Necessary and sufficient conditions for the conditional central limit theorem

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1 Necessary ad sufficiet coditios for the coditioal cetral it theorem Jérôme DEDECKER ad Florece MERLEVÈDE Uiversité Paris 6 Abstract Followig Lideberg s approach, we obtai a ew coditio for a statioary sequece of square-itegrable ad real-valued radom variables to satisfy the cetral it theorem. I the adapted case, this coditio is weