Pointwise ergodic theorems with rate and application to the CLT for Markov chains

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1 Aales de l Istitut Heri Poicaré - Probabilités et Statistiques 009, Vol. 45, No. 3, DOI: 0.4/08-AIHP80 Associatio des Publicatios de l Istitut Heri Poicaré, Poitwise ergodic theorems with rate ad alicatio to the CLT for Markov chais Christohe Cuy a ad Michael Li b a Uiversité de Nouvelle Calédoie, Nouméa, New Caledoia. b Be-Gurio Uiversity, Beer-Sheva, Israel. Received 5 Setember 007; revised 5 May 008; acceted May 008 Dedicated to Yves Derrieic o the occasio of his 60th birthday Abstract. Let T be Duford Schwartz oerator o a robability sace Ω, μ).forf L μ), >, we obtai growth coditios o which imly that / 0 μ-a.e. I the articular case that = adt is the isometry iduced by a robability reservig trasformatio we get better results tha i the geeral case; these are used to obtai a queched cetral limit theorem for additive fuctioals of statioary ergodic Markov chais, which imroves those of Derrieic Li ad Wu Woodroofe. Résumé. Soit T u oérateur de Duford Schwartz sur u esace de robabilité Ω, μ). Pour f L μ), >, ous obteos des théorèmes ergodiques du tye / 0 μ-.s. sous des coditios ortat sur la croissace de. Lorsque T est iduit ar ue trasformatio réservat la mesure et que =, ous obteos de meilleurs résultats. Ces deriers sot alors utilisés our obteir le théorème cetral limite queched our les sommes artielles associées aux foctioelles de chaîes de Markov statioaires et ergodiques. Nous amélioros aisi des résultats atérieurs de Derrieic Li et Wu Woodroofe. MSC: Primary 60F05; 60J05; 37A30; 37A05; secodary 47A35; 37A50 Keywords: Ergodic theorems with rates; Cetral limit theorem for Markov chais; Duford Schwartz oerators; Probability reservig trasformatios. Itroductio The motivatio for this aer was the search for a queched cetral limit theorem CLT) for additive fuctioals of Markov chais which will iclude the results of [7] ad [7]. We obtai the followig: Theorem. Let {X } 0 be a statioary ergodic Markov chai with state sace S, S), trasitio robability P, ivariat iitial distributio m, ad corresodig Markov oerator P o L S,m). For x S deote by P x the robability of the chai startig from x, defied o the roduct σ -algebra of Ω := S N. Let f L S,m)with f dm = 0. If there exists τ>such that log ) 5/ log log ) τ su 3 P k f < ) the for m-almost every oit x S the sequece fx k ) coverges i distributio, i the sace Ω, P x ), to a ossibly degeerate) Gaussia distributio N 0,σf) ) with variace σf) ideedet of x). Moreover, also the ivariace ricile holds.

2 Poitwise ergodic theorems with rate 7 The first geeral queched CLT of this tye seems to be that of Gordi ad Lifshitz i Sectio IV.8 of [6], which assumed f I P)L m). Our theorem imroves that of Derrieic ad Li [7], who assumed that P k f =O α ) for some 0 <α</, ad that of Wu ad Woodroofe [7], roved for f satisfyig log ) su β P k f < for some β>5/, uder the additioal assumtio that f L m) for some >. Both these results imly that of [3] obtaied ideedetly). Durig the rearatio of the reset mauscrit, after comletig our research, we discovered the rerit of Zhao ad Woodroofe [8]; their mai theorem imlies the queched CLT whe ) holds with τ>3/ which also imroves [7] ad [7]); however, the result of [8] does ot imly our result whe <τ3/ see Chater 5). Our strategy follows that of Derrieic ad Li. We first rove some ergodic theorems with rates, the use them to show that the error term i the martigale aroximatio teds to 0; the CLT for the martigale follows as show i [6]) from Brow s CLT. Our mai oitwise ergodic theorem with rate, used for rovig the queched CLT, may be of ideedet iterest: Theorem. Let T be the isometry iduced o L μ) by a ergodic robability reservig trasformatio. If f L μ) satisfies log ) 3/ log log ) τ su < 3 for some τ>, the 0 μ-a.e. The referee oited out that a differet sufficiet coditio was obtaied by Wu [6]. Wu s coditio does ot imly ours, ad i Chater 3 we will exhibit a examle i which our coditio holds while Wu s does ot.. O rates i the mea ergodic theorem It is well kow that i geeral there is o seed of covergece i the mea ergodic theorem for a ower-bouded oerator T o a reflexive Baach sace X, ot eve for isometries of L iduced by robability reservig trasformatios; a fixed rate for a give T imlies that the averages coverge i oerator orm, ad the I T)Xis closed ad we have a rate of / e.g., see [4]). I geeral, oly coboudaries the elemets of I T)X) have covergece of the averages to 0 with rate of /. For 0 <α<, covergece to 0 of the averages with rate of / α was obtaied i [5] forα-fractioal coboudaries, which are the elemets of I T) α X, with the oerator I T) α defied there by I T) α = I j= a α) j T j, where a α) = α ad a α) j = j! α j k α) are the coefficiets of the ower-series exasio t) α = j= a α) j t j for t. It is show i Corollary.5 ad Theorem.7 of [5] that f I T) α X imlies 0 for X reflexive), but i geeral the latter covergece imlies oly f I T) γ X for γ<α. α It is therefore of iterest to fid a growth coditio o, better faster) tha O α ɛ ), which still yields f I T) α X; we would like also to have i this case a rate i the ergodic theorem for h which satisfies f = I T) α h. Note also that sice there is o rate i the mea ergodic theorem, kowig oly f = I T) α h will ot give ay rate for h; a excetio is whe i fact f I T) α+ɛ X, which imlies that we ca take h I T) ɛ X. It was show i [5], Proositio.0, that covergece of j=0 b α) j coefficiets of t) α = j=0 b α) I T) α X. We will use the asymtotic behavior [9], vol. I,. 77) bα) Ɣα) α C α j T j f, with b α) j = j + )a α) j+ / α) the t j for t <, is sufficiet ad ecessary whe X is reflexive) for f to be i for some C>0, where Ɣ is Euler s fuctio. Whe α is uderstood, it will be coveiet to deote b α) simly by b. )

