Almost Sure Invariance Principles via Martingale Approximation

Transcription

1 Almost Sure Ivariace Priciples via Martigale Approximatio Florece Merlevède a, Costel Peligrad b ad Magda Peligrad c a Uiversité Paris Est, Laboratoire de mathématiques, UMR 8050 CNRS, Bâtimet Coperic, 5 Boulevard Descartes, Champs-Sur-Mare, FRANCE. florece.merlevede@uiv-mlv.fr b Departmet of Mathematical Scieces, Uiversity of Ciciati, PO Box 20025, Ciciati, Oh , USA. address: peligrc@ucmail.uc.edu c Departmet of Mathematical Scieces, Uiversity of Ciciati, PO Box 20025, Ciciati, Oh , USA. address: peligrm@ucmail.uc.edu Abstract I this paper we estimate the rest of the approximatio of a statioary process by a martigale i terms of the projectios of partial sums. The, based o this estimate, we obtai almost sure approximatio of partial sums by a martigale with statioary differeces. The results are exploited to further ivestigate the cetral limit theorem ad its ivariace priciple started at a poit, the almost sure cetral limit theorem, as well as the law of the iterated logarithm via almost sure approximatio with a Browia motio, improvig the results available i the literature. The coditios are well suited for a variety of examples; they are easy to verify, for istace, for liear processes ad fuctios of Beroulli shifts. Key words: martigale approximatio, queched CLT, almost sure CLT, ormal Markov chais, fuctioal CLT, law of the iterated logarithm, almost sure approximatio. AMS 2000 Subject Classificatio: Primary 60F05, 60F5, 60J05. Itroductio ad otatios I recet years there has bee a itese effort towards a better uderstadig of the structure ad asymptotic behavior of stochastic processes. For processes with short memory there are two basic techiques: approximatio with idepedet radom variables or with martigales. Each of these methods have its ow stregth. O oe had the classes that ca be treated by couplig with a idepedet sequece exhibit faster rates of covergece i various limit theorems; o the other had the class of processes that ca be treated by a martigale approximatio is larger. There are plety of processes that beefit from approximatio with a martigale. Examples are: liear processes with martigale iovatios, fuctios of liear processes, reversible Markov chais, ormal Markov chais, various dyamical systems, discrete Fourier trasform of geeral statioary sequeces. A martigale approximatio provides importat iformatio about these structures, because martigales ca be embedded ito Browia motio, they satisfy the fuctioal cetral limit theorem started at a poit, the law of the iterated logarithm, ad the almost sure cetral limit theorem. Moreover, martigale approximatio provides a simple ad uified approach to asymptotic results for may depedece structures. For all these reasos, i recet years martigale approximatio, couplig with a martigale, has gaied a promiet role i aalyzig depedet data. This is also due to importat developmets by Liverai (996), Maxwell ad Woodroofe (2000), Derrieic ad Li (200-a), Wu Supported i part by a Charles Phelps Taft Memorial Fud grat ad the NSA grat H

2 ad Woodroofe (2004) ad recet developmets by Peligrad ad Utev (2005), Peligrad, Utev ad Wu (2007), Merlevède ad Peligrad (2006), Peligrad ad Wu (200) amog others. May of these ew results, origially desiged for Markov operators, (see Kipis ad Varadha (986) ad Derrieic ad Li (2007) for a survey) have made their way ito limit theorems for stochastic processes. So far this method has bee show to be well suited to trasport from the martigale to the statioary process either the coditioal cetral limit theorem or coditioal ivariace priciple i mea. As a matter of fact, papers by Dedecker-Merlevède-Volý (2006), Volý (2007), Zhao ad Woodroofe (2008-a), Gordi ad Peligrad (200), obtai characterizatios of stochastic processes that ca be approximated by martigales i quadratic mea. These results are useful for treatig evolutios i aealed media. I this paper we address the questio of almost sure approximatio of partial sums by a martigale. These results are useful for obtaiig almost sure limit theorems for depedet sequeces ad also limit theorems started at a poit. Limit theorems for stochastic processes that do ot start from equilibrium is timely ad motivated by evolutios i queched radom eviromet. Moreover recet discoveries by Ouchti ad Volý (2008) ad Volý ad Woodroofe (200) show that may of the cetral limit theorems satisfied by classes of stochastic processes i equilibrium, fail to hold whe the processes are started from a poit, so, ew sharp sufficiet coditios should be poited out for the validity of these types of results. Recet steps i this directio are papers by Wu ad Woodroofe (2004), Zhao ad Woodroofe (2008-b), Cuy (2009 ad 20), Cuy ad Peligrad (2009). The techical challege is to estimate the rest of approximatio of partial sums by a martigale which leads to almost sure results, ragig from the almost sure cetral limit theorems, almost sure approximatio with a Browia motio ad the law of the iterated logarithm. We shall develop our results i the framework of statioary processes that ca be itroduced i several equivalet ways. We assume that (ξ ) Z deotes a statioary Markov chai defied o a probability space (Ω, F, P) with values i a measurable space (S, S). Let π deote the margial distributio of ξ 0 ad suppose that there is a regular distributio of ξ give ξ 0, say Q(x, A) = P(ξ A ξ 0 = x). Next let L 2 0(π) be the set of fuctios o S such that f 2 dπ < ad fdπ = 0, ad for a f L 2 0(π) deote X i = f(ξ i ), S = X i (i.e. S = X 0, S 2 = X 0 +X,...). I additio Q deotes the operator o L 2 (π) actig via i=0 (Qf)(x) = S f(s)q(x, ds). Deote by F k the σ field geerated by ξ i with i k. For ay itegrable variable X we deote E k (X) = E(X F k ). I our otatio E 0 (X ) = Qf(ξ 0 ) = E(X ξ 0 ). Notice that ay statioary sequece (X k ) k Z ca be viewed as a fuctio of a Markov process ξ k = (X i ; i k), for the fuctio g(ξ k ) = X k. The statioary stochastic processes may also be itroduced i the followig alterative way. Let T : Ω Ω be a bijective bi-measurable trasformatio preservig the probability. Let F 0 be a σ-algebra of F satisfyig F 0 T (F 0 ). We the defie the odecreasig filtratio (F i ) i Z by F i = T i (F 0 ). Let X 0 be a radom variable which is F 0 -measurable, cetered, i.e. E(X 0 ) = 0, ad square itegrable E(X0 2 ) <. We the defie the statioary sequece (X i ) i Z by X i = X 0 T i. I this paper we shall use both frameworks. The followig otatios will be frequetly used. We deote by X the orm i L 2 (Ω, F, P), the space of square itegrable fuctios. We shall also deote by X p the orm i L p (Ω, F, P). For ay two positive sequeces a b meas that for a certai umerical costat C ot depedig o, we have a Cb for all ; [x] deotes the largest iteger smaller or equal to x. For the law of the iterated logarithm we use the otatio log 2 = log(log(max(e, ))). The otatio a.s. meas almost surely, while deotes covergece i distributio. The mai questio addressed is to fid sufficiet projective coditios such that there is a martigale M with statioary differeces such that either S M = o( /2 ) a.s., () 2

