Exact convergence rates in the central limit theorem for a class of martingales

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1 xact covergece rates i the cetral limit theorem for a class of martigales M. l Machkouri L. Ouchti Laboratoire de Mathématiques Paul Pailevé, UMR 854 CNRS-Uiversité des Scieces et Techologies de Lille, Cité Scietifique Villeeuve d Ascq Mohamed.l-Machkouri@math.uiv-lille.fr Laboratoire de Mathématiques Raphaël Salem, UMR 6085 CNRS-Uiversité de Roue, Aveue de l Uiversité, BP. Techopôle du Madrillet F7680 Sait-Étiee-du-Rouvray lahce.ouchti@uiv-roue.fr We give optimal covergece rates i the cetral limit theorem for a large class of martigale differece sequeces with bouded third momets. The rates deped o the behaviour of the coditioal variaces for statioary sequeces the rate / log is reached. AMS Subject Classificatios 000 : 60G4, 60F05 Keywords : cetral limit theorem, martigale differece sequece, rate of covergece, Lideberg s decompositio. Itroductio otatios The optimal rate of covergece i the cetral limit theorem for idepedet rom variables X i i Z is well kow to be of order / if the X i s are cetered have uiformly bouded third momets cf. Berry [] ssee [8]. For depedet rom variables the rate of covergece was also most fully ivestigated but i may results the rate is ot better tha /4. For example, Philipp [9] obtais a rate of /4 log 3 for uiformly mixig sequeces, Lers Rogge [5] obtai a rate of /4 log /4 for a class of Markov chais see also Bolthause [3] Suklodas [3] obtais a rate of /4 log for strog mixig sequeces. However, Rio [] has show that the rate / is reached for uiformly mixig sequeces of bouded rom variables as soo as the sequece φ p p>0 of uiform mixig coefficiets satisfies p>0 pφ p <. Ja [3] also established a / rate of covergece i the cetral limit theorem for bouded processes takig values i R d uder some mixig coditios recetly, usig a modificatio of the proof i Rio [], Le Borge Pèe [6] obtaied the rate / for statioary processes satisfyig a strog decorrelatio hypothesis. For bouded martigale differece sequeces, Ibragimov [] has obtaied the rate of /4 for some stoppig partial sums Ouchti [8] has exteded Ibragimov s result to a class of martigales which is related to the oe we are goig to cosider i this paper. Several results o the rate of covergece for the martigale cetral limit theorem have bee obtaied for the whole partial sums, oe ca refer to Hall Heyde [0] sectio 3.6., Chow Teicher [5] Theorem 9.3., Kato [4], Bolthause [], Haeusler [], Riott Rotar [0] []. I fact, Kato obtais the rate / log 3 for uiformly bouded variables uder the assumptio assumptio that the coditioal variaces are almost surely costat. I this paper, we are most iterested i results by Bolthause [] who obtaied the better i fact optimal rate / log uder somewhat weakeed coditios. I this paper, we shall ot aim to improve

2 the rate / log but rather itroduce a large class of martigales which leads to it. Fially, ote that l Machkouri Volý [7] have show that the rate of covergece i the cetral limit theorem ca be arbitrary slow for statioary sequeces of bouded strog mixig martigale differece rom variables. Let be a fixed iteger. We cosider a fiite sequece X = X,..., X of martigale differece rom variables i.e. X k is F k -measurable X k F k = 0 a.s. where F k 0 k is a icreasig filtratio F 0 is the trivial σ-algebra. I the sequel, we use the followig otatios σ kx = X k F k, τ k X = X k, k, v X = τ k X V X = v X σkx. We deote also S X = X + X X. The cetral limit theorem established by Brow [4] Dvoretzky [6] states that uder some Lideberg type coditio X = sup µ S X/v X t Φt 0. t R + For more about cetral limit theorems for martigale differece sequeces oe ca refer to Hall Heyde [0]. The rate of covergece of X to zero was most fully ivestigated. Here, we focus o the followig result by Bolthause []. Theorem Bolthause, 8 Let γ > 0 be fixed. There exists a costat Lγ > 0 depedig oly o γ such that for all fiite martigale differece sequece X = X,..., X satisfyig V X = a.s. X i γ the log X Lγ. We are goig to show that the method used by Bolthause [] i the proof of the theorem above ca be exteded to a large class of ubouded martigale differece sequeces. Note that Bolthause has already give extesios to ubouded martigale differece sequeces which coditioal variaces become asymptotically orom cf. [], Theorems 3 4 but his assumptios caot be compared directly with ours cf. coditio below. So the results are complemetary. Mai Results We itroduce the followig class of martigale differece sequeces: a sequece X = X,..., X belogs to the class M γ if X is a martigale differece sequece with respect to some icreasig filtratio F k 0 k such that for ay k, F k γ k X k F k a.s. where γ = γ k k is a sequece of positive reals umbers. Our first result is the followig. v 3

