regular ad M 0 -measurable, ad P 0 (ε i <. i 0 i 0 Now, if (a i i Z doeot belog to l, Theorem ca still be successfully applied. For istace, if the iovatios are square itegrable martigale differeces, we obtai the followig result. Corollary 4. Let (a i i Z be a sequece of real umbers i l. Let ε 0 be a radom variable i H 0 H ad let ε k = ε 0 T k. Defie the X 0 = i Z a iε i. Let = a + +a. If the two followig coditios hold, i= ( lim sup a i <, i= a i ( either a i = o(, or i Z a i <, k= i k the {s S [t], t [0, ]} coverges i distributio i (D([0, ], d to E(ε 0 IW, where W is a stadard Browia motio idepedet of I. 0

12 Remark. Uder the assumptios of Corollary 4, C 0 ( holds. Hece, accordig to Remark 3, σ / coverges to ε 0. It follows that we ca take = σ i Corollary 4 ad cosequetly {σ S [t], t [0, ]} coverges i distributio to ηw where η = E(ε 0 I/E(ε 0 (i particular, η = if P is ergodic. Note that, i Corollary 3 ad 4 we have oly required that E(ε 0 <. Now, if we assume that E( ε 0 +δ < for some δ > 0, the the coditios of Corollary 4 ca be weakeed. For istace, for causal liear processes X 0 = i 0 a iε i, Wu ad Mi (005, Theorem, ad idepedetly Merlevède ad Peligrad (005, Propositio, have proved that {σ S [t], t [0, ]} coverges i distributio to ηw as soo as (4. ( i k=0 a k ad i 0 ( +i k=i+ ( ( i a k = o The coditio (4. meas exactly that σ, ad E(S M 0 = o(σ. However, (4. together with E(ε 0 < iot sufficiet for the WIP (see the discussio i Wu ad Mi (005 ad the couterexample give i Merlevède ad Peligrad (005, Sectio 3.. To be complete o this questio, ote that i Wu ad Mi (005, the WIP is proved uder (4. ad E( ε 0 +δ < for iovatios which are ot ecessarily i H 0 H, but which satisfy both ε i = F (..., ζ i, ζ i for some iid sequece (ζ i i Z, ad k 0 P 0(ε k +δ < (i particular, the first coditio implies that P is ergodic, so that the limitig process is a stadard Browia motio. k=0 a k. Remark 3. Accordig to Remark 7, if = h( where h( is a svf, the (4.3 a i = o( i is equivalet to the first part of the coditio ( of Corollary 4. Remark 4. The coditio ( of Corollary 4 doeot allow the followig possibility: i= a i diverges but i= a i coverges. For istace if, for < 0, a = 0, ad for, a = ( u for some sequece (u of positive coefficiets decreasig to zero, such that u =, the Corollary 4 caot be applied sice the coditio ( fails to hold. However, for this selectio of (a Z, the coditio give by Heyde (975 ( (4.4 a k < = k is satisfied as soo as u <, which is a miimal coditio.

13 Remark 5. Notice that the coditios ( ad ( of Corollary 4 are satisfied for sequeces (a i i Z such that for i 0, a i = i h( i where h( is a svf (this class of sequeces obviously doeot satisfy (4.4. The coditio ( of Corollary 4 excludes sequeces (a i such that for i 0, a i = i α, for / < α <. However, for iid iovatios, we kow that for such sequeceeither {σ S [t], t [0, ]} or {s S [t], t [0, ]} ca coverge i distributio to a Wieer process, sice they both coverge to a fractioal Browia motio of idex α (see Giraitis ad Surgailis (989. I fact, if S [t] /σ coverges weakly to the Browia motio, the ecessarily σ has the represetatio σ = h( with h( a svf. This is obviously ot the case here sice σ ε 0 3/ α. Remark 6. The coditio ( of Corollary 4 was used by Wag et al (00 to prove the ivariace priciple for liear processes (see their Theorem. uder the ormalizatio s = a 0 + j= s j/j. However istead of usig i additio the coditio ( of our corollary, they used (for oe-sided liear processes such that a 0 0 the followig coditio: ( j (4.5 lim max a k = 0. j It appears that this coditio combied with the coditio ( iot eough to esure the weak ivariace priciple, so that their theorem is false. Ideed, Merlevède ad Peligrad (005 have poited out the followig fact (see the costructio of their example : there exists a oe-sided liear process for which a i coverges, a /( log ( ad such that s S [t] caot satisfy the weak ivariace priciple. I this couterexample, = / log( + ad k= ( i >k a i / C / log. It follows that the coditio ( of Corollary 4 is satisfied, as well as (4.5. However, the coditio ( of Corollary 4 fails to hold sice this coditio imposes that lim if > 0. As already metioed i Wu ad Mi (005, the wrog argumet i the proof of Theorem. i Wag et al (00 lies o page 34 betwee the equatios (36 ad (37 (the weak ivariace priciple (6 caot follow from (36 ad (37 oly; to derive (6 from (37, the equality i (36 eedot oly be true for ay t [0, ], but also for ay fiite dimesioal k=0 margials of the two processes, which is clearly false.

14 5 Applicatio to fuctios of Liear processes Let Ω = X Z ad P = µ Z, where µ is a probability measure o X. If x is a elemet of X Z, let T be the shift defied by (T (x i = x i+. Let ε i = ε 0 T i be the projectio from X Z to X defied by ε i (x = x i. The sequece ε = (ε i i Z is a sequece of iid radom variables with margial distributio µ. I this sectio, we assume that X 0 is a square itegrable radom variable, which ca be writte as (5. X 0 = G(ε, so that X k = X 0 T k = G(ε T k. Note that, sice P = µ Z, the probability P is ergodic: for ay A I, P(A = 0 or. Moreover, X 0 is regular with respect to the σ-algebras (5. M i = σ(ε j, j i. For such sequeces, the coditio C 3 may be writte as (5.3 E(G(ε T k M 0 E(G(ε T k M <. k Z I this sectio, we shall focus o fuctios of real-valued liear processes ( ( ( (5.4 X k = G(ε T k = f a i ε k i E f a i ε k i, i Z ad we shall give sufficiet coditios for the weak ivariace priciple i terms of the regularity of the fuctio f. As usual, we defie the modulus of cotiuity of f o the iterval [ M, M] by w,f (h, M = i Z sup f(x + t f(x. t h, x M, x+t M Corollary 5. Let X = R, (a i i Z be a sequece of real umbers i l, ad assume that i Z a iε i is defied almost surely. Let X k ad M k be defied as i (5.4 ad (5. respectively. Let (ε i i Z be a idepedet copy of (ε i i Z, ad let { } M k = max a i ε i, a i ε i. i Z a k ε 0 + i k If the followig coditio holds ( (5.5 w,f ak ε 0, M k X0 <, k Z the C 3 holds. I particular, 3

15 . if f is γ-hölder o ay compact set, with w,f (h, M Ch γ M α for some C > 0, γ ]0, ], ad α 0, the (5.5 holds as soo as a k γ < ad E( ε 0 (α+γ <.. if ε 0 = m <, the (5.5 holds as soo as ( w,f m ak, X 0 <. k Z Now, for fuctios of causal liear processes, that is ( ( ( (5.6 X k = G(ε T k = f a i ε k i E f a i ε k i, we ca apply Theorem to the case where i 0 a i =. i 0 Corollary 6. Let X = R ad assume that E(ε 0 = 0 ad that ε 0 is fiite. Let (a k k 0 l, be such that k 0 a k =, ad let = a a. Assume that (5.7 lim sup a i <, ad a i = o(. a i Let X k, M k be defied as i (5.6 ad (5. respectively. If f is Lipschitz ad f is cotiuous, the the process {s S [t], t [0, ]} coverges i distributio i the space D([0, ], d to ηw, where W is a stadard Browia motio, ad ( E (5.8 η = ε0 (f. a i ε i Cosiderig (5.8, we see that the ormalizatio = a a may be too large i 0 k= i all the cases where ( (5.9 E (f a i ε i = 0. i 0 Notice that (5.9 arises i may situatios such as: ε 0 is symmetric ad f is eve. I the followig corollary, we give sufficiet coditios for the coditio C 3 whe (5.9 holds ad k 0 a k iot ecessarily fiite. Corollary 7. Let X = R ad assume that E(ε 0 = 0 ad that ε 0 4 is fiite. Let (a k k 0 l, be such that (5.0 a k k 0 i=k+ 4 a i <. i k i 0

16 Let X k, M k be defied as i (5.6 ad (5. respectively. If f is differetiable, f is Lipschitz ad (5.9 holds, the C 3 holds. Remark 7. Let a i = i for i > 0. The the coditio (5.7 holds, ad Corollary 6 applies. Now, if i additio (5.9 holds, the Corollary 7 applies. Note also that (5.0 holds as soo as i>0 ia i is fiite. I particular, for f(x = x, we obtai the weak ivariace priciple as soo as E(ε 0 = 0, E(ε 4 0 < ad i>0 ia i <. I all the results above, o assumptio was made o the law of ε 0, except momet assumptios. Now, if we assume that ε 0 has a desity bouded by C, the, for the sequeces defied by (5.6, the regularity assumptio o f i the coditio (5.5 may be weakeed by cosiderig the L p -modulus of cotiuity. As usual, we defie the L p - modulus of cotiuity of f by ( /p w p,f (h = sup f(x + t f(x p dx. t h Corollary 8. Let (a i i 0 be a sequece of real umbers i l, ad assume that i 0 a iε i is defied almost surely. Let X k ad M k be defied as i (5.6 ad (5. respectively. Assume that ε 0 has a desity bouded by C. If there exists p [, ] such that (5. the C 3 holds. w p,f ( a k ε 0 <, k 0 Remark 8. I particular (5. holds for ay fuctio f of bouded variatio as soo as there exists p [, [ such that k 0 a k /p < ad E( ε 0 /p <. 6 Other types of depedece We have see that coditios based o the sequece (P 0 (X i i Z ca be verified for certai fuctios X 0 = G((ε i i Z of statioary processes. However, i may situatios (for istace whe we kow some property of a Markov kerel, we rather have iformatios o the decrease of E(X k M 0 ad of X k E(X k M 0. I the followig propositio, we give sufficiet coditios, based o such quatities, for C 3 to hold. Propositio 5. Cosider the two coditios 5