3 7 C. Cuy ad M. Li I the aer we will make use of regularly varyig fuctios. Followig [],. 76, we say that a ositive fuctio L, defied o a half lie [A, ), A 0, is slowly varyig at ifiity) if for every x>0, Ltx)/Lt), ad t + we say that a ositive fuctio Φ is regularly varyig with exoet ρ <ρ<+ ) ifφx) = x ρ Lx) for some slowly varyig fuctio L. The regularly varyig fuctios of articular iterest i this aer are Φx) = x ρ logx + c) γ. Lemma.. Let T be a ower-bouded oerator o a Baach sace X ad let f X. If there exist 0 <α< ad a o-decreasig regularly varyig fuctio Φ with exoet β>such that Φlog + )) α < +, 3) su the m 0 bα) m T m f coverges i X, to a elemet h I T)Xwhich satisfies f = I T) α h, ad for every, b m α) T m f C Φlog + )), m where Φx) := x du Φu) ), which is a o-decreasig regularly varyig fuctio with exoet β. Proof. For the give α, we will deote b m α) Abel s summatio, k =j α T k ) f = S α + ) α + k α S k j α S j. =j By 3) ad ), there exists C such that k k b T f C =j =j k α + =j by b m.for, write S = m= T m f.fork>j> we have, by α C j α + Φlogj)) + = C j α + Φlogj)) + ) α Φlog + )) + Φlogj)) + Φlogk + )) k j logk ) logj ) ) dx xφlog x) du Φu) ). 4) Sice /Φ is regularly varyig with exoet β <, the lemma o. 80 of [] yields that Φ x) := x defies a fiite-valued regularly varyig fuctio of exoet β +, so Φ as defied i the theorem is regularly varyig with exoet β. The covergece of the itegral defiig Φ show that the right-had side of 4) coverges to 0 as k>j, so the series 0 b T f coverges. Lettig k i 4) gives a estimate for the tail of the series. By Lemma i [],. 77, /j α C 3 /Φlog j) for large j; by Theorem a) i [],. 8, comarig /Φ ad Φ yields that the middle term i the estimate is bouded by a costat mutile of the last term. Hece the asserted estimate for the tail holds. Remark. For Φx) = x β we have Φx) = β )x β ; i this case the roof of the lemma is direct. It is a atural questio whether we must have β> say i the case where Φx) = x β ). I the roof of Proositio 5.3a) see also c)), we give a ormal cotractio o L ad a fuctio f for which the result holds for α = /, with β>/. This motivates a imrovemet of the revious lemma whe the oerator is a ormal cotractio V i a comlex Hilbert sace H, which is give below. dt Φt)

4 Poitwise ergodic theorems with rate 73 Let f H. By the sectral theorem e.g., [9,0,4]), there exists a uique ositive measure σ f o the Borel sets of the uit disk D, called the sectral measure of f, such that V f,f = D z σ f dz) for every 0, ad V k f = D z σ k f dz) = D z z z σ f dz). 5) We will use the reresetatio z = re iπθ with θ.for z, we clearly have z k z k mi, / r)). Covexity of x siπx/) o [0, ] yields x siπx/) for 0 x, so siπθ) θ for θ. Hece, for / r wehave z = r cosπθ)+ r = r) + 4r si πθ) 8θ. For z we have z k < < θ, so we fially obtai z k = z z z {, mi r, } z ). 6) θ As i [] ad [7], we wat to relate the rate V k f =O α Φlog+)) ), to the cocetratio of σ f at. We eed some otatio. For every defie { D := z = re iπθ : r, θ }. Notice that D is the uit disk D, ad m,, 6) yields z k mi, m) z D D m. 7) We ca ow reset the sectral characterizatio of coditio 3). Theorem.. Let V be a ormal cotractio o a comlex Hilbert sace H, let Φ be a mootoe regularly varyig fuctio ad 0 α<. The followig are equivalet for f H. i) There exists C > 0, such that, V k f C α Φlog + )) ii) There exists C > 0, such that, σ f D ), 8) C α Φlog + ))). 9) Proof. i) ii). Assume 8) holds. Let. Sice ) decreases to /e, for / r wehave r r ) >r/3, r = r) r k r)r r)/3. k=0

5 74 C. Cuy ad M. Li O the other had, siπθ) θ π siπθ) for θ For z = re iπθ D,, sice r, we thus obtai z k which yields, by 5), σ f D ) 36 = z z z = r r cosπθ) + r r cosπθ)+ r r ) + 4r si πθ) 4 r) + 4r si πθ) 36, D. z k σ f dz) 36 V k f, 0) which roves 9), by 8). ii) i). Assume 9) holds. Sice D = D,usig7) we obtai V k f = = z σ k f dz) D D z σ k f dz) + j= D j D j+ z σ k f dz) σ f D ) + j + ) σ f D j ) σ f D j+ ) ) j= σ f D ) + σ f D j ) j + ) j ) σ f D ) + 4σ f D ) j= j + C j α + f Φlogj + ))) j= 3 C Φlog )) + ) 3x x α Φlogx + ))) dx + f. ) Sice /Φ is also a mootoe regularly varyig fuctio, x /Φlogx + ))) is slowly varyig. Sice α >, it follows from [], Theorem b),. 8 with = α ad γ = 0), that the last itegral is O ). This cocludes the roof of the theorem. α Φlog+))) Lemma.3. Let T be a isometry or a ormal cotractio of a comlex Hilbert sace H ad let f H. If there exist 0 <α< ad a o-decreasig regularly varyig fuctio Φ with exoet β>/ such that Φlog + )) K := su α < +, ) the m 0 bα) m T m f coverges i H to a elemet h I T)H, which satisfies f = I T) α h, ad for every b m α) T m f C Φlog + )), 3) m

6 where Φ := x Poitwise ergodic theorems with rate 75 du Φu)) ) /, which is a o-decreasig regularly varyig fuctio with exoet β /. Proof. Assume first that T is a ormal cotractio. Let σ f be the sectral measure of f as above. Assumtio ) imlies 0, so f I T)H ad σ f {}=0. By ) ad the revious theorem there exists C>0 such that σ f D ) C α Φlog + ))). 4) By Proositio.0 of [5], if we rove that the series m 0 b mt m f coverges, the f I T) α H. For k>j>wehave k b T f =j = k b z σ f dz). D =j Defie the argumet fuctio o C {0} by arg z = θ for z = re iπθ, r>0, / <θ /. It follows from ) ad [9], vol. I,. 9, that there exists C α >, such that for every ad z = with θ = arg z 0wehave b k e iπkθ C α arg z α. O the other had, for every 0 z <, m=0 b m z m 0 b z = z ) α. Hece k b z C α mi { arg z α, z ) α} =j k>j, 0 z. 5) Sice b is decreasig e.g., [5], Lemma.5), Abel summatio ad 6) yield k { } b z Cb j mi arg z,, 0 < z. 6) z =j Usig 6) ad 5), ad the ) ad the defiitio of D r, we obtai k D =j b z σ f dz) = r C j r= D r D r+ =j b j + C Cj α r j+ k b z σ f dz) { } mi D r D r+ arg z, σ f dz) z { } mi D r D r+ arg z, α σ f dz) z j r + ) σ f D r ) σ f D r+ ) ) r= + C r + ) α σ f D r ) σ f D r+ ) ). r j+

7 76 C. Cuy ad M. Li The first term is O/Φlog )) ), by the estimate for the sum whe it aeared i ). For the series, usig 4) ad Abel summatio by art oe ca see that the residual term at ifiity is 0 by 4)), we deduce r + ) α σ f D r ) σ f D r+ ) ) C r j+ r j+ C j rφlog r)) dx xφlog x)) = C log j du Φu)), where the last itegral is coverget by the lemma i [],. 80, VIII.9, sice β >. Hece k =j b T f C du log j ) / 0, so by Cauchy s criterio Φu)) m 0 b mt m f coverges i H. The estimatio for the tail of the series follows from the last iequality, sice by the lemma i [],. 80, x du Φu)) is regularly varyig with exoet β + ; hece Φ is regularly varyig of exoet β /. I case T is a isometry, we ote that by the uitary dilatio theorem [4], there exist a larger Hilbert sace H cotaiig H ad a uitary oerator o H such that for f H we have = EU k f for every k 0, where E is the orthogoal rojectio of H oto H. Sice T is a isometry, EU k f = f = U k f shows that U k f H, so i fact = U k f, ad the result for the isometry follows from alyig the above result to the uitary U. Remarks.. For Φx) = x β we have Φx) = β x β /.. It is ot ossible to imrove the coditio o β i Lemma.3. Ideed, the examle i [5],. 7, has a symmetric cotractio T o L [0, ] ad f L such that f / I TL, ad checkig the comutatios i the examle we have that 3) holds with Φx) = x so β = ) ad α =. The costructio i the roof of Proositio 5.3 may be alied to show the same heomeo with a Markovia symmetric cotractio. Theorem.4. Let T be a ower-bouded oerator o a Baach sace X ad let f X. If there exist 0 <α< ad a o-decreasig regularly varyig fuctio Φ with exoet β> such that su Φlog + )) α < +, the f I T) α X, there is a uique elemet h I T)Xsuch that f = I T) α h, ad h satisfies Φlog + )) T k h < +, su 7) where Φ := x su du Φu) ) is a o-decreasig regularly varyig fuctio with exoet β. Theorem.5. Let T be a isometry or a ormal cotractio of a comlex Hilbert sace H ad let f H. If there exist 0 <α< ad a o-decreasig regularly varyig fuctio Φ with exoet β>/ such that Φlog + )) α < +, the f I T) α H, there is a uique elemet h I T)H such that f = I T) α h, ad h satisfies Φlog + )) T k h < +, su 8) where Φ := x du Φu)) ) / is a o-decreasig regularly varyig fuctio with exoet β /.

8 Poitwise ergodic theorems with rate 77 Proof of Theorems.4 ad.5. We will rove both theorems together. I the roof X will stad for a Baach sace or a Hilbert sace. I either case, by Lemma. or.3, the series h := 0 b T f coverges. It follows from Proositio.0 ad Theorem. of [5] that f I T) α X, ad that h is the uique elemet of I T)Xsatisfyig f = I T) α h. Moreover, by Lemma. or.3, b m T m f C Φ log + )), m 9) where accordigly Φ is a o-decreasig regularly varyig fuctio, either Φ with exoet β, or Φ with exoet β /. Let us rove the estimates 7) ad 8). For wehave k T k h = b m T m+k f = b m T m+k f + b m T m+k f. 0) m 0 m=0 m + k Let us deal with the last sum. Usig that T is a cotractio ad 9)wehave m + k b m T m+k f m + k b m T m f C Φ log + k)) = C Φ logk + )) C log + C dx Φ logx + )) C 3 Φ log + )), by Theorem b), i [],. 8 with = γ = 0), sice /Φ log) is slowly varyig. It gives the desired boud for the secod sum i 0). Let us deal with the first sum i 0). Writig S 0 := 0, ad usig b from ), we obtai k b m T m+k f = m=0 m=k m= m b m k T m f m = b m k )T m f = = C m= [ m= m= m C α b k )T m f k=0 b k )S m S m ) = b m S m ) + S b k k=0 m= ] Φlogm + )) + α Φlog + )) α k=0 C Φlog + )). Usig the costructios of the corresodig Φ ad Theorem a) i [],. 8, we obtai the Φx)/Φ x) as x, which yields the desired boud also for the first sum of 0).

9 78 C. Cuy ad M. Li 3. Poitwise ergodic theorems with rates Let T be a Duford Schwartz oerator o a robability sace Ω, μ). Theorem 3. of [5] shows that for > with dual idex q = / ) ad f I T) /q L μ) we have / log ) /q k=0 0 a.s. We wat to have k=0 0 a.s., so additioal hyotheses are eeded [5], Proositio 3.8, for examle, < +. / su / ɛ I Theorem 3. we obtai the desired a.s. covergece uder a weaker hyothesis; its roof uses the followig roositio. Proositio 3.. Fix > ad let T be a ower-bouded oerator o L μ) of a robability sace Ω, μ). Let h L μ) ad assume that there exist δ / ad τ>/ such that log + )) δ log log + )) τ su T k h < +. The for every τ <τ / we have log + )) δ / log log + )) τ T k h 0 + a.s. log+)) Moreover, su δ / log log+)) τ T k h L μ). Proof. For ay atural umber defie Ψ):= )/ log + κ)) δ log logx + κ)) τ, where κ is large eough, so Ψ is o-decreasig. Sice T is ower-bouded, the hyothesis o h yields k+ j=k+ T j h CΨ ). Usig the defiitio Λ) = [log ] k=0 Ψ ), with log k+ x beig the logarithm to base ad x the uer itegral art of x, we ca comute for our Ψ that [log ] Λ) = k=0 ) [/log ] Ψ k+ k=0 ) Ψ k+ + C )/ CΨ ). It follows from [], Theorem 4, that there exists C such that for k 0 ad, max k+l l j=k+ T j h CΛ ) C Ψ ) C log + )) δ log log + )) τ.