3 or S M = o(( log 2 ) /2 ) a.s. (2) These types of approximatios are importat to study for istace the limit theorems stated at a poit (queched) ad the law of the iterated logarithm. The so called queched CLT, states that for ay fuctio f cotiuous ad bouded E 0 (f(s / )) E(f(cN)) a.s., (3) where N is a stadard ormal variable ad c is a certai positive costat. By the queched ivariace priciple we uderstad that for ay fuctio f cotiuous ad bouded o D[0, ] edowed with uiform topology we have E 0 (f(s [t] / )) E(f(cW (t))) a.s. (4) where W is the stadard Browia motio o [0, ]. We shall also refer to these types of covergece also as almost sure covergece i distributio uder P 0 a.s., where P 0 (A) = P(A F 0 ). 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 (999), Rootzé (976), ad Hall ad Heyde (980). I the Markov chai settig the almost sure covergece i (3) or (4) are preseted i a slightly differet termiology. Deote by P x ad E x the regular probability ad coditioal expectatio give X 0 = x. I this cotext the queched CLT is kow uder the ame of CLT started at a poit i.e. the CLT or its fuctioal form holds for π almost all x S, uder the measure P x. Here is a short history of the queched CLT uder projective criteria. A result i Borodi ad Ibragimov (994, Ch 4) states that if E 0 (S ) is bouded, the the CLT i its fuctioal form started at a poit holds. Later work by Derrieic ad Li (200-b) improved o this result imposig the coditio E 0 (S ) /2 ɛ with ɛ > 0 (see also Rassoul-Agha ad Seppäläie, 2008 ad 2009). This coditio was improved i Zhao ad Woodroofe (2008-a) ad further improved by Cuy (20) who imposed the coditio E 0 (S ) /2 (log ) 2 (log log ) δ with δ > 0. A result i Cuy ad Peligrad (2009) shows that the coditio k= E 0(X k ) /k /2 <, is sufficiet for (3). We shall prove here that the coditio imposed to E 0 (S ) ca be improved, by requirig less restrictive coditios o the regularity of E 0 (S ) tha the result i Cuy (20). The we shall poit out that the coditio ca be further weakeed if we are iterested i a result for averages or if fiite momets of order larger tha 2 are available. To prove the law of the iterated logarithm we shall develop sufficiet coditios for almost sure approximatio with a Browia motio; that is we shall redefie X, without chagig its distributio, o a richer probability space o which there exists a stadard Browia motio (W (t), t 0) such that for a certai positive costat c > 0, S W (c) = o(( log 2 ) /2 ) a.s. We shall also develop sufficiet coditios i terms of E 0 (S ) for the validity of the almost sure cetral limit theorem, amely: for a certai positive costat c > 0 ad ay real t, lim log k= k {S k / k t} = P(cN t) a.s. Our method of proof is based o martigale approximatio that is valid uder the Maxwell- Woodroofe coditio: E 0 (S k ) (X 0 ) = <. (5) k 3/2 k= 3

4 The key tool i obtaiig our results is the estimate of the rest of the martigale approximatio i terms of E 0 (S k ). We shall establish i Sectio 2 that there is a uique martigale with statioary ad square itegrable differeces such that S M /2 k E 0 (S k ) k 3/2. (6) We the further exploit the estimate (6) to derive almost sure martigale approximatios of the types () ad (2). Our paper is orgaized as follows: I Sectio 2 we preset the martigale approximatio ad estimate its rest. I Sectio 3 we preset the almost sure martigale approximatio results. Sectio 4 is dedicated to almost sure limitig results for the statioary processes. Sectio 5 poits out some examples. Several results ivolvig maximal iequalities ad several techical lemmas are preseted i the Appedix. 2 Martigale approximatio with rest Propositio For ay statioary sequece (X k ) k Z ad filtratio (F k ) k Z described above with (X 0 ) <, there is a martigale (M k ) k with statioary ad square itegrable differeces (D k ) k Z adapted to (F k+ ) k Z, M = i=0 D i, satisfyig (6). To prove this propositio we eed two preparatory lemmas. It is coveiet to use the otatio Y m k = m E k(x k X k+m ). (7) As i Zhao ad Woodroofe (2008-b), we shall also use the followig semi-orm otatio. For a statioary process (X k ) k Z defie the semi-orm X = lim sup E(S2 ). (8) Lemma 2 Assume Y m The, there is a martigale (M k ) k with statioary ad square itegrable differeces adapted to (F k+ ) k Z satisfyig S M /2 E 0 (S k ) max + Y k / Proof of lemma 2. The costructio of the martigale decompositio is based o averages. It was itroduced by Wu ad Woodroofe (2004; see their defiitio 6 o the page 677) ad further developed i Zhao ad Woodroofe (2008-b), extedig the costructio i Heyde (974) ad Gordi ad Lifshitz (98); see also Theorem 8. i Borodi ad Ibragimov (994), ad Kipis ad Varadha (986). We give the martigale costructio with the estimatio of the rest. We itroduce a parameter m (kept fixed for the momet), ad defie the statioary sequece of radom variables: θ0 m = m E 0 (S i ), θk m = θ0 m T k. m Set i= Dk m = θk+ m E k (θk+) m ; M m = Dk m. (9) k=0 4

5 The, (Dk m) k Z is a sequece of statioary martigale differeces such that Dk m ad (M m ) is a martigale. So we have is F k+-measurable X k = D m k + θ m k θ m k+ + m E k(s k+m+ S k+ ), ad therefore S k = M m k + θ m 0 θ m k + k j= = M m k + θ m 0 θ m k + R m k, m E j (S j+m S j ) (0) where we implemeted the otatio Observe that With the otatio we have Notice that R m k = k j= m E j (S j+m S j ). R m k k = Yj m. () j=0 R m k = θ m 0 θ m k + R m k, (2) S k = M m k + R m k. (3) S M 3 max i E 0(S i ). (4) Gordi ad Peligrad (2009) have show that if Y0 m + 0, the D0 coverges i L 2 to a martigale differece we shall deote by D 0. Moreover max l m E(S l F 0 ) 2 /m 0. Deote D i the limit of Di ad costruct the martigale M = j=0 D j. Let ad m be two strictly positive itegers. By the fact that both D0 ad D0 m are martigale differeces ad usig (2) ad (3) we deduce D 0 D m 0 2 = M m M m m 2 m m (θ 0 θ m + R m) (θ m 0 θ m m + R m m) 2. So for fixed, by the fact that sup l m E(S l F 0 ) 2 /m 0 we have that D0 D 0 = lim m D 0 D0 m lim m m /2 R m = Y0 +. (5) We cotiue the estimate i the followig way S M 2 2( S M 2 2( S M 2 + M M 2 ) + D 0 D 0 2 ). The lemma follows by combiig the estimates i (4) ad (5). Next we estimate Y

6 Lemma 3 Uder the coditios of Propositio, for every ad ay m, we have max j Y m E 0 (S k ) /2 k, (6) j k 3/2 ad k=0 k=m+ Y0 m + E 0 (S k ). (7) k 3/2 k m Proof of Lemma 3. I order to prove the iequality (6), we apply the maximal iequality i Peligrad ad Utev (2005) to the statioary sequece Y0 m defied by (7), where m. The where max j j k=0 (Y m 0 ) := Yk m /2 ( Y0 m + (Y0 m )), k= k 3/2 E 0(Y m Y m k ). We first otice that Y0 m m E 0 (S m ). We estimate ow (Y0 m ). With this aim, it is coveiet to use the decompositio (Y m 0 ) m k= k 3/2 E 0(Y m Y m k ) + k=m+ k 3/2 E 0(Y m Y m k ). To estimate the first sum otice that, by the properties of the coditioal expectatio, we have E 0 (Y m Y m k ) k E 0 (Y m 0 ), ad the, sice E 0 (Y0 m ) E 0 (S m ) /m we have m k E 0(Y m 3/ Yk ) m m k= k=m+ m k= E 0 (S m ) k /2 m /2 E 0(S m ). To estimate the secod sum we also apply the properties of the coditioal expectatio ad write this time E 0 (Y0 m Yk ) m E 0 (S k ). The, k E 0(Y m 3/ Yk ) m E 0 (S k ), k 3/2 ad overall (Y m 0 ) m /2 E 0(S m ) + k=m+ We coclude that for ay strictly positive itegers ad m max j j k=0 Yk m 2 E 0(S m ) + m /2 k=m+ E 0 (S k ) k 3/2. k=m+ E 0 (S k ) k 3/2. The estimate (6) of this lemma follows ow by usig Lemma 9 from the Appedix with p = 2 ad γ = /2. With the otatio (8), by passig to the limit i the iequality (6), we obtai (7). Proof of Propositio. Notice that (5) implies Y0 m + 0. We combie the estimate i Lemma 2 with the estimate of Y0 m + i Lemma 3 to obtai the desired result, via Lemma 9 i Appedix applied with p = 2 ad γ = /2. 6

7 3 Almost sure martigale approximatios I this sectio we use the estimate (6) obtaied i Propositio to approximate a partial sum by a martigale i the almost sure sese. Propositio 4 Assume (b ) is ay odecreasig positive, slowly varyig sequece such that b ( k E 0 (S k ) ) 2 <. (8) k 3/2 The, there is a martigale (M k ) k (F k+ ) k Z satisfyig with statioary ad square itegrable differeces adapted to S M b 0 a.s. (9) where b := k= (kb k). As a immediate cosequece of this propositio we formulate the followig corollary: Corollary 5 Assume that for a certai sequece of positive umbers (b ) that is slowly varyig, odecreasig ad satisfies (b ) <, the coditio (8) is satisfied. The there is a martigale (M k ) k with statioary ad square itegrable differeces adapted to (F k+ ) k Z satisfyig: S M 0 a.s. (20) Example: I Corollary 5 the sequece (b ) 3 ca be take for istace b = (log )(log 2 ) γ, for some γ >. Selectig i Propositio 4 the sequece b = log, we obtai: Corollary 6 Assume that log ( E 0 (S k ) ) 2 <. (2) k 3/2 k The there is a martigale (M k ) k with statioary ad square itegrable differeces adapted to (F k+ ) k Z satisfyig: S M 0 a.s. (22) ( log 2 ) /2 Proof of Propositio 4. By Corollary 4.2 i Cuy (20), give i Appedix for the coveiece of the reader (see Propositio 20), i order to show that (9) holds, we have to verify that By Propositio we kow that S M /2 b S M 2 2 <. k E 0 (S k ) k 3/2. Therefore the coditio (8) implies the desired martigale approximatio. 7