3 Theorem There exists a costat L > 0 ot depedig o such that for all fiite martigale differece sequece X = X,..., X which belogs to the class M γ the u l X L miv, } + V X / V X /3 where u = γ k. Theorem There exists a costat L > 0 ot depedig o such that for all fiite martigale differece sequece X = X,..., X which belogs to the class M γ which satisfies V X = a.s. the u l X L miv, } For ay rom variable Z we deote δz = sup t R µz t Φt. We eed also the followig extesio of Lemma i Bolthause [] which is of particular iterest. Lemma Let X Y be two real rom variables. If there exist real umbers l > 0 r such that Y belogs to L lr µ the δx + Y δx + 3 Y l X l+ r Y X / δx δx + Y + 3 Y l X l+ r Y X /. 3 The proofs of various cetral limit theorems for statioary sequeces of rom variables are based o approximatig the partial sums of the process by martigales see Gordi [9], Volý [4]. More precisely, if f T k k is a p- itegrable statioary process where T : Ω Ω is a bijective, bimeasurable measure-preservig trasformatio i fact, each statioary process has such represetatio the there exists ecessary sufficiet coditios cf. Volý [4] for f to be equal to h + g g T where h T k k is a p-itegrable statioary martigale differece sequece g is a p-itegrable fuctio. The term g g T is called a coboudary. The followig theorem gives the rate of covergece i the cetral limit theorem for statioary processes obtaied from a martigale differece sequece which is perturbed by a coboudary. Theorem 3 Let p > 0 be fixed let F = f T k k be a statioary process. If there exist m g i L p µ such that H = h T k k is a martigale differece sequece f = h + g g T the If p = the F H + 4 g p/p+ p. p/p+ F H + 4 g /. 3

4 3 Proofs 3. Proof of theorem I the sequel, we are goig to use the followig lemma by Bolthause []. Lemma Bolthause, 8 Let k 0 f : R R be a fuctio which has k derivatives f,..., f k which together with f belog to L µ. Assume that f k is of bouded variatio f k V, if X is a rom variable if α 0 α are two real umbers the f k α X +α f k V sup t R where φx = π / exp x /. µx t Φt + α k+ f sup φ k x x Cosider u = u defied by u = γ k. Clearly the class M γ is cotaied i the class M u. For ay u, v R N + R +, we cosider the subclass L u, v = X M u V X =, v X = v a.s. } we deote β u, v = sup X X L u, v}. I the sequel, we assume that X = X,..., X belogs to L u, v, hece X = X,..., X, X + X belogs to L 4u, v cosequetly, β u, v β 4u, v. Let Z, Z,..., Z be idepedet idetically distributed stard ormal variables idepedet of the σ-algebra F which cotais the σ-algebra geerated by X,..., X ξ be a extra cetered ormal variable with variace θ > u which is idepedet of aythig else. Notig that i= σ ixz i /v is a stard ormal rom variable, ideed P } e it i= σ i XZ i v = P j= σ e t j X } v Accordig to Iequality 3 i Lemma, = exp t Sice V X = a.s.. X sup Γ t 6θ + t R v. 4 where Γ t µ S X + ξ /v t µ σ i XZ i + ξ /v t. For ay iteger k, we cosider the followig rom variables Y k k X i, W k σ i XZ i + ξ, v v i= 4 i= i=k+

5 H k v i=k+ σ i X + θ T k t t Y k, t R H k with the usual covetio i=+ σ i X = i=+ σ ixz i = 0 a.s. Moreover, oe ca otice that coditioed o G k = σx,..., X, Z k, the rom variable W k is cetered ormal with variace Hk. Accordig to the well kow Lideberg s decompositio cf. [7], we have Γ t = = = = µ Y k + W k + X k v Wk µ T k t X k H k W k H T k t X k G k k t µ Y k + W k + σ kxz k v Wk µ t T k t σ kxz k H k W k H T k t σ k XZ k G k k Φ T k t X k Φ T k t σ kxz k Now, for ay iteger k ay rom variable ζ k, there exists a rom variable ε k < a.s. such that Φ T k t ζ k = Φ T k t ζ k Φ T k t+ ζ k T Φ k t ζ3 k Φ T k t ε k ζ k a.s. 6 So, we derive Γ t = X k + σ kxz k Φ X T k t + k v H σ k XZ k k v Hk X 3 k 6v 3 Hk 3 Φ T k t ε kx k σ 3 + k XZ 3 k 6v 3 Hk 3 Φ Φ T k t T k t ε k σ kxz k Sice V X = a.s. we derive that H k T k t are F k -measurable, hece Γ t = 6v 3 X3 k Hk 3 Φ T k t ε } kx k + σ3 k XZ3 k Hk 3 Φ T k t ε k σ kxz k cosequetly where S = S = Γ t 6v 3 S + S 5 Xk 3 σ 3 k X Z k 3 T k t ε } kx k T k t ε k σ kxz k }. }. 5

6 Cosider the stoppig times νj j=0,.., defied by ν0 = 0, ν = for ay j < νj = if k k i= σ i X jv } a.s.. Notig that,..., } = j=νj +,..., νj} a.s. we derive S = j= νj k=νj + moreover, for ay νj < k νj we have Hk v σi X + θ i=νj+ i= i= T k t ε } kx k, = νj v σi X σi X σνj X + θ v v jv u + θ m j a.s. Similarly, Hk v i=νj + i= σi X + θ = νj v σi X σi X + θ i= v v j v + θ M j a.s. Now, for ay νj < k νj put R k v k i=νj + X i, Rk A k m j t Y } νj + for ay positive iteger q cosider the real fuctio ψ q defied for ay real x by ψ q x sup Φ y ; y x q}. O the other h, o the set 6

7 A k X k q} we have Thus Tk t ε kx k = t Y νj + R k ε kx k H k H k t Y νj + R k X k H k H k t Y νj + R k q m j θ t Y νj + q a.s. sice θ. T k t ε kx k t Yνj + Ak Xk q ψ q So, for ay j we have νj = k=νj + νj k=νj + νj k=νj + T k t ε kx k Ak Xk q}} } t ψ Yνj + q O the other h, for ay j we have νj k=νj + = = = = k=νj + l= k=l+ l= k=l+ νj Ak X k q. } } t Yνj + Hk 3 F νj ψ q. 6 } Hk 3 F νj } Hk 3 F νj Xk 3 k=νj + X k 3 k=νj+ } Hk 3 F νj } Xk 3 } νj =l F νj νj=l F νj } F k νj =l F νj X k 3 } Hk 3 F k νj=l F νj X k 3 Hk 3 F k F νj }. 7 7

8 By usig the iequality 6, 7 the fact that X L u, v, we have νj = k=νj + νj u m 3 j k=νj + Moreover, ote that νj νj k=νj + k=νj + Thus, for each j, νj k=νj + u v m 3 j T k t ε kx k Ak Xk q}} Xk 3 } } t Yνj + Hk 3 F k F νj ψ q } } t σkx F Yνj + νj ψ q. νj νj σkx = σkx σkx j + v ψ q t Yνj + j v = v T k t ε kx k Ak Xk q}} } a.s. 8 Usig Lemma, otig that ψ q keepig i mid the otatio δz sup t R µz t Φt there exists a positive costat c 3 such that } t Yνj + ψ q δy νj + + c 3. Now, usig Lemma the iequality X k F νj v k=νj + j a.s. we obtai δy νj + δs X/v + 3 v = X + 3 v k=νj + k=νj + X k X k } / Y νj + } / Y νj + β 4u, v + 3 j / 9 8

9 so } t Yνj + ψ q β 4u, v + 3 j / + c. Usig this estimate the domiated covergece theorem, we derive for ay iteger j, νj = c 4u m 3 j k=νj + v β 4u, v + T k t ε kx k Ak} j / +. 0 O the other h, for ay iteger νj < k νj A c R i k B j max > t Y } νj +. νj <i νj m j Sice the set A k is F k, we have νj = k=νj + Φ u u νj k=νj + νj k=νj + νj k=νj + σ k X σ k X T k t ε kx k A c k A c k Bj. A c k By usig iequality 8 the fact that H k m j for ay k νj +,, νj}, we have u m 3 j u m 3 j u m 3 j v µb j v µ max R i > m j t Y νj + νj <i νj v 4M j mi, m j t Y νj + Noticig that the sequece of rom variables Ri, si νj + i νj; R i = R νj, si νj + i. } max νj <i νj R i F νj }. 9