17 C 5 : There exist two sequeces (a k k>0 ad (b k k>0 of positive umbers such that (6. ( i a k <, i= k= ( i b k <, i= k= ad C 6 : k a k E(X k M 0 <, k E(X k M 0 k < ad k We have the implicatios C 6 C 5 C 3. b k X k E(X k M 0 <. k X k E(X k M 0 k <. Remark 9. The coditio C 5 is a mixigale type coditio, i the sese of McLeish (975. I the case where X 0 is M 0 -mesurable, the fact that C 5 implies the WIP with the ormalizatio = has bee established i Propositio of Dedecker ad Merlevède (00. I the same cotext, Peligrad ad Utev (005 have proved that the WIP holds uder the ormalizatio = provided that (6. >0 E(S M 0 3/ <. I that case, sice X k E(X k M 0 = 0, we have the implicatio C 6 (6.. Let us give a simple applicatio of Propositio 5 to fuctios of adapted sequeces. Defiitio 3. Let Y 0 be a M 0 -measurable real valued radom variable, ad let Y k = Y 0 T k. Let F Yk M 0 be the coditioal distributio fuctio of Y k give M 0, ad let F be the distributio fuctio of the Y i s. For ay p [, ], defie the depedece coefficiets β p (i of the sequece (Y k k Z by β p (i = sup F Yi M 0 (t F (t. t R p For p =, we shall use the otatio φ(i = β (i. Corollary 9. Let Y 0 be a M 0 -measurable real valued radom variable, ad assume that (6.3 X 0 = (f g(y 0 E((f g(y 0, 6

18 where f ad g are two o decreasig fuctios. If both f(y 0 ad g(y 0 belog to L p for some p, ad if the depedece coefficiets of the sequece (Y k k Z satisfy (6.4 k ( β(p /(p (k (p /p k <, the the coditio C 3 holds. I particular, for p = ad p =, the coditio (6.4 becomes respectively φ(k k k < ad k β (k k <. Remark 0. Usig the otatios of Defiitio 3, defie the depedece coefficiets α(i of the sequece (Y k k Z by α(i = sup F Yi M 0 (t F (t. t R From Dedecker ad Rio (000, we kow that, if X 0 is M 0 -measurable ad the sequece X 0 E(S M 0 coverges i L, the { / S [t], t [0, ]} coverges i distributio i (D([0, ], d to ηw, where W is a stadard Browia motio idepedet of I ad η = k Z E(X 0X k I. If X 0 is defied by (6.3, we ifer from iequality (.c i Rio (000 that X 0 E(S M 0 coverges i L as soo as (6.5 k 0 α(k Q (udu <, where Q = Q f Q g, ad Q f is the geeralized iverse of x P( f(y 0 > x. Sice α(i β (i, it follows that if both f(y 0 ad g(y 0 belog to L p for some p >, the (6.5 holds as soo as (6.6 k /(p β (k <. k Of course, (6.6 caot be compared to (6.4, sice the coefficiets β (i are smaller tha β (p /(p (i for ay p. However, if β (i is of the same order tha β (p /(p (i, the the rate give i (6.4 is better. Example. Liear processes. Assume that X 0 is defied by (6.3, with Y 0 such that Y 0 = i 0 a iε i ad (ε i i Z is the iid sequece defied i Sectio 5. Let M 0 = σ(ε i, i 0. 7

19 If Y 0 has a desity bouded by K, the oe ca prove that (see Dedecker ad Prieur (005, Sectio 4. β (i ε 0 K k i This leads us to cosider the coditio a k ad φ(i K ε 0 a k. k i (6.7 k i a k k k <. If (6.7 holds, it follows from Corollary 9 that the coditio C 3 is satisfied as soo as. ε 0 < ad f(y 0, g(y 0 belog to L. This holds i particular if X 0 = h(y 0 for some fuctio h of bouded variatio. Note that the coditio (6.7 is stroger tha the coditio k 0 a k give i Remark 8, but we have ot assumed here that ε 0 has a desity.. ε 0 < ad f(y 0, g(y 0 belog to L. Here, the momet assumptios o f(y 0 ad g(y 0 are sharp, ad this result caot be deduced from ay results give i Sectio 5. This result applies i particular to the well kow example where a i = i ad ε 0 is a Beroulli-distributed radom variable with parameter /. I that case, Y 0 is uiformly distributed over [0, ], so that C 3 holds as soo as the icreasig fuctios f, g satisfy λ(f < ad λ(g <, for the Lebesgue measure λ over [0, ]. Note that, for this particular example, it follows from Lemma i Woodroofe (99 that the coditio C 3 holds for X 0 = f(y 0 E(f(Y 0 if ad oly if the Fourier coefficiets ˆf(k of f are such that ˆf((p + k <. k= p=0 Example. Uiformly expadig maps. Let τ be a Borel-measurable map from [0, ] to [0, ]. If the probability µ is ivariat by τ, the sequece (τ i i 0 of radom variables from ([0, ], µ to [0, ] is strictly statioary. Defie the operator K from L ([0, ], µ to L ([0, ], µ via the equality (Kh(xk(xµ(dx = h(x(k τ(xµ(dx