8 Remark. Notice that our coditio (8) is implied by the coditio i Corollary 5.8 i Cuy (20). He assumed for the same result E 0 (S ) /2 (log ) 2 (log 2 ) δ with δ > 0, that clearly implies (8). Also (2) is implied by the result i Corollary 5.7 i Cuy (20) who obtaied the same result uder the coditio E 0 (S ) /2 (log ) 2 (log 2 ) τ with τ > /2. I the ext two subsectios we propose two ways to improve o the rate of covergece to 0 of E 0 (S k ) / k that assure a almost sure martigale approximatio i some sese. 3. Averagig I the ext propositio we study a Cesàro-type almost sure martigale approximatio. Propositio 7 Assume that ( E 0 (S k ) ) 2 <. (23) k 3/2 k The there is a martigale (M k ) k with statioary ad square itegrable differeces adapted to (F k+ ) k Z satisfyig: S k M k 0 a.s. (24) k /2 k= Before provig this propositio we shall formulate the coditio (23) i a equivalet form that is due to mootoicity: ( E 0 (S l ) ) 2 <. (25) l 3/2 r 0 l 2 r Proof of Propositio 7. We otice that the coditio (23) implies by Propositio the existece of a martigale (M ) with statioary differeces such that that further implies Whece, by Kroecker lemma, S M 2 2 <, (26) (S M ) 2 < a.s. ad the, by Cauchy-Schwarz iequality Therefore k= 2 (S k M k ) 2 0 a.s. k k= S k M k k /2 k= We ca also formulate the followig result: ( (S k M k ) 2 ) /2. k k= S k M k k /2 0 a.s. 8

9 Propositio 8 Assume that ( E 0 (S k ) ) 2 <. (27) log k 3/2 k The there is a martigale (M k ) k with statioary ad square itegrable differeces adapted to (F k+ ) k Z satisfyig: S k M k 0 a.s. (28) log k 3/2 k= Proof of Propositio 8. The coditio (27) implies by Propositio the existece of a martigale (M ) with statioary differeces such that which, by Kroecker lemma, implies log (S M ) 2 2 log (S k M k ) 2 0 a.s. k= ad the (28), by Cauchy-Schwarz iequality. k 2 < a.s., (29) This idea of cosiderig the average approximatio ca be also applied to Markov chais with ormal operators (i.e. QQ = Q Q o L 2 (π)). For this case we ca replace our Propositio by a result stated i Cuy (20) for ormal Markov chais, amely S M 2 E 0 (S k ) 2 + E 0 (S k ) 2 k k 2. (30) k k> The we ca replace i the proof of Propositios 7 ad 8, the iequality give i our Propositio by the iequality (30). We ca the formulate: Propositio 9 Let (ξ ) Z be a Markov chai with ormal operator ad statioary distributio π. Let f L 2 0(π) ad X 0 = f(ξ 0 ). If the coditio log E 0 (S ) 2 2 is satisfied, the (24) holds. If the coditio is satisfied, the (28) holds. log 2 E 0 (S ) <, (3) 2 <, (32) We poit out that the coditio (3) by itself does ot imply (20) so the averagig is eeded. As a matter of fact, Cuy ad Peligrad (2009) commeted that there is a statioary ad ergodic ormal Markov chai ad a fuctio f such that ad such that (20) fails. 2 log log 2 E 0 (S ) 2 2 <, 9

10 3.2 Higher momets Aother way to improve o the rate of covergece to 0 of E 0 (S k ) /k /2, i order to establish limit theorems started at a poit, is to cosider the existece of momets larger tha 2. Propositio 0 Assume that for some δ > 0, E( X 0 2+δ ) <, ad that the coditio (23) is satisfied. The, there is a martigale (M k ) k with statioary ad square itegrable differeces adapted to (F k+ ) k Z satisfyig for every ε > 0 ad therefore S M = o( /2 ) a.s. P(max j S j M j ε ) <, (33) Proof of Propositio 0. The sequece (max j S j M j ) beig odecreasig, the property (33) is equivalet to: for every ε > 0, P( max S j M j ε2 N/2 ) <, j 2 N N which implies that S M = o( /2 ) almost surely. It remais to prove (33). By assumptio (23) it follows that j j 3/2 E 0 (S j ) <. Therefore, accordig to Propositio, there exists a martigale (M k ) k with statioary ad square itegrable differeces (D k ) k Z adapted to (F k+ ) k Z such that (6) is satisfied. Applyig the Corollary 8 with ϕ(x) = x 2, p = 2, Y i = X i, Z i = D i ad G i = F i, ad takig ito accout (6), we get that for every x > 0 ad ay α [0, ), P(max S j M j 4x) ( ) 2 j x 2 j E 0(S 3/2 j ) + x E ( ) X 0 { X0 x α } (34) + x 2 ( k [ α ]+ j E 0 (S k ) k 3/2 ) 2. Choosig ow α = δ/(2 + 2δ) ad x = ε, we get by usig Fubii theorem that E ( ) X /2 0 { X0 ε /2 α } ε +δ E( X 0 2+δ ). (35) Therefore, startig from (34) ad usig (35), we ifer that (33) holds provided that ( E 0 (S j ) ) 2 <. (36) j 3/2 j [ δ/(2+2δ) ] Now, by the usual compariso betwee the series ad the itegrals, we otice that for ay oicreasig ad positive fuctio h o R + ad ay positive γ, h() <. (37) h( γ ) < if ad oly if Applyig this result with h(y) = ( j [y] j 3/2 E 0 (S j ) ) 2, it follows that the coditios (23) ad (36) are equivalet. This eds the proof of the theorem. Next propositio will be useful to trasport from the martigale to the statioary sequece the law of iterated logarithm. 0