10 is a martigale adapted to the filtratio F i i, thus = max νj <i νj R i F νj max νj <i R i F νj 4 R F νj = 4 R νj F νj. By the iequality 8,, we have u m 3 v j 6Mj mi, m j t Y νj + } R νj F νj u m 3 j v mi, 3M j m j t Y νj + By applyig lemma with fx = mi; x, we have } 3M j 3 mi, m j t Y νj + δy νj + + π mj } δy νj + + c 3, where c 3 is a strictly positive costat. By the iequality 9, we have } 3M j mi, m j t Y νj + β 4u, v+3 j / +c 3.. Thus there exists a positive costat c 4 such that c 4u m 3 j v β 4u, v + j / + 3 From 0 3, there exists a positive costat c 5 such that + c 5u m 3 v j β 4u, v + j / +. Fially, we obtai the followig estimate S Xk 3 T k t ε } kθ k c 5 u v β 4u, v m 3 j= j + m 3 j= j j / + j= m 3 j. 0

11 O the other h, m 3 j= j = j= j + u v + θ u v 3/ v j= j + u v θ u v l c 5 sice v u, θ u m 3 j= j j / j= j= j j + u v 3/ / sice θ > u j c 5 l, sice v u j= m 3 j = m 3 j= j m 3 j= j j c 5 l + + θ v / j / + θ v θ θ u. m 3 j= j Hece S c 5 u v v l β 4u, v + l + θ u θ. 4 θ u Note that to obtai the above estimates of S, we have oly use the fact that the martigale differece sequece X belogs to the class L u, v. Sice the sequece σz σ XZ,..., σ XZ belogs to L 4u/ π, v, we are able to reach a similar estimate for S : S c 6 u v β 6u/ v l π, v + l + θ u θ. θ u where c 6 is a positive costat. Usig 4, 5, 4 5 there exist a positive costat c such that 5 β u, v 6 c u β 6u/ l π, v + l θ u v + θ + 6θ θ u v. 7

12 Puttig β u, v D v sup u log ; u RN + θ + 4c l u, by the iequality 6, we have D v D v + C v }. 8 where C is a positive costat which does ot deped o. Fially, we coclude that Thus The proof of Theorem is complete. 3. Proof of Theorem D v C v + 4C miv;. β u, v 4C u l miv;. Let X = X,..., X i M u. Followig a idea by Bolthause [], we are goig to defie a ew martigale differece sequece ˆX which satisfies V ˆX = a.s. Deote for each d R +, ˆd = + [d/u ], ˆkd = v + d v V /u, kd = [ˆkd], d = v V X v, d = v V X v ui, for i ; û i = u, for + i ˆd. where [.] deotes the iteger part fuctio. ˆX,..., ˆXˆ+ defied as follows: We put Cosider the rom variables ˆX i = X i a.s. i µ ˆXi = ± u F = a.s. + i + kd µ ˆX+kd+ = ± [ˆkd kd] u F = a.s. ˆX i = 0 else. V ˆd ˆX = ˆd v ˆd ˆX i ˆF i, i= ˆd ˆv ˆd = ˆX i ˆF l = σ ˆX,, ˆX l. i= Lemma 3 For each i ˆd, we have ˆv ˆd v = d, V ˆd ˆX = ˆX i 3 ˆF i û i X i ˆF i a.s.

13 Proof of Lemma 3: By defiitio of ˆX, we have ˆv ˆd = v + = v + ˆd i=+ ˆd i=+ = v + u [ˆkd] = v + d [ ˆXi F ] [ u i +kd + u [ˆkd kd] i=+kd+ ] V ˆd ˆX = ˆd ˆv ˆd ˆX i ˆF i = v ˆd V X + = ˆv ˆd = ˆv ˆd i= v V X + ˆd i=+ ˆd i=+ ˆX i ˆF i u i +kd + u [ˆkd kd] i=+kd+ vv X + u kd + u [ˆkd kd] = ˆv ˆd vv X + v + d vv X = v + d ˆv ˆd =. O the other h, for each + i ˆd, we obtai ˆX i 3 ˆF u 3, if i + kd; i = u 3 [ˆkd kd] 3/, if i = + kd + ; 0, else. ˆX i ˆF u, if i + kd; i = u [ˆkd kd] 3/, if i = + kd + ; 0, else. Thus, for each 0 i ˆd, we obtai The proof of lemma 3 is complete. Oe ca easily check that X sup t R ˆX i 3 ˆF i û i ˆX i ˆF i a.s. µs X/ˆvˆd t Φt + sup Φ t R v t ˆvˆd Φt. 3