10 Hece for biary blocks we have l max T j m h C m l m+ log m + )) δ log log m + )) τ. j= m Now let 0 <τ <τ. The m δ / log m) τ m max m C m m l m+ Poitwise ergodic theorems with rate 79 l T j h j= m m δ log m) τ m m log + m )) δ log log m < +. + )) τ The assertios of the theorem ow follow easily, sice also m δ / log m) τ m T j h m C m m j= m δ log m) τ m m log + m )) δ log log m < +. + )) τ Remark. Poitwise ergodic theorems with rates as cosequece of rates i the mea ergodic theorem were obtaied by Gaoshki [] for geeral uitary oerators i a comlex Hilbert sace, by Derrieic Li [5] for Duford Schwartz oerators, ad by Weber [5] for ower-bouded oerators o L. A theorem of this tye for the isometries iduced i L by robability reservig trasformatios is i fact roved i [8]. Related results are i Assai Li []. For more results ad refereces see []. Theorem 3.. Let T be a Duford Schwartz oerator o a robability sace Ω, μ). Let f L μ), >. Assume that there exists τ>, such that log + )) log log + )) τ su / < +. ) The / 0 a.s. ad + k / coverges a.s. Moreover, there exists K f > 0 such that for every λ>0, { λ μ su T k } { } f / >λ K f ad λ μ su k / >λ K f. Proof. Let Φx) = x log x) τ, which is o decreasig o [, + ) ad regularly varyig with exoet. Estimatig du x C u log u) τ xlog x) τ, we obtai from Theorem.4, with α = / ad X = L μ), that there exists h L μ) with f = I T) α h such that log + )log log + )) τ su T k h < +. )

11 70 C. Cuy ad M. Li Hece f = h a T h coverges i L μ), where a = a α) are the coefficiets i the exasio t) α = a t. Sice a is absolutely coverget, the series a T h is μ-almost everywhere absolutely coverget. By [5] roof of Theorem 3.), we have k=0 = A + B + C, where A := h + a k )T j h, B := j= j + k j+ a k )T j h, T j= k=j C := ) a j T j T k h. k=0 It is roved i stes I ad II of the roof of Theorem 3. i [5],., that A / / 0 ad C / / 0a.s. uder the oly coditio f = I T) α h, sice α =. Moreover, it is roved i [5],. 3, that su C / L μ), for every h L μ). The corresodig statemet for A / / follows by Abel summatio ad the classical iequality su T k h L μ) for every h L μ). Let us rove that B / / 0 a.s. ad that λ μ{su >λ}k / f for every λ>0. By [5], Lemma.5, α k j a k = ja j,sowehavesee3.6)i[5]) B = T j= α ja j T j h + α + )a + T k h. 3) For the last term i 3), we use α + )a + = c α with c bouded to obtai / α + )a + T k h k= ) k= = c T k h 0 By the classical maximal iequality, we also have a maximal iequality for this term. It remais to rove that Let q := + q / T ja j T j h 0 j= a.s. k= a.s. be the dual idex, ad let q <τ <τ here we use τ>). Alyig Proositio 3. with δ = = to ) we obtai h := su log + )) /q log log + )) τ T k h L μ). For k ut S k := k j= T j h, ad defie S 0 := 0. Notice that for every k, T k S k = S k S k k logk + )) /q log logk + )) τ h k + logk + )) /q log logk + )) τ h 3k logk + )) /q log logk + )) τ h.

12 Deote by T the liear modulus of T.For wehave T ja j T j h = T ja j S j S j ) j= j= Poitwise ergodic theorems with rate 7 T j T j ) S j jaj j + )a j+ + a T S j= ) 3j jaj j + )a j+ logj + )) /q log logj + )) τ T j h + a T S. j= a We have already roved that T S / 0 a.s., with a maximal iequality, whe dealig with the last term of 3). Sice ja j j + )a j+ = αa j = 0/j + ) +α ), the roof will be fiished if we show that for every g L μ), / j α logj + )) /q log logj + )) τ T j g 0 a.s. 4) j= We roceed as i [5], with the required slight modificatios. Let g L μ). Usig Hölder s iequality with / + /q = ) ad Tg T g ) [9],. 65), we obtai / j α logj + )) /q log logj + )) τ T j g j= ) /q T j / j qα g / logj + ))log logj + )) qτ j= ) /q T j g ) /. 5) jlogj + ))log logj + )) qτ j= j=0 The series is coverget by our choice of τ, ad the secod term is bouded a.s., by the ergodic theorem for g L μ). Hece, for every g L μ) we have su / j α logj + )) /q log logj + )) τ T j g < + a.s. 6) j= By the Baach ricile it suffices to rove 4)forg i a dese subset of L μ). We rove it for every g L μ): / j α logj + )) /q log logj + )) τ T j g j= / j α logj + )) /q log logj + )) τ g j= C α / log + )) /q log log + )) τ g 0. Hece 4) holds for every g L μ).

13 7 C. Cuy ad M. Li Sice g L μ), the classical iequality λμ{su T k g >λ} g = g for every λ>0 yields the asserted weak-tye maximal iequality by the estimate 5). Now deote R 0 = 0 ad R k := k j= T j f for k. The k / = R k R k k / = k / k + ) / The last term teds to 0 as as we have see, ad ) k / R k k + ) / R k k +/ k +)/ k= k= coverges a.s. by Beo Levi, sice by ) wehave R k R k k +)/ k= ) R k + R. 7) / Ck / k +)/ logk + )) log logk + )) τ <. The above ad weak maximal iequality for the last term i 7) yield the a.s. covergece of existece of K f > 0 such that λ μ{su >λ}k k / f for every λ>0. T f / ad the Remarks.. The series b k, with the coefficiets b Ɣ q k = b /) )k/ k, coverges a.s. by ) ad Beo Levi s theorem, so the theorem yields that h = k=0 b k also with a.s. covergece.. Wu [6], Proositio iii)) showed that for T iduced by a robability reservig trasformatio ad f L i.e., {T f } L is strictly statioary), the coditio ) /+) < 8) is sufficiet to obtai 0 a.s. Actually it ca be show that Wu s coditio yields also a.s. / + covergece of the series. Hece, for strictly statioary sequeces our result follows from Wu s, sice ) k imlies 8). Thus, the ovelty i our / theorem is its alicatio to all Duford Schwartz oerators. We ca imrove the theorem, by weakeig the assumtio ), whe T is a isometry of L μ) iduced by a robability reservig trasformatio. Theorem 3.3. Let ϑ be a measure reservig trasformatio of Ω, F,μ). Let f L μ). Assume that there exists τ>, such that log + )) 3/ log log + )) τ su f ϑ k < +. 9) The f ϑ k 0 a.s. ad + f ϑ k k coverges a.s. Moreover, there exists K f > 0 such that for every λ>0, { λ μ su f ϑ k } { >λ K f ad λ μ su } f ϑ k >λ K f. k