11 Propositio Assume that E( X 0 2+δ ) < for some δ > 0, ad that 3 ( E 0 (S j ) ) 2 <. (38) log 2 j 3/2 j The there is a martigale (M k ) k with statioary ad square itegrable differeces adapted to (F k+ ) k Z satisfyig for every ε > 0 ad therefore S M = o(( log 2 ) /2 ) a.s. P(max j S j M j ε( log 2 ) /2 ) <, Proof of Propositio. We follow the lies of Propositio 0 with the differece that we select i (34) x = ε( log 2 ) /2 ad apply (37) with h(y) = ( (log 2 y) /2 j [y] j 3/2 E 0 (S j ) ) 2. We shall poit ow two sets of coditios that satisfy the coditios of these last two propositios. Assume that E 0 (S ) /2 (log ) 3/2 (log 2 ) β for a certai β > /2. The coditio (23) is satisfied. If E 0 (S ) /2 (log ) 3/2 (log 2 ) γ for a γ > 0, the the coditio (38) is satisfied. 4 Applicatios to almost sure limit theorems We shall formulate here a few applicatios of the almost sure martigale approximatios to queched fuctioal CLT, LIL ad almost sure CLT. For simplicity we assume i this sectio that the statioary sequece is ergodic to avoid radom ormalizers. Theorem 2 Assume that the statioary sequece is ergodic ad the coditios of Corollary 5 or Propositio 0 hold. The S [t] / σw (t) uder P 0 a.s., where σ = D 0 ad D 0 is defied by (9). Proof of Theorem 2. The coditios of Corollary 5 or Propositio 0 imply that for every ε > 0 that further implies P 0 ( max k S k M k > ε ) 0 a.s. P 0 ( sup S [t] M [t] > ε ) 0 a.s. 0 t Accordig to Theorem 3. i Billigsley (999), the limitig distributio of S [t] / is the same as of M [t] / uder P 0 a.s. It was show i Derrieic ad Li (200-a) i details that ad the result follows. M [t] / σw (t) uder P 0 a.s., Theorem 3 Assume that the statioary sequece is ergodic ad the coditios of Propositio 7 are satisfied. The we have S k 2 k /2 3 σn uder P 0 a.s., (39) k= where σ = D 0 ad D 0 is defied by (9).

12 Proof of Theorem 3. Uder the coditio (23) we kow there is a ergodic martigale (M ) with statioary ad square itegrable differeces (D ) satisfyig k= S k M k k /2 0 a.s. The, by Theorem 3. i Billigsley (999), the limitig distributio of k= S k k /2 coicides to the limitig distributio of By chagig the order of summatio we ca rewrite k= M k/k /2 as ( i=0 k=i+ ) D k /2 i, k= M k k /2 uder P 0 a.s. ad, accordig to the Raikov method for provig the cetral limit theorem for martigales, we have to study the limit of the sum of squares. We first write that 4 2 ( + i) 2 Di 2 ( 2 i= i=0 The, by the Birkhoff ergodic theorem, we have k=i+ Di 2 E(D 2 0) = σ 2 a.s. ad i L. i=0 ) 2D 2 k /2 i 4 2 ( i) 2 Di 2. (40) Hece, applyig the geeralized Toeplitz lemma (see Lemma 2) to the both sides of the iequality (40) with x i = D 2 i ad c i = i ad the with c i = i, we get that 2 ( i=0 k=i+ i=0 k /2 )2 D 2 i 2 3 σ2 a.s. ad i L. The, by Theorem 3.6 i Hall ad Heyde (980) we easily obtai the covergece i (39). Theorem 4 Assume that either the coditios of Corollary 6 or of Propositio hold ad i additio the sequece is ergodic. The we ca redefie (X ) Z, without chagig its distributio, o a richer probability space o which there exists a stadard Browia motio (W (t), t 0) such that S W ( D 0 2 ) = o(( log 2 ) /2 ) a.s. Therefore, the LIL holds: lim sup S ± (2 log 2 ) = D 0 a.s. /2 Proof of Theorem 4. Sice by Corollary 6 or by Propositio we have S M = o(( log 2 ) /2 ) a.s. the result follows by the almost sure ivariace priciple for statioary, ergodic ad square itegrable martigales (see Strasse, 967). 2