20 where h L ([0, ], µ ad k L ([0, ], µ. It is easy to check that (τ, τ,..., τ has the same distributio as (Y, Y,..., Y where (Y i i Z is a statioary Markov chai with ivariat distributio µ ad trasitio kerel K. Hece, we ca obtai iformatios o the distributio of S (h = h τ + + h τ i by studyig that of h(y + + h(y. Assume ow that τ is uiformly expadig, that is: it satisfies the coditios give i Broise, Sectio., page, with a uique ivariat probability µ which is mixig i the ergodic-theoretic sese (ote that uder Broise s coditios, µ is absolutely cotiuous with respect to the Lebesgue measure, with a bouded desity. For such maps, Dedecker ad Prieur (005 have proved that the coefficiets φ(k of the Markov chai (Y i i Z satisfy φ(k Cρ k for some C > 0 ad ρ ]0, [. It follows from Corollary 9, that if h = (f g µ(f g for two o decreasig fuctios f, g such that µ(f < ad µ(g <, the the process { / S [t] (h, t [0, ]} coverges i distributio i the space (D([0, ], d to ηw, where W is a stadard Browia motio ad η = µ(h + k µ(h h τ k. This result seems to be ew, although these dyamical systems have bee widely studied. The momet assumptios o f ad g are sharp. Usually, the cetral limit theorem for / S (h is give for h belogig to some class of bouded fuctios of [0, ], such as bouded variatio fuctios or γ-hölder fuctios for some γ > 0. 7 Proofs Proof of Theorem. We first show that C 0 ( implies C (. To this aim, defie M = i= m T i, ad otice that E(M M 0 = 0. The E ( S M0 = ( S E M M0 S M which proves that C 0 ( implies the first part of C ( (a. Notice ow that E(M M = M. It follows that S E(S M = S M ( S E M M S M, 9

21 which proves that C 0 ( implies the secod part of C ( (a. Noticig ow that the followig decompositio holds (7. S = S E(S M + E(S M 0 + E(S M E(S M 0, we write that S M = S E(S M + E(S M 0 + E(S M E(S M 0 M. Next by orthogoality, we derive that (7. S M = S E(S M + E(S M 0 + E(S M E(S M 0 M. Cosequetly, if C 0 ( holds, lim E(S M E(S M 0 M = 0. Sice E(S M E(S M 0 = i= k= P i(x k, we have, by orthogoality ad statioarity, (7.3 E(S M E(S M 0 M = ( k= P i(x k m T i s i= = k= P i(x k m T i s i= = k= P 0(X k i m, which eds the proof of C 0 ( C (. The fact ow that C ( C 0 ( follows directly from (7. ad (7.3. Proof of Propositio. The fact that (.3 implies C ( (b is straightforward. Now, if X 0 is M 0 -measurable, the C ( (a reduces to E(S M 0 = o(. I the same way, (.3 reduces to (7.4 P 0 (X i m i L, ad i= l= P 0 (X k = o ( s. Let = h(. Usig the decompositio l l l l P 0 (X i = P 0 (X i P 0 (X i + i=l s l 0 k=l s l ( P 0 (X i h(l, h(

22 we see that the first part of (7.4 implies the secod part of (7.4 provided that (7.5 l= ( h(l coverges to 0. h( Now (7.5 is true as soo as h( is a svf. To see this, ote that h ( is a svf also, ad that, for ay svf sequece g(, g( g(l coverges to. l= Proof of Corollary. Clearly, if the first part of C holds, the (.3 holds with = ad m = i Z P 0(X i, ad cosequetly C ( (b holds. Now, from the decompositio (7., we obtai that (7.6 S = S E(S M + E(S M 0 + E(S M E(S M 0. By assumptio S coverges to m. Sice C ( (b holds, it follows that E(S M E(S M 0 coverges to m. Cosequetly, we ifer from (7.6 that C ( (a holds also, so that C C (. Clearly, if C 3 holds, the the first part of C does. Next, we shall prove that (7.7 E(S I E(X 0 X k I a.s., ad E(m I = E(X 0 X k I a.s., k Z k Z which clearly implies the secod part of C, so that C 3 C. To prove (7.7, ote that, sice X 0 is regular, the decompositio (. is valid. Hece, by orthogoality ad statioarity, E(X 0 X k I = E(P i (X 0 P i (X k I = E(P 0 (X i P 0 (X k+i I. i Z i Z Hece E(X 0 X k I i Z P 0 (X i P 0 (X i+k, so that ( E(X 0 X k I P 0 (X i, k Z i Z

23 which is fiite uder C 3. It follows that, almost surely, the series k Z E(X 0X k I coverges absolutely ad that (7.8 Cosequetly k Z E(X 0 X k I = i Z E(S I = k= E(P 0 (X i P 0 (X j I a.s.. j Z ( k E(X 0 X k I coverges almost surely to k Z E(X 0X k I ad the first part of (7.7 is proved. Now E(m I = E(P 0 (X i P 0 (X j I a.s., i Z j Z ad the secod part of (7.7 follows from (7.8. Proof of Propositio 3. k, We first cosider the followig decompositio: for every (7.9 S k = S k E(S k M + E(S k M 0 + E(S k M E(S k M 0. The, due to the coditio C 4 ( (a, {s soo as (7.0 lim lim sup λ s E(( S + λ + = 0 ad lim max k Sk } will be uiformly itegrable as λ lim sup s E(( S λ + = 0, with S k = E(S k M E(S k M 0, S + = max(0, S,..., S ad S = max(0, S,..., S. We shall oly prove the first part of (7.0, the secod part beig similar. First, ote that S k = k j j= i=j P j i (X j = k k (+i i= j= (i+ P j i (X j. For ay positive iteger i, let (Y i,k, k be the martigale Y i, k, = k (+i j= (i+ P j i (X j ad defie Y + i,j, = max{0, Y i,,,..., Y i,j, }. With these otatios, S k = k i= Y ( i, k, ad therefore settig b i, = u, i l= u l we have for all k, ( S k λ + k i= (Y i, k, λb i, +.