14 Poitwise ergodic theorems with rate 73 Proof. Note that ϑ iduces a isometry T of L μ), hece we ca use Theorem.5 istead of Theorem.4 at the begiig of the roof of Theorem 3., this time with Φx) = x 3/ log x) τ, to obtai ). From that oit o the roof is exactly the same, takig = q = ad usig 9) istead of ). Remarks.. Oe ca see that Theorem 3.3 is valid for ay Duford Schwartz oerator T that is a isometry of L μ). However, there is a examle i [3],. 58, showig that Theorem 3.3 is ot true if oe assumes oly that T is uitary.. As metioed above, Wu [6] obtaied a differet sufficiet coditio, amely 8), imlyig the coclusio of Theorem 3.3. Whe τ>3/, 9) imlies 8). I the ext roositio we show a examle satisfyig coditio 9), with <τ<3/, but ot Wu s 8). Let {ε } Z L Ω, μ) be a sequece of strictly statioary martigale differeces e.g., i.i.d. cetered radom variables with fiite variace). For a sequece {a } l N), we defie the movig average sequece X := k 0 a kε k, which is strictly statioary, i.e., X = X 0 ϑ, with ϑ the shift associated with {ε }. Proositio 3.4. There exists a movig average {X } such that ad su log + )) 3/ log log + )) 5/4 X 0 ϑ k ) /3 =. X 0 ϑ k < 30) 3) I articular, Theorem 3.3 alies while Wu s coditio does ot hold. Proof. Let a = a = 0, for every 3 ut a =, ad let a log ) 5/ log log ) 5/4 0 = k a k,so k 0 a k = 0. Fix a statioary martigale differece sequece with uit variace {ɛ } Z, ad defie a movig average as above. By orthoormality of {ε } we have X k = a m ε k m = a k m ε m m 0 = a k m )ε m + = m= k=m m m= Usig k 0 a k = 0 we obtai ) a k + k=0 m 0 mk 0 m= m+ k=m+ ) a k m ε m a k ). X k = a k) + m+ ) a k. 3) m=0 k m m 0 k=m+ We will use the estimate k j+ a C k for j 3. log j) 3/ log log j) 5/4 For the first sum o the right-had side of 3) we have, for large, m=0 k m a k ) c + 4 m=0 C log m )) 3 log log m )) 5/ c + C log ) 3 log log ) 5/.

15 74 C. Cuy ad M. Li For the secod sum, we obtai slittig the sum, accordig to m / log ) m+ ) ) a k a k + logm + )) 5 log logm + )) 5/ log + m ) m m 0 k=m+ 0m/ log k m+ m>/ log C log ) 4 log log ) 5/ + log + )) 5 log log + )) 5/ = o log ) 3 log log ) 5/ ). m>/ log ) m Hece, X k log ) 3/ log log ) 5/4. I articular, 30) holds, while 8) does ot. Theorem 3.3 raises the questio whether the assumtio o the ower of the logarithm i Theorem 3. ca be imroved i geeral. Ideed, if we kow that the fuctio f is also i some L r, r>, we ca assume a smaller ower of the logarithm i ), ad the roof is also somewhat simler. The result below is isired by the argumets of Wu ad Woodroofe [7] whe = ) i the roof of their queched CLT. Theorem 3.5. Let T be a Duford Schwartz oerator o a robability sace Ω, μ). Let f L μ), >. Assume that there exists τ>/, such that log + )) +/ log log + )) τ su / < +. 33) If i additio f L r μ) for some r>, the / 0 a.s. ad + k / coverges a.s. Moreover, su ad su k / are i L μ). / Proof. For the claims about it suffices to rove the stroger results) that / max m/ T k f 0 ad su max m m + m m/ m L μ). Let 0 <γ </ /r. Defie u m := [ γm ]+ block size) ad v m := [ γ)m ]+ uer boud o umber of blocks). Sice γ<, we have [ lu m max max l )u m ] +j + max. m lv m ju m k=l )u m + Sice for every l v m, l )u m +j max u m max um max, l )u m +klu m ku m v m ju m k=l )u m + we obtai max max lu m T l f + u m max. 34) ku m v m m lv m

16 Poitwise ergodic theorems with rate 75 Let T be the liear modulus of T, which is also a Duford Schwartz oerator. Sice T k f ) r T k f r ) [9],. 65) ad u m v m < 3 m, we obtai u m This yields 3 m max ) r u r m max T k f r) u r m T k f r). ku m v m ku m v m m m/ max ) r dμ ku m v m um which coverges by our choice of γ. Hece m u r 3 m m rm/ T k f r) dμ 3 m ur m f r r mr/ 3 r f r r m+γr r/), m m su m u m m/ max ku m v m L r L ad u m m/ max ku m v m m 0 a.s. We ow deal with the first term o the right-had side of 34). Fix m, ad for k v m defie R k = kum i=k )u m + T i f.for0 j<l v m,33) yields l k=j+ R k = lu m i=ju m + T i f l j)u m i= T i f l j)u m C logl j)u m + )) + log logl j)u m + )) τ Cu m logu m + )) + l j). log logu m + )) τ Sice we have a liear boud, we ca use [], Theorem 3 see also [], Proositio.3), which yields the maximal iequality max j R k Hece jv m ju m max m/ jv m i= C u m logu m + )) + log logu m + )) τ log v m) v m. T i f = max j R k m jv m C mlog m) τ, which is the term of a coverget series by our assumtio o τ. The assertios cocerig the series k / are roved as i Theorem 3.,usig33) istead of ). Remarks.. I view of the revious theorem, it would be iterestig to kow whether Theorem 3.3 remais true if we take oly τ>, without assumig f Lr for some r>.. Whe f L r, r>, the revious theorem requires a smaller ower of the logarithm tha i ), amely + istead of. For T iduced by a measure-reservig trasformatio ad, is the L aalogue of Theorem 3.3, with the ower + i the logarithm, true without the additioal coditio f Lr for some r>)?