13 Theorem 5 Assume that the statioary sequece is ergodic ad Coditio (27) is satisfied. The, for ay real t, lim log k {S k / k t} = P(σN t) a.s., (4) k= where σ = D 0 ad D 0 is defied by (9). Proof of Theorem 5. Accordig to the step (a) of the proof of Theorem i Lacey ad Philipp (990), (4) is equivalet to: for ay Lipschitz ad bouded fuctio f from R to R, lim log k= k f(s k/ k) = E(f(σN)) a.s. (42) Next, by Propositio 8, if the coditio (27) is satisfied, there is a martigale (M k ) k with statioary ad square itegrable differeces satisfyig (28). Therefore, for ay Lipschitz fuctio f, lim log f(s k / k) f(m k / k) = 0 a.s. k k= Notice ow that (M k ) k is ergodic sice (X k ) k Z is. The proof is the completed by the fact that (42) holds with M k replacig S k (see Lifshits (2002)). 5 Examples We shall metio two examples for which the quatity E 0 (S ) is estimated. The, these estimates itroduced i our results will provide ew asymptotic results started at a poit ad LIL, that improve the previous results i the literature.. Liear processes. Let (ε ) Z be a sequece of ergodic martigale differeces ad cosider the liear process X k = i a i ε k i, where (a i ) i is a sequece of real costats such that i a2 i <. We defie S = X i. i= Deote by The b j = a j a j+. E 0 (S ) 2 = j 0 b 2 j. For the particular case a /(L()), where L( ) is a positive, odecreasig, slowly varyig fuctio, computatios i Zhao ad Woodroofe (2008-a) show that E 0 (S ) /L(). 3

14 2. Fuctios of Beroulli shifts. Let (ε k ) k Z be a i.i.d. sequece of Beroulli variables, that is P(ε = 0) = /2 = P(ε = ) ad let Y = 2 k ε k, X = g(y ) g(x)dx, ad S = X k, k=0 where g L 2 (0, ), (0, ) beig equipped with the Lebesgue measure. The trasform Y j is usually referred to as the Beroulli shift of the i.i.d. sequece (ε k ) k Z. The, followig Maxwell ad Woodroofe (2000), as i Peligrad, Utev ad Wu (2007), E(g(Y k ) Y 0 ) 2 2 k ad the E(S Y 0 ) k= E(g(Y k) Y 0 ). 6 Appedix 6. Maximal iequalities k= { x y 2 k } g(x) g(y) 2 dydx, Followig the idea of proof of the maximal iequality give i Propositio 5 of Merlevède ad Peligrad (200), we shall prove the followig result: Propositio 6 Let (Y i ) i 2 r be real radom variables where r is a positive iteger. Assume that the radom variables are adapted to a icreasig filtratio (G i ) i 2 r. Let (Z i ) i 2 r be real radom variables adapted to (G i ) i 2 r ad such that for every i, E(Z i G i ) = 0 a.s. Let S = Y + + Y ad T = Z + + Z. Let ϕ be a odecreasig, o egative, covex ad eve fuctio. The for ay positive real x, ay real p ad ay iteger u [0, r ], the followig iequality holds: P( max i 2 r S i T i 4x) ϕ(x) E(ϕ(S 2 r T 2 r)) + x l=u k= 2 r i= + ( r ( 2 r l /p ) p x p E(S (k+)2 l S k2 l G k2 l) p) p. E( Y i { Yi x/2 u }) Remark 7 Whe the sequece (Y ) Z is statioary as well as the filtratio (G ) Z, the iequality has the followig form: P( max S i 2 r i T i 4x) ϕ(x) E(ϕ(S 2 r T 2r 2r)) + x E( Y { Y x/2 u }) ( + 2r r ) p x p 2 E(S l/p 2 l G 0) p. l=u Notice ow that for ay iteger [2 r, 2 r ), where r is a positive iteger, E(ϕ(S 2 r T 2 r)) ( max <k<2 E(ϕ(S ) k T k )). I additio, due to statioarity ad the subadditivity of the sequece E(S G 0 ) p, we have that 2 k E(S 2 k G 0 ) p 2 2 k j= E(S j G 0 ) p, 4

15 implyig that for ay iteger [2 r, 2 r ), where r is a positive iteger, ad ay iteger u [0, r ], ad the that r i=u r i=u 2 2 E(S i/p 2 i G r 0) p 2 ( 2 E(S i/p 2 i G 0) p 22+/p 2 +/p j= 2 u(+/p) E(S j G 0 ) p 2, i(+/p) i:2 i j 2 u 2 u k= E(S k G 0 ) ) p E(S k G 0 ) p +. k +/p k=2 u It remais to apply Lemma 9 below (with γ = /p) to obtai the followig corollary: Corollary 8 Let (Y i ) i Z be a statioary sequece of real radom variables. Assume that the radom variables are adapted to a icreasig ad statioary filtratio (G i ) i Z. Let (Z i ) i Z be a sequece of real radom variables adapted to (G i ) i Z ad such that for all i, E(Z i G i ) = 0 a.s. Let S = Y + +Y ad T = Z + + Z. Let ϕ be a odecreasig, o egative, covex ad eve fuctio. The for ay positive real x, ay positive iteger, ay real p ad ay real α [0, ], the followig iequality holds: P( max i S i T i 4x) ϕ(x) + c p x p ( max E(ϕ(S k T k )) + 2 <k<2 x E( Y { Y x/ α }) k=[ α ]+ where c p is a positive costat depedig oly o p. So, Proof of Propositio 6. E(S k G 0 ) p k +/p ) p, Usig the fact that E(T T k G k ) = 0 for 0 k, we get, for ay m [0, 2 r ], that max S i 2 r i T i S 2r m T 2r m = E(S 2 r T 2 r G 2r m) E(S 2 r S 2r m G 2r m). max E(S 0 m 2 r 2 r T 2 r G 2 r m) + max E(S 0 m 2 r 2 r S 2 r m G 2 r m). (43) Sice (E(S 2 r T 2 r G k )) k is a martigale, we shall use Doob s maximal iequality (see Theorem 2. i Hall ad Heyde, 980) to deal with the first term i the right had side of (43). Hece, sice ϕ is a odecreasig, o egative, covex ad eve fuctio, we get that Write ow m i basis 2 as follows: P ( max E(S 0 m 2 r 2 r T 2 r G 2 r m) x ) ϕ(x) E(ϕ(S 2 r T 2r)). (44) r m = b i (m)2 i, where b i (m) = 0 or b i (m) =. i=0 Set m l = r i=l b i(m)2 i. With this otatio m 0 = m. Let 0 u r ad write that E(S 2 r S 2r m G 2r m) E(S 2r m u S 2r m G 2r m) + E(S 2 r S 2r m u G 2r m). 5