24 Next applyig Hölder s iequality, ad takig the maximum o k o both sides, we get ( S + λ + ( l= ( u l i= (Y + i,, u λb i, + i Takig the expectatio ad applyig Propositio (a of Dedecker ad Rio (000 to the martigale (Y i,k, k, we get that ( s E ( S + λ + ( 4s l= ( u l i= ( (+i u i j= (i+ where Γ(i, j, λb i, = {Y + i,j, > λb i, }. Sice {u i } i Z is such that the first part of (7.0 will hold if we ca prove that (7. lim lim sup λ s i= ( u i j= With this aim, otice that for ay positive A, we have E(Pj i(x j I Γ(i,j,λbi,, u i is bouded, i= E(Pj i(x j I Γ(i,,λbi, = 0. E(P j i(x j I Γ(i,,λbi, E(P j i(x j I P j i (X j >Au i + Au i P(Γ(i,, λb i,. Usig this iequality ad the statioarity, we get that for ay positive A s i= ( E(P u j i(x j I Γ(i,,λbi, i j= i= + A (E(P0 (X i I u P 0 (X i >Au i i i= u i P(Γ(i,, λb i, The coditio C 4 ( (b esures that the first term i the right had side coverges to zero by first lettig ted to ifiity ad after A. Now to treat the secod oe, we use Doob s iequality followed by statioarity which leads to P(Γ(i,, λb i, 4 λ b i, j= (i+ E(P j i(x j By takig ito accout the choice of b i,, it follows that 4 E(P λ b i, s 0 (X i. (7. A i= u i P(Γ(i,, λb i, 4 A λ ( i= E(P0 (X i ( u i u l. l= 3

25 Now otice that i= E(P 0 (X i u i i= ( P E 0 (X i I u P 0 (X i >Au + A i i u i. i= The by takig ito accout the selectio of {u i } i Z ad the coditio C 4 ( (b, it follows that (7.3 sup i= E(P 0 (X i u i < +. Startig from (7. ad usig (7.3 together with the selectio of {u i } i Z, it follows that for ay positive A, A lim sup lim sup λ which eds the proof of (7.. u i P(Γ(i,, λb i, = 0, i= Proof of Remark 7. Notice first that (3. is equivalet to (7.4 P 0 (X i = o(. k= i k The (3. implies (3.3. Now otice that, for r < r, we have that r P 0 (X i k= i k k=0 Hece, usig (3.3, we derive that k+ l= k r P 0 (X i k P 0 (X i. i l k=0 i k P 0 (X i P 0 (X i k= i k i 0 N k + ɛ N s r k=0 r s k s k=n+ r, where ɛ N is such that lim N ɛ N = 0. Now if = h( with h( a svf, we ifer from the properties of the slowly varyig fuctios that r k=n+ s ks < C, where C r is a costat ot depedig o N or r. The, by first lettig r ad ext N, it follows easily that P 0 (X i = o(s r. k= i k 4

26 Now, sice r < r, it follows that if = h( where h( is a svf, the s r = O(. This completes the proof. Proof of Theorem. Sice C 4 ( holds, the sequece s max k Sk is uiformly itegrable, ad the process {s S [t], t [0, ]} is tight (apply (8.7 i Billigsley (968 ad Markov iequality. It remais to prove that for ay 0 t < t < < t k, the k-tuple ( [t ] X k, [t ] X k,..., [t k ] X k k= k= coverges i distributio to E(m I (W (t, W (t,..., W (t k. Clearly, this will follow from the ivariace priciple for statioary martigale differece sequeces, provided that, for ay t [0, ], S [t] (7.5 lim [t] k= k= m T k = 0. To prove (7.5, ote that C 4 ( (a together with C ( (b implies that C 0 ( holds (see Theorem. Two cases arise: either E(m > 0, ad the / is a svf (cf. Remark 3, so that (7.5 is equivalet to C 0 ( ; or m = 0 almost surely, ad (7.5 follows from C 0 ( ad the fact that s [t] / is bouded. Proof of Corollary. From Corollary, we kow that C 3 implies C (. From Theorem, it remais to prove that C 3 implies C 4 (. The fact that C 3 implies C 4 ( (b is clear, by takig u i = P 0 (X i. To prove that C 3 implies C 4 ( (a, ote first that, sice X 0 is regular, Let E(S k M 0 = Obviously k i P j i (X j ad S k E(S k M = i= j= k i Y i,k = P j i (X j ad Z i,k, = j= k j= (i++ k k i= j= (i++ P j i (X j. P j i (X j. sup E(S k M 0 k i= sup k Y i,k ad sup S k E(S k M k sup i= k Z i,k, 5

27 Next, takig u i = P 0 (X i ad deotig C = i Z P 0(X i, we obtai that sup E(S k M 0 C k sup S k E(S k M C k u i= i u i= i sup Y i,k, ad k sup Z i,k,. k Applyig Doob s maximal iequality to the martigales (Y i,k k ad (Z i,k, k, we ifer that (7.6 ad (7.7 sup k sup k E(S k M 0 C S k E(S k M C (i P 0 (X i, i= i= ( ( i P 0 (X i. From (7.6 ad (7.7, we easily ifer that, uder C 3, lim sup E(S k M 0 = 0 ad lim sup S k E(S k M = 0, k k which is exactly C 4 ( (a. Proof of Propositio 4. Notice that it is sufficiet to fid a cetered radom variable X 0, a trasformatio T, ad a sigma-algebra M 0 such that (/ i= X i 0, P 0(X i coverges i L to a costat zero, but the tightess coditio i the Dosker ivariace priciple iot satisfied. Let (Ω, A, P = ( [0, ], B([0, ], λ Z, where λ is the Lebesgue measure, ad let T be the left shift i.e. (T (ω i = ω i+ for all i Z. For all i Z, let π i : Ω [0, ] be the projectios such that π i (ω = ω i, ad let M k = σ(π i, i k. For k ad j k, let Ā k,j be idepedet subsets of [0, ] such that λ(āk,j = /(k4 k, ad let A k,j = π 0 (Āk,j. Notice that for all k, j k ad i Z, the sets A k,j T i are idepedet hece the radom variables I Ak,j T i, i Z are mutually idepedet. We defie ẽ k,j = k j I Ak,j ad e k,j = ẽ k,j E(ẽ k,j, j k j = i, m k = l+, i= 6 l=

28 f k,j = ( j e k,j T i e k,j T i T (m k+ j, f k = i= j j+ ad fially X 0 = k f k. k j= f k,j Note that k k j= f,j k < +, so that X 0 is well defied i L.. We prove that X i 0. Notice first that each of the fuctios f k,j is a coboudary: where g k,j = = ( j We the have that (7.8 Sice j l=0 ( j i= f k,j = g k,j g k,j T ẽ k,j T (i+l T (m k+ j (i + ẽ k,j T i + ( j+ i ẽ k,j T i T (m k+ j. i= j j+ f k,j T l = g k,j T g k,j. l= g k,j E(g k,j j (i + e k,j j+ k. The, for all (ad i particular for j, (7.9 O a other had, f k,j T l = l= = f k,j T l l= j+3 k. ( j j+ e k,j T l i e k,j T l i T (m k+ j l= l= i= j ( j (e k,j T m a(, m, j (e k,j T m b(, m, j T (m k+ j, m= j m= j+ 7

29 where a(, m, j = l= I m l m+ j ad b(, m, j = l= I m+ j l m+ j+. The j l= f k,j T l (e k,j T m a(, m, j + m= j (e k,j T m b(, m, j. m= j+ Now usig the idepedece of the radom variables (e k,j T m m Z, we get that (e k,j T m a(, m, j ( = e k,j a(, m, j ( + j e k,j m= j m= j ad j (e k,j T m b(, m, j = m= j+ It follows that if j the (7.0 f k,j T l l= j ( e k,j b(, m, j ( + j e k,j. m= j+ 4 ( + j e k,j 8 k j. Cosequetly, from (7.9 ad (7.0, we get that for > k : f k,j T l k k/ (j k/ l= k (j k/ for j < j k : f k,m T l k (j / (m j+/ k (m j+/ for m j l= f k,m T l m/ < k k (j m/ for m j. l= From these last cosideratios, it follows that uiformly i k ad i, there exists a positive costat C such that (7. Usig (7., we the derive that (7. X i i= f k T l C. k l= N f k T l + C k= l= k=n+ Sice each of the fuctios f k is a coboudary, it follows that for all, l= f T l k C(k, where C(k is a costat oly depedig o k. The startig from (7. ad lettig ted to ifiity ad after N, we get that (/ 0. 8 i= X i k/

30 . We prove that N P 0 (X l coverges i L to zero. l=0 Notice first that for l = (m k + j + i ad 0 i j, we have that P l (X 0 = e k,j T l, ad that for l = (m k + j + i ad j i j+, we have P l (X 0 = e k,j T l. I additio if l (m k + j + i, for 0 i j+, k ad j k, the we get P l (X 0 = 0. The sequece of P 0 (X l is the a sequece where j terms equal e k,j are followed by j terms equal e k,j. The sice E( j e k,j k, the sum N l= P 0(X l coverges i L to zero. Notice first that m 3. We prove that m { [t] } X i, 0 t iot tight. i= j= g m,j is almost surely fiite ad that X 0 = m m j= g m,j ( m j= m g m,j T. The X 0 is a coboudary ad accordig to Theorem i Volý ad Samek (000, i order for the process { / [t] i= X i, 0 t } to be tight, it iecessary that / max l l i= X i coverges to zero i probability. Usig (7.8 ad the fact that the fuctios g k,j are oegative, otice first that for all k, l ( max k X i max g l l k,j T l + i= max l j= k j= g k,j T l m m m,m k j= m j= g m,j. g m,j T l m m g m,j j= 9