17 76 C. Cuy ad M. Li 4. A queched CLT for Markov chais We ow use our results to obtai a queched cetral limit theorem for additive fuctioals of statioary ergodic Markov chais. Let Px,A) be a trasitio robability o S, S) with Markov oerator Pgx) = gy)px,dy) defied o bouded measurable fuctios, ad let m be a ivariat robability for P, assumed ergodic. The ivariace of m ad the iequality Pgx) Pg )x) yield that P exteds to a cotractio of L m). LetΩ := S N be the sace of trajectories ad {X } 0 the corresodig Markov chai with trasitio robability P. The robability law of the chai is deoted by P m whe the iitial distributio is m, ad by P x whe the chai starts at the oit x S. We deote by ϑ the shift o Ω, which is measure reservig ad ergodic i Ω, P m ). For f L m), cosider S f ) = k=0 fx k). Theorem 4.. Let f L m), with fx)dmx) = 0. If there exists τ>such that log + )) 5/ log log + )) τ su P k f < +, 35) k=0 the σ f ) := lim E m S f ) ) exists ad is fiite, ad for m-almost every oit x S, the sequece / S f ) coverges i distributio, uder the robability measure P x, to the Gaussia distributio N 0,σ f )) if σ f ) = 0, it is the Dirac measure at 0); furthermore, also the ivariace ricile holds. Proof. We basically follow the roof of [7] with the corresodig modificatios. Defie Φx) := x 5/ log x) τ, x>. For 0 <t< defie the Gree kerel G t = k 0 tk P k. By Abel summatio, G t = t) k 0 tk k j=0 P j, ad by assumtio G t f C t) k Φlogk + )) tk. k 0 By a Tauberia theorem see Theorem 5 i Sectio XIII.5 of []), with the slowly varyig Lx) := Φlog x)),for every 3 t< π G t f C t) / Φ log t) ). 36) Write ϕ t X 0,X ) := G t fx ) PG t fx 0 ).Wehave E m [ ϕs ϕ t ) ] = {[I + P )Gs f G t f) ][ I P )G s f G t f) ]} dm. Sice I P)G t = I t)pg t for 0 t<, we have I P )G s f G t f)= t)p G t f PG s f)+ s t)pg s f. By the Cauchy Schwarz iequality ad the triagle iequality, we obtai E m [ ϕs ϕ t ) ] I + P )Gs f G t f) I P )Gs f G t f) G s f + G t f ) [ t) Gs f + G t f ) + s t Gs f ]. Hece, usig 36) to estimate G t f for t,wehave [ su E m ϕs ϕ t ) ] / C s [t,+t)/] Φ log t) ).

18 Poitwise ergodic theorems with rate 77 Fix t< ad aly the above iequality with t = + t )/ istead of t to obtai su s [t,) E m [ ϕs ϕ t ) ] / su 0 s [t,t + ) 0 E m [ ϕs ϕ t ) ] / C Φ log t ) ) = C Φ log t)/ ) ) C Φ log t) ) + C 0 0 C 3 log t) 3/ log log t) ) τ. dx Φ log t) +x log ) By Cauchy s criterio, there exists M L P m ) such that lim t ϕ t M = 0. The sums M := k=0 M ϑk defie a martigale with statioary icremets, sice M t) := k=0 ϕ tx 0,X ) ϑ k is a martigale ad P m is ϑ-ivariat. By orthogoality ad statioarity of the martigale differeces, E m [ M t) M ) ] = Em [ ϕt M) ] D Φ log t) ), 37) where, Φx) := x 3/ log x) τ, x>. To obtai the CLT we eed to estimate the residual term W := S f ) M. By costructio, I tp)g t f = f for 0 t<. Hece, for every 0 t<wehave W = Gt fx k ) tpg t fx k ) ) ϕ t X k,x k+ ) + M t) M k=0 = ) M t) M + Gt fx 0 ) G t fx ) + t) PG t fx k ). Hece, usig 37), estimatig G t f by 36), ad the takig t = /, we obtai k=0 )) Em W / D / / Φ log t) ) + C π t) / Φ log t) ) + t)c π t) / Φ log t) ) C / Φlog ) = C / log ) 3/ log log ) τ. Sice W = k=0 f X 0) M) ϑ k ad τ>, we aly Theorem 3.3 to the fuctio fx 0 ) M L Ω, P m ) ad obtai that W 0 P m -a.s., so for m-a.e. x we have k=0 W 0 P x -a.s. The ed of the roof is ow similar to [7],. 75. Remarks.. Sice P m = P x dmx), the queched CLT, with the variaces of the limitig Gaussia equal a.s. to σ ideedetly of x), imlies the aealed CLT for {fx )}: I the sace Ω, P m ), the sequece / S f ) coverges i distributio to the Gaussia distributio N 0,σ ) if σ = 0, it is the Dirac measure at 0); furthermore, also the ivariace ricile holds.

19 78 C. Cuy ad M. Li. Imrovig the result of [4], Maxwell ad Woodroofe [0] roved the aealed CLT, with variace of the limit σ f ) := lim E m S f ) ), uder the assumtio that 3/ P k f <. = The mai questio is whether this coditio is sufficiet for the queched CLT. 3. Sice 35) imlies 38), σ f ) is the variace i Ω, P m ) of the statioary martigale differeces M ϑ k. Corollary of [8] ad its roof show that 35) imlies also 38) lim su S f ) log log = σf) P m -a.s. 5. O coditios for the CLT for Markov chais I this sectio we comare some of the coditios for the CLT. We use the otatios of the revious sectio: Px,A)is a trasitio robability o S, S) with ivariat robability m, assumed ergodic. The Markov oerator P the exteds to a cotractio of L m). We deote by {X } 0 the corresodig Markov chai o the sace of trajectories. For f L m) we defie S f ) = k=0 fx k). Let us recall: Proositio 5.. Let P be a Markov oerator as described. Let f L m). Assume that oe of the followig coditios is satisfied i) P is ormal ad f I PL m) [5,6]. ii) 3/ P k f < + [0]. iii) fp f coverges i L m) [3]. The {fx )} satisfies the aealed Cetral Limit Theorem. Remarks.. It is kow that for P ormal) ii) imlies i); i fact, ii) always imlies f I PL [8]. We will aswer the questio asked i [8], whether there exists a ormal Markov oerator P ad f L such that i) is satisfied but ot ii).. Coditio ii) was itroduced by Maxwell Woodroofe [0]. For geeral strictly statioary rocesses, the coditio reads 3/ ES X 0 ) < +, were S := k=0 fx k). It was roved by Peligrad Utev [] that it is sufficiet for the fuctioal CLT i that case. 3. Coditio iii) is due to Dedecker Rio [3], ad also has a aalogous sufficiet coditio for the geeral statioary case which esures the fuctioal CLT. Of course, iii) imlies P f,f 0, so if P has eigefuctios with uimodular eigevalues, they are coboudaries which do ot satisfy iii). 4. Coditios ii) ad iii) look differet i ature. We will rovide a examle of P mixig where ii) is satisfied but iii) is ot. Proositio 5.. Let P be a Markov oerator as above. Let f L m). Assume that oe of the followig coditios is satisfied: i log ) ) su 5/ log log ) τ / P k f < +, for some τ>. ii ) su α P k f < +, for some α</. iii ) There exists a ositive o-decreasig slowly varyig fuctio l such that l) < + ad l) log) P k f < +. 3/ The {fx )} satisfies the queched Cetral Limit Theorem.