16 Notice first that P ( max E(S 0 m 2 r 2 r m u S 2 r m G 2 r m) 2x ) P ( max 2 r m u 0 m 2 r j=2 r m+ E(Y j G 2r m) 2x ). Therefore, by usig the fact that m m u 2 u implies 2 r m u j=2 r m+ Y j x + 2 r m u j=2 r m+ Y j { Yj x/2 u }, we derive that P ( max E(S 0 m 2 r 2 r m u S 2r m G 2r m) 2x ) P ( P ( max 2 r m u 0 m 2 r j=2 r m+ max 2 r 0 m 2 r j= E( Y j { Yj x/2 u } G 2r m) x ) E( Y j { Yj x/2 u } G 2 r m) x ). Noticig the that ( 2 r j= E( Y j { Yj x/2 u } G k ) ) is a martigale, Doob s maximal iequality implies k that P ( max E(S 0 m 2 r 2 r m u S 2r m G 2r m) 2x ) x 2 r i= E( Y i { Yi x/2 u }), (45) O the other had, followig the proof of Propositio 5 i Merlevède ad Peligrad (200), for ay m {0,..., 2 r } ad ay p, we get that where ad λ l = r E(S 2 r S 2 r m u G 2 r m) p λ p l E(A r,l G 2 r m) p, (46) α l r l=u α l A r,l = with α l = ( 2 r l k= l=u E(S (k+)2 l S k2 l G k2 l) p p) /p, max E(S 2 r (k )2 S l 2 r k2 l G 2 r k2 l). k 2 r l,k odd Notice ow that by Jese s iequality, E(A r,l G 2 r m) p E(A p r,l G 2 r m). Hece startig from (46), we get that for ay p, P ( max E(S 0 m 2 r 2 r S 2 r m u G 2 r m) x ) P ( max r 0 m 2 r l=u λ p l E(A p r,l G 2 r m) x p). 6

17 Next, sice ( r l=u λ p l E(A p r,l G k) ) is a martigale, Doob s maximal iequality etails that k P ( max E(S 0 m 2 r 2 r S 2 r m u G 2 r m) x ) r x p λ p l E(A p r,l ). Takig ito accout the fact that E(A p r,l ) αp l together with the defiitio of α l ad λ l, we the derive that for ay p, P ( max E(S 0 m 2 r 2 r S 2 r m u G 2r m) x ) ( r ( 2 r l /p ) p x p E(S (k+)2 l S k2 l G k2 l) p) p. (47) l=u k= Startig from (43) ad cosiderig the bouds (44), (45) ad (47), the propositio follows. 6.2 Techical results Lemma 9 I the cotext of statioary sequeces, for every γ > 0, ad p, where c γ = 2 3γ+3. γ max E 0(S k ) p c γ k 6 k=+ l=u k γ+ E 0(S k ) p, Proof of lemma 9. Let k be a positive iteger ad otice first that E 0 (S ) E 0 (S k+ ) + E 0 (S k+ S ). The, by the properties of the coditioal expectatio ad statioarity, So, for ay, Therefore E 0 (S ) p E 0 (S k+ ) p + E 0 (S k ) p. γ E 0(S ) p = γ+ E 0(S ) p ( 2 γ+ 2 2γ+2 2 k=+ 2 k=+ 2 k=+ k γ+ E 0(S k+ ) p + 2 γ+ (k + ) γ+ E 0(S k+ ) p + 2 γ+ γ E 0(S ) p 2 2γ+2 By writig ow, for ay positive iteger k, 3 l=+ ) 2 γ+ E 0 (S ) p 2 2 k=+ 2 k=+ E 0 (S k ) E 0 (S k+ ) + E 0 (S k+ S k ), k=+ k γ+ E 0(S k ) p k γ+ k γ+ E 0(S k ) p. l γ+ E 0 (S l ) p. (48) 7

18 ad by usig statioarity we obtai max E 0(S k ) p k max E 0(S k ) p + E 0 (S ) p 2 max E 0(S k ) p, k 2 k 2 ad the result follows by the iequality (48) applied for each k, k 2. Next result we formulate is Corollary 4.2 i Cuy (20). Propositio 20 Assume (X ) Z is a statioary sequece of square itegrable radom variables ad (b ) a positive odecreasig slowly varyig sequece. Assume b S 2 2 <. The S b 0 a.s. where b := k= (kb k). We give here a geeralized Toeplitz lemma, which is Lemma 5 i M. Peligrad ad C. Peligrad (20). Lemma 2 Assume (x i ) i ad (c i ) i are sequeces of real umbers such that i= x i L, c ad c c c C <. The, i= c ix i i= c i L. 7 Ackowledgemet Magda Peligrad would like to thak Dalibor Volý for suggestig this topic ad for umerous very helpful discussios. The authors are also idebted to the referee for carefully readig the mauscript ad for helpful commets that improved the presetatio of the paper. Refereces [] Billigsley, P. (999). Covergece of probability measures. Wiley, New York. [2] Borodi, A. N. ad Ibragimov, I. A. (994). Limit theorems for fuctioals of radom walks. Trudy Mat. Ist. Steklov. 95. Trasl. ito Eglish: Proc. Steklov Ist. Math. (995). [3] Cuy, C. (2009). Some optimal poitwise ergodic theorems with rate. C. R. Acad. Sci. Paris. 347, [4] Cuy, C. (20). Poitwise ergodic theorems with rate with applicatios to limit theorems for statioary processes. Stoch. Dy.,