31 The, sice the fuctios g k,j are oegative, to show that { / [t] i= X i, 0 t } iot tight, it suffices to prove that there exists a subsequece (k such that (7.3 (k max l (k k j= g k,j T l doeot coverge to 0 i probability. Take (k = 4 k ad otice that ( P max l (k k j= g k,j T l (k ( 4 k = P l= { k j= } g k,j T l k. Now let B k be the sets defied by B k = {ω Ω such that ω is i oly oe of the sets A k,j T (j (m k + j +l for l 4 k ad j k } O B k, oe of the fuctios g k,j T l atteits its maximum which is equal to k others are oegative. The it follows that ad the ( P max l (k k j= g k,j T l (k P(B k > ( k k4 k 4 k /e, which proves (7.3. Proof of Corollary 3. Clearly, if ε 0 is regular, the so is X 0. It remais to see that k Z P 0(X k is fiite. Clearly ( ( P 0 (X k a i P 0 (ε k, k Z ad C 3 follows from (4.. The approximatig martigale is give by m = ( P 0 (X k = a i P 0 (ε k. k Z i Z The idetificatio of the variace follows by applyig Corollary (for the secod equality, ote that, by assumptio, C 3 holds for (ε i i Z. Proof of Corollary 4. First ote that if (a i i Z is a sequece of real umbers i l, Corollary 4 follows easily from Corollary 3, sice, accordig to the coditio (, 30 i Z k Z k Z

32 / coverges to i Z a i > 0 (i that case the approximatig martigale is m = sig( i Z a iε 0. We shall ow focus o the case where i Z a i =. Accordig to Theorem, it is eough to prove that C ( (b ad C 4 ( hold (the fact that s [t] / is bouded follows from the coditio (. Sice the coditio ( holds, we ca take u i = a i i C 4 ( (b. We first prove C 4 (. From Remark 7, C 4 ( (a follows from (3., which is equivalet to (sice X 0 is regular (7.4 P 0 (X i = o( ad P 0 (X i = o(. k= i k k= i k Sice P 0 (X i = a i ε 0, (7.4 follows from (7.5 a i = o(. k= i k Now, with u i = a i, C 4 ( (b holds as soo as i= lim lim sup a i A E( ε 0I ε 0 >A = 0, i= a i which follows from the coditio (. It remais to prove C ( (b. By Corollary, it is eough to prove (.3. Sice i Z a i =, we ifer from the coditio ( that a + + a coverges to + or to. Hece i= P 0 (X i = ε a i 0 i= i= a i coverges i L to ε 0 or to ε 0. Now, accordig to the coditio (, the secod coditio i (.3 will hold as soo as (7.6 lim ( l k a k k a = 0. k l= We shall prove that (7.6 holds without the square, which is clearly sufficiet. Applyig Hölder s iequality, we have that l= l k a k k a k k a k l= l k a k, 3

33 ad the right had term coverges to 0 accordig to the coditio ( ad (7.5. This completes the proof. Proof of Corollary 5. Sice X 0 is regular, we oly have to prove that k Z P 0(X k is fiite. Let ε be a idepedet copy of ε, ad deote by E ε ( the coditioal expectatio with respect to ε. Clearly (7.7 ( P 0 (X k = E ε (f a i ε k i + a k ε 0 + ( a i ε k i f a i ε k i + a k ε 0 + i<k i>k i<k i>k Sice w,f (t + t, M w,f (t, M + w,f (t, M, it follows that (7.8 P 0 (X k E ε ( X 0 a i ε k i. ( w,f ( a k ε 0, Y Y + w,f ( a k ε 0, Y Y, where Y = i k a iε k i + i>k a iε k i ad Y = i<k a iε k i + i k a iε k i. The result follows by otig that (ε 0, Y + Y ad (ε 0, Y + Y are both distributed as (ε 0, M k, ad by takig the L -orm i (7.8. Items ad are straightforward. Proof of Corollary 6. Startig from (7.7 (with a i = 0 for i < 0, we obtai that (7.9 P 0 (X k = a k E ε ((ε 0 ε 0 Sice f 0 ( k f a i ε k i + ta k (ε 0 ε 0 + i>k a i ε k i dt. is cotiuous ad bouded, ad a k ( ε 0 + ε 0 + i>k a iε k i coverges i probability to 0, it follows that (7.30 Z k = E ε ((ε 0 ε 0 0 ( k (f a i ε k i + ta k (ε 0 ε 0 + ( k a i ε k i f a i ε k i dt i>k coverges to 0 i L. Sice k a iε k i is idepedet of (ε 0 ε 0 ad coverges i distributio to a iε i, it follows that (7.3 ( lim E ε (ε 0 ε 0 k 0 ( k f a i ε k i + ta k (ε 0 ε 0 + i>k a i ε k i dt = ε 0 E i L. Sice / teds to ifiity ad, by the first coditio i (5.7, (f ( a i ε i, lim k=0 a k k=0 a k = a with a =, 3