20 Poitwise ergodic theorems with rate 79 Remark. Coditio ii ) obviously imlies coditio i ), which clearly imlies ii). The queched CLT uder iii ) was obtaied by Zhao ad Woodroofe [8]. It is robably ot comarable to our coditio i ). We will rovide a examle where i ) is satisfied but iii ) is ot. Note that covergece of the secod series of iii ) ad mootoity of l imly ii). Coditio ii ) imlies iii ) with l) = log + )) +ε. I order to comare the revious coditios we will use the same symmetric Markov oerator P o L [0, ) := L [0, ), λ), where λ deotes the Lebesgue measure which geerates a reversible chai). Proositio 5.3. There exists a symmetric ositive defiite Markov oerator P o L [0, ) such that: a) There exists f L such that i) is satisfied but ii) ad iii) are ot. b) There exists f L satisfyig ii) but ot iii). c) There exists f L such that i ) is satisfied but ii ) ad iii ) are ot. Proof. We first costruct P.Letα R Q ad take P = 4 I + R α + R α ), where R α deotes the rotatio of the uit circle by the agle α. The irratioality of α makes P ergodic. Let f L [0, ], with Fourier exasio fx)= Z c e iπx. The for 0 x wehave P k fx)= Z + e iπα + e iπα ) k c e iπx = c cos k πα)e iπx. 39) 4 Z We will take α := e. I the roof of each art of the roositio, the aroriate fuctio f will be defied by its Fourier coefficiets {c } Z. I all these defiitios we take c = 0 if there is o k with =k!, ad c k! = c k! R, which makes f real valued. We will eed the followig lemma. Lemma 5.4. For every k, there exists l k N such that k ek! l k k ad for every k π, π k π cosπek!) 3k. Proof. For every k, defie l k := k! k j=0 j! N. Sice e = + j=0 j!,wehave Hece k + ek! l k! k k + )! + j=k+ k + )! j! k ek! l k k + ) /k + )) = k. = k + + s=0 ) s. k + Hece the first estimatio is true. Let k π. By the above, we have π k πk!e l k) π k that for every x [0, ],. Usig the fact that cos is decreasig o [0, ] ad x cos x x + x4 4 x + x 4 x 3, we obtai the secod estimatio.

21 730 C. Cuy ad M. Li Lemma 5.5. Let P be the above Markov oerator. Let f L [0, ) with Fourier exasio fx):= k Z c ke iπkx. Assume that c k = 0 if there is o N such that k =! ad c! = c!. The there exists K>0, such that for every o-decreasig sequece {u l } with u, ad every m, we have m 8 π m c! m P k f Proof. Let m. By 39)wehave K + 8 π 4 m P k f = m c cos πe)) k. Z 7u m 4 c! + m Hece, by the symmetry ad usig Lemma 5.4 for the o-zero coefficiets, m P k f = m ) c! cos k π!e) c! m cos 4m π!e) m π m π m c! ) 4m. m Usig Beroulli s iequality: + u) m + mu) for u, we obtai m P k f m 3 π m c!. >u m c!. Let us rove the secod iequality. By 39), the symmetry, ad Lemma 5.4, m P k f = m ) c! cos k π!e) + m c! cos k π!e) π >π K + c! cos π!e)) + c! m 7u m >u m K + 9 π 4 4 c! + m c!. 7u m >u m ) Proof of Proositio 5.3a). Fix/ <β. For ut c! = c! := Z c!e iπ!x. Usig Lemma 5.5 we obtai m P k f m 8 π m which roves that ii) is ot satisfied. c! m 8 π m dt t 3 log t) β C m log m) β, 3/ log ) β, ad defie fx) :=

22 Poitwise ergodic theorems with rate 73 Let us rove that f satisfies i). Sice P is ositive defiite, by [5] it suffices to rove the covergece of the series m 0 P m f,f. Usig39), the defiitio of {c } ad Lemma 5.4, we obtai P m f,f = c cos m eπ) = k m 0 m 0 Z k 3 log β k) cos ek!π)) k 3 log β k) cos ek!π)) + 6 π k log β k < +. kπ k π The roof that iii) is ot satisfied follows from the roof of art b) below, sice c! >. Proof of Proositio 5.3b). Takec! = c! :=,, ad ut fx):= Z c!e iπ!x. Aly Lemma 5.5 to f with u m := m /4.Wehave m P k f K + 9 π 4 4 c! + m c! K + 9 π 7u m >u 4 u m + m 3u 3 Cm /4, m m which roves ii). Let us rove that iii) is ot satisfied. Assume that {g = m= fp m f } coverges i L [0, ) to a fuctio g L [0, ). Forl Z ad h L [0, ) defie the Fourier coefficiet γ l h) := 0 ht)e πlt dt. The, γ l g ) + γ l g) for every l Z. O the other had, for every ad x [0, ) 39) yields ) cos k πje) e iπjx. P k fx)= j Z c j Hece, sice by the choice of {c } the revious series are absolutely summable, g x) = ) c j c m cos k πje) e iπj+m)x. j Z m Z Let r. By ositivity of the coefficiets, γ r!+! g ) 4r m= cos m πr!e). Hece γ r!+ g) = lim γ r!+g ) + By Lemma 5.4, we obtai γ r!+ g) lim + 4r cos m cos πr!e) πr!e) = 4r cos πr!e)). m= π /r ) 4r π /r ) π /3r ) r 6π. Hece {γ r!+ } does ot coverge to zero whe r teds to ifiity, which cotradicts the fact that g belogs to L [0, ), by the Riema Lebesgue lemma. Proof of Proositio 5.3c). Takec! = c! := for 3, ad ut fx):= 3/ log ) 5/ log log ) 3/ Z c j e πjx. By Lemma 5.5,wehave m P k f m 8 π m c! m C log m) 5 log log m) 3, 40)