19 [5] Cuy, C. ad Peligrad, M. (2009). Cetral limit theorem started at a poit for additive fuctioal of reversible Markov Chais; to appear i J. Theoret. Probab. preprit. arxiv: [6] Dedecker J., Merlevède F. ad Volý, D. (2007). O the weak ivariace priciple for o-adapted statioary sequeces uder projective criteria. J. Theoret. Probab. 20, [7] Derrieic, Y. ad Li, M. (200 a). The cetral limit thorem for Markov chais with ormal trasitio operators started at a poit, Probab. Theory Relat. Fields, 9, [8] Derrieic, Y. ad Li, M. (200 b). Fractioal Poisso equatios ad ergodic theorems for fractioal coboudaries. Israel J. Math. 23, [9] Derrieic, Y. ad Li, M. (2007). The cetral limit theorem for radom walks o orbits of probability preservig trasformatios, Cotemp. Math. 444, 3-5. [0] Gordi, M. I. ad Lifshitz, B. (98). A remark about a Markov process with ormal trasitio operator. Third Vilius Cof. Proba. Stat., Akad. Nauk Litovsk, (i Russia), Vilius, [] Gordi, M. I. ad Peligrad, M. (20). O the fuctioal CLT via martigale approximatio. Beroulli. 7, [2] Hall, P. ad Heyde, C. C. (980). Martigale limit theory ad its applicatio. Academic Press, New York-Lodo. [3] Heyde, C. C. (974). O the cetral limit theorem for statioary processes. Z. Wahrsch. verw. Gebiete. 30, [4] Kipis, C. ad Varadha, S.R.S. (986). Cetral limit theorem for additive fuctioals of reversible Markov processes ad applicatios to simple exclusios. Comm. Math. Phys. 04, -9. [5] Lacey, M. ad Philipp, W. (990). A ote o the almost sure cetral limit theorem. Statist. Probab. Lett. 9, [6] Lifshits, M.A. (2002). Almost sure limit theorem for martigales. I: Limit theorems i probability ad statistics, Balatolelle, 999, vol. II, Jáos Bolyai Math. Soc., Budapest, [7] Liverai, C. (996). Cetral limit theorem for determiistic systems. I: Iteratioal Coferece o Dyamical systems (Motevideo, 995), Pitma Res. Notes Math. Ser. 362, Logma, Harlow, [8] Maxwell, M., Woodroofe, M. (2000). Cetral limit theorems for additive fuctioals of Markov chais. A. Probab. 28, [9] Merlevède, F. ad Peligrad, M. (2006). O the weak ivariace priciple for statioary sequeces uder projective criteria. J. Theoret. Probab. 9, [20] Merlevède, F. ad Peligrad, M. (200). Rosethal iequalities for martigales ad statioary sequeces ad examples. To appear i A. Probab. arxiv: v [2] Ouchti, L. ad Volý, D. (2008). A coditioal CLT which fails for ergodic compoets. J. Theoret. Probab. 2, [22] Peligrad, M. ad Utev, S. (2005). A ew maximal iequality ad ivariace priciple for statioary sequeces. A. Probab. 33,

20 [23] Peligrad, M., Utev, S. ad Wu, W. B. (2007), A maximal L(p)- Iequality for statioary sequeces ad applicatio. Proc. AMS 35, [24] Peligrad, M. ad Utev, S. (2006). Cetral limit theorem for statioary liear processes. A. Probab. 34, [25] Peligrad, M. ad Peligrad, C. (20). O the ivariace priciple uder martigale approximatio; to appear i Stoch. Dy., [26] Peligrad M. (200). Coditioal cetral limit theorem via martigale approximatio. Depedece i aalysis, probability ad umber theory (The Phillipp memorial volume), Kedrick Press [27] Peligrad, M. ad Wu, W. B. (200). Cetral limit theorem for Fourier trasform of statioary processes. A. Probab. 38, [28] Rassoul-Agha, F. ad Seppäläie, T. (2008). A almost sure ivariace priciple for additive fuctioals of Markov chais. Statist. Probab. Lett. 78, [29] Rassoul-Agha, F. ad Seppäläie, T. (2009). A almost sure fuctioal cetral limit theorem for ballistic radom walks i a radom eviromet, A. Ist. H. Poicaré Probab. Statist. 45, [30] Rootzé, H. (976). Fluctuatios of sequeces which coverge i distributio. A. Probab. 4, [3] Strasse, V. (967). Almost sure behavior of sums of idepedet radom variables ad martigales. Proc. Fifth Berkeley Sympos. Math. Statist. ad Probability (Berkeley, Calif., 965/66) II Uiv. Califoria Press, Berkeley, Calif. [32] Volý, D. (2007). A oadapted versio of the ivariace priciple of Peligrad ad Utev. C. R., Math., Acad. Sci. Paris 345, [33] Volý, D. ad Woodroofe, M. (200). A example of o-queched covergece i the coditioal cetral limit theorem for partial sums of a liear process. Depedece i aalysis, probability ad umber theory (The Phillipp memorial volume), Kedrick Press [34] Zhao, O. ad Woodroofe, M. (2008-a). Law of the iterated logarithm for statioary processes. A. Probab. 36, [35] Zhao, O. ad Woodroofe, M. (2008-b). O Martigale approximatios. A. Appl. Probab., 8, [36] Wu, W.B. ad Woodroofe, M. (2004). Martigale approximatios for sums of statioary processes. A. Probab. 32,