34 it follows from (7.9 ad (7.3 that ( (7.3 lim P 0 (X k = aε 0 E (f a i ε i k=0 i L. Now, sice (5.7 implies that / is a svf (see Remark, it follows from Corollary that C ( (b holds. Accordig to Theorem, it remais to prove that C 4 ( holds. Sice, from (7.9, P 0 (X k f a k ( ε 0 + ε 0, the proof may be doe as i Theorem 4, by choosig u i = a i. To coclude, ote that the limitig variace is give by the variace of the right had term i (7.3. Proof of Corollary 7. Startig from (7.9 ad usig (5.9, we obtai that ( k ( P 0 (X k = a k (Z k + ε 0 E (f a i ε i f a i ε i, where Z k is defied i (7.30. Sice f is Lipschitz, we obtai P 0 (X k a k f ( ε 0 + ε 0 + f ( ε 0 + ε 0 ε 0 a k a i ε i. i>k Sice a k i>k the coditio C 3 follows from (5.0. a i ε i = ε 0 a k Proof of Corollary 8. Assume without loss that a 0 0. Let Y k = k a iε k i. The desity of Y k is give by f Yk = a 0 f ε0 ( /a 0 a k f ε0 ( /a k ad hece, it is bouded by C a 0. Startig from (7.7 (with a i = 0 for i < 0, we have that ( P 0 (X k = f(y + a k ε 0 + a i ε k i f(y + a k x + a i ε k i f Yk (yf ε0 (xdydx. i>k i>k Cosequetly ( f(y+ak P 0 (X k ε 0 + a i ε k i f(y+a k x+ a i ε k i p /pfε0 f Yk (ydy (xdx. i>k i>k Now sice f Yk is bouded by C a 0 ad w p,f ( t + t w p,f ( t + w p,f ( t, we obtai that (7.33 P 0 (X k (C a 0 /p (w p,f ( a k ε 0 + E (w p,f ( a k ε i>k a i,

35 The result follows by takig the L -orm i (7.33. Proof of Propositio 5. Note first that C 5 implies that E(X 0 M = 0 ad that E(X 0 M = X 0 almost surely, so that X 0 is regular. Cosequetly the decompositio (. is valid. It follows that a k E(X k M 0 = a k P i (X k = ( i a k P 0 (X i k>0 k>0 i 0 i>0 k= b k X k E(X k M 0 = b k P i (X k = ( i b k P 0 (X i. k>0 i>0 i< k= k>0 Now, by Hölder s iequality i l, P 0 (X i i>0 P 0 (X i i< ( ( i / ( ( i a k a k P 0 (X i i k= i>0 ( ( i / ( b k i k= i< k= ( i k= / b k P 0 (X i /, which proves that C 5 implies C 3. To prove that C 6 implies C 5, it suffices to prove that, uder C 6, the sequeces a k = k E(X k M 0 ad b k = k X k E(X k M 0 satisfy (6.. Sice the sequeces E(X k M 0 ad X k E(X k M 0 are o icreasig, we have that i a k k= i E(X [i/] M 0 ad i b k k= i X [i/] E(X [i/] M 0, ad (6. follows easily from C 6. Proof of Corollary 9. Let S (M 0 be the set of all M 0 -measurable radom variables Z such that E(Z =. Clearly, if X 0 is defied by (6.3, the (7.34 E(X k M 0 = sup cov(z, (f g(y k Z S (M 0 Let b(m 0, Y k = sup t R F Yk M 0 (t F (t. Applyig Corollary. i Dedecker (004, we have, for ay cojugate expoets p, q, cov(z, (f g(y k ( f(y 0 p + g(y 0 p { E( Z q b(m 0, Y k } /q. 34

36 Let p. Applyig Hölder s iequality o the last term of the right had side, we obtai, for Z such that E(Z =, (7.35 cov(z, (f g(y k ( f(y 0 p + g(y 0 p ( β (p /(p (k (p /p. Combiig (7.34 ad (7.35, we obtai that E(X k M 0 ( f(y 0 p + g(y 0 p ( β (p /(p (k (p /p. Hece, C 6 follows from (6.4, ad Corollary 9 follows from Propositio 5. Refereces [] Billigsley, P. (968. Covergece of probability measures. Wiley, New-York. [] Bradley, R. C. (00. ititroductio to strog mixig coditios. Volume, Techical Report, Departmet of Mathematics, Idiaa Uiversity, Bloomigto. Custom Publishig of I.U., Bloomigto. [3] Broise, A. (996. Trasformatios dilatates de l itervalle et théorèmes limites. Études spectrales d opérateurs de trasfert et applicatios. Astérisque [4] Dedecker, J. (998. Pricipes d ivariace pour les champs aléatoires statioaires. Thèse 555, Uiversité Paris Sud. [5] Dedecker, J. (004. Iégalités de covariace. C. R. Acad. Sci. Paris Ser. I [6] Dedecker, J. ad Merlevède, F. (00. Necessary ad sufficiet coditios for the coditioal cetral limit theorem. A. Probab [7] Dedecker, J. ad Merlevède, F. (003. The coditioal cetral limit theorem i Hilbert spaces. Stoch. Processes Appl [8] Dedecker, J. ad Prieur, C. (005. New depedece coefficiets. Examples ad applicatios to statistics. Probab. Theory ad Relat. Fields [9] Dedecker, J. ad Rio, E. (000. O the fuctioal cetral limit theorem for statioary processes. A. Ist. H. Poicaré

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