14 Notig that ˆv v = d usig Lemma with l = r =, if d d there exist a positive costat c such that X ˆd ˆX [ + ˆd ˆvˆd i=+ ˆd ˆX + d/3 + + v d / /3 π π v ˆd ˆX + c d/3 v /3 ] S /3 ˆX i X + ˆv d v π v oe ca suppose that d v, where c is a positive costat. Usig Lemma 3 applyig Theorem, we derive ûˆd l ˆd d/3 X L + mi ˆvˆd; ˆd v /3 u l[ + v ] L mi d/3 v ; + v /3 u l 4L mi d/3 v ; + because d v. where L is a strictly positive costat. Similarly if d d the v /3 ûˆd l ˆd X L mi u l 4L mi d/3 v ; + d/3 + ˆvˆd ; ˆd v /3 v /3 Fially, we have u l X 4L miv ; + mi. d /3 v /3 }, d/. v The proof of Theorem is complete. 3.3 Proof of Theorem 3 Lemma Applyig the iequality 3.3 i Lemma for Y = / g g T, l = p r = F H + g g T p /p+ h T i / H + g g T p/p+ p p/p+ H + 4 g p/p+ p. p/p+ i= 4

15 If p = +, we obtai F H + g g T / h T i H + 4 g /. The proof of the theorem 3 is complete. / Let X Y be two real rom variables. We put for each k > 0 r, deote β = Y k X r cosider q R } such that /r + /q =. Let λ > 0 t be two real umbers we have Sice we obtai µ X + Y t µx t λ, Y t X i= = µx t λ µx t λ, Y > t X µx t λ X t λ µ Y > t X X}. X t λ µ Y > t X X} t X k Y k X X t λ } Cosequetly β X t λ t X k } q βλ k, µx + Y t µx t λ βλ k. µx + Y t Φt µx t λ Φt λ λ π βλ k takig λ = β π /k+, there exists a positive costat c such that O the other h δx + Y δx cβ /k+. 9 µx + Y t µx t + λ + µx t + λ, Y t X = µx t + λ + X>t+λ µ Y t X X} X>t+λ µ Y t X X} X>t+λ Y k X t X k} Cosequetly β X>t+λ t X k q βλ k. µx + Y t µx t + λ + βλ k µx + Y t Φt µx t + λ Φt + λ + λ π + βλ k. 5

16 Takig λ = β π /k+, there exists a positive costat c such that δx + Y δx + c β /k+. 0 Combiig 9 0 with Lemma i Bolthause [] completes the proof of Lemma. Refereces [] Berry, A. C., The accuracy of the Gaussia approximatio to the sum of idepedet variates, Tras. Amer. Math. Soc., 49, -36, 94. [] Bolthause,., xact covergece rates i some martigale cetral limit theorems, A. Probab., 0, 3, , 98. [3] Bolthause,., The Berry-ssee theorem for fuctioals of discrete Markov chais, Probab. Theory Relat. Fields., 54, 59-73, 980. [4] Brow, B.M., Martigale Cetral limit theorems, A. Math. Statist., 4, 59-66, 97. [5] Chow, Y. Teicher, H., Probability Theory: Idepedece, Iterchageability, Martigales, Spriger-Verlag, Berli, New-York, 978. [6] Dvoretzky, A., Asymptotic ormality for sums of depedet rom variables, Proc. Sixth Berkeley Symp. o Math. Statist. Probability,, , 970. [7] l Machkouri, M. Volý, D., O the fuctioal cetral limit theorem for idepedet rom fields, To appear i Stochastics Dyamics. [8] ssee, C. G., O the Liapuov limit of error i the theory of probability, Ark. Math. Astr. och Fysik, 8A, -9, 94. [9] Gordi, M. I., The cetral limit theorem for statioary processes, Soviet Math.Dokl., 74-76, 969. [0] Hall, P. Heyde, C. C., Martigale limit theory its applicatio, Academic Press, New York, 980. [] Haeusler,., A Note o the Rate of Covergece i the Martigale Cetral Limit Theorem rich Haeusler, A. Probab.,, No., , 984. [] Ibragimov, I. A. A cetral limit theorem for a class of depedet rom variables, Theory Probab. Appl., 8, 83-89, 963. [3] Ja, C., Vitesse de covergece das le TCL pour des chaîes de Markov et certais processus associés à des systèmes dyamiques, C. R. Acad. Sci. Paris, t. 33, Série I, , 000. [4] Kato, Y., Rates of covergeces i cetral limit theorem for martigale differeces, Bull. Math. Statist., 8, -8, 978. [5] Lers, D. Rogge, L., O the rate of covergece i the cetral limit theorem for Markov chais, Z. Wahrsch. Verw. Gebiete, 35, 57-63,

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