23 73 C. Cuy ad M. Li which roves that ii ) is ot satisfied. Let us rove that iii ) is ot satisfied either. Let l be ay ositive fuctio of the itegers. By Hölder s iequality with cojugate exoets 3 ad 3/, ad 40), we have, for every 3 m=3 m log m log log m = ) /3 lm)) /3 ) mlm) m /3 log m log log m m=3 m=3 m=3 ) /3 mlm) m=3 ) /3 mlm) m=3 ) /3 lm) mlog m log log m) 3/ ) lm) log m m /3 P k f. Cm 3/ Hece oe of the series o the right must diverge ad iii ) caot be satisfied. However, takig u m = m, i Lemma 5.5, we obtai m P k f K + 8 π 4 4 c! + m c! 7u m >u m u m K + C log u m ) 5 log log u m ) 3 + m ) m u m log u m) 5 log log u m ) 3 C log u m ) 5 log log u m ) 3, which roves i ) with τ = 3/. Remark. The examle of Proositio 5.3a) resets P symmetric with a fuctio f I PL m) which does ot satisfy ay of the other coditios, i articular oe of the coditios for the queched CLT. However, sice this is a examle of a symmetric) radom walk o orbits of a rotatio, the queched CLT holds for {fx )} by [8]. We metio that the geeral questio of Kiis ad Varadha [8], whether for every P symmetric ad f I PL m) the queched CLT holds for {fx )}, is still oe. By lookig at the two-sided Markov shift, we see that the aealed CLT holds for f i the forward chai, govered by P, if ad oly if it does for f i the backward chai, which is govered by P. Whe P is ormal, all the coditios o f L, excet for coditio iii) of Proositio 5., hold with resect to P if ad oly if they hold with resect to P. We ow show that i geeral this is ot so. Let ϑ be the trasformatio of [0, ], defied by ϑx) = x mod for x [0, ), which reserves Lebesgue s measure. Let P be defied by Pg = g ϑ, for every measurable g; the P is a Markov oerator, ad P is give by P gx) = g x ) + gx+ )). Letf be defied by fx):= [0,/)x). Proositio 5.6. Let ϑ ad f as above. We have: i) m m 3/ P ) k f <, so f I P L. m ii) f/ I PL, so m m 3/ P k f =. m iii) 0 fp ) f coverges i L m), but 0 fp f does ot. Proof. Oe ca see that P f = 0, hece the series i i) coverges i L [0, ) ad f I P L. To rove ii) just ote that the rocess {f ϑ } 0 is the Rademacher sequece, hece the series P m f m m does ot coverge i L [0, ),sof/ Im I P. Obviously 0 fp ) f coverges i L m), sice P f = 0. Sice P is a isometry of L ad f, we have fp f = f =, so 0 fp f does ot coverge i L.

24 Poitwise ergodic theorems with rate 733 Ackowledgmet The secod author is grateful to the Uiversity of New Caledoia, where this research was carried out, for its hositality ad suort. Refereces [] I. Assai ad M. Li. O the oe-sided ergodic Hilbert trasform. Cotem. Math ) MR3333 [] G. Cohe ad M. Li. Extesios of the Mechoff Rademacher theorem with alicatios to ergodic theory. Israel J. Math ) MR94 [3] J. Dedecker ad E. Rio. O the fuctioal cetral limit theorem for statioary rocesses. A. Ist. H. Poicaré Probab. Statist ) 34. MR [4] Y. Derrieic. Some asects of recet works o limit theorems i ergodic theory with secial emhasis o the cetral limit theorem. Discrete Coti. Dy. Syst ) MR9389 [5] Y. Derrieic ad M. Li. Fractioal Poisso equatios ad ergodic theorems for fractioal coboudaries. Israel J. Math. 3 00) MR83590 [6] Y. Derrieic ad M. Li. The cetral limit theorem for Markov chais with ormal trasitio oerators, started at a oit. Probab. Theory Related Fields 9 00) MR86405 [7] Y. Derrieic ad M. Li. The cetral limit theorem for Markov chais started at a oit. Probab. Theory Related Fields 5 003) MR95457 [8] Y. Derrieic ad M. Li. The cetral limit theorem for radom walks o orbits of robability reservig trasformatios. Cotem. Math ) 3 5. MR436 [9] N. Duford ad J. Schwartz. Liear Oerators, Part I. Wiley, New York MR0096 [0] N. Duford ad J. Schwartz. Liear Oerators, Part II. Wiley, New York, 963. MR00963 [] W. Feller. A Itroductio to Probability Theory ad Its Alicatios, Vol. II, d editio. Wiley, New York, 97. MR [] V. F. Gaoshki. O the deedece of the covergece rate i the SLLN for statioary rocesses o the rate of decay of correlatio fuctio. Theory Probab. Al. 6 98) MR [3] V. F. Gaoshki. Sectral criteria for the existece of geeralized ergodic trasformatios i Russia). Teor. Veroyatost. i Primee. 4) 996) 5 7. Traslatio i Theory Probab. Al ) ).) MR [4] M. Gordi ad B. Lifshitz. A cetral limit theorem for Markov rocess. Soviet Math. Doklady 9 978) MR05077 [5] M. Gordi ad B. Lifshitz. A remark about a Markov rocess with ormal trasitio oerator. Proc. Third Vilius Cof. Probab. Statist Akad. Nauk Litovsk., Vilius, 98 i Russia). [6] M. Gordi ad B. Lifshitz. The cetral limit theorem for Markov rocesses with ormal trasitio oerator, ad a strog form of the cetral limit theorem. I Limit Theorems for Fuctioals of Radom Walks Sectios IV.7 ad IV.8. A. Borodi ad I. Ibragimov Eds). Proc. Steklov Ist. Math. 95, 994. Eglish traslatio Amer. Math. Soc., Providece, RI, 995.) [7] A. G. Kachurovskii. The rate of covergece i ergodic theorems. Russia Math. Surveys 5 996) MR48 [8] C. Kiis ad S. R. Varadha. Cetral limit theorem for additive fuctioals of reversible Markov rocesses ad alicatios to simle exclusios. Comm. Math. Phys ) 9. MR [9] U. Kregel. Ergodic Theorems. De Gruyter, Berli, 985. MR07974 [0] M. Maxwell ad M. Woodroofe. Cetral limit theorems for additive fuctioals of Markov chais. A. Probab ) MR787 [] F. Moricz. Momet iequalities ad the strog laws of large umbers. Z. Wahrsch. Verw. Gebiete ) MR [] M. Peligrad ad S. Utev. A ew maximal iequality ad ivariace ricile for statioary sequeces. A. Probab ) MR30 [3] F. Rassoul-Agha ad T. Seäläie. A almost sure ivariace ricile for additive fuctioals of Markov chais. Statist. Probab. Lett ) MR [4] F. Riesz ad B. Sz.-Nagy. Leços D aalyse Foctioelle, 3rd editio. Akadémiai Kiadó, Budaest, 955. MR [5] M. Weber. Uiform bouds uder icremet coditios. Tras. Amer. Math. Soc ) MR77045 [6] W. B. Wu. Strog ivariace riciles for deedet radom variables. A. Probab ) MR [7] W. B. Wu ad M. Woodroofe. Martigale aroximatios for sums of statioary rocesses. A. Probab ) MR06034 [8] O. Zhao ad M. Woodroofe. Laws of the iterated logarithm for statioary rocesses. A. Probab ) 7 4. MR [9] A. Zygmud. Trigoometric Series, corrected d editio. Cambridge Uiv. Press, Cambridge, UK, 969. MR036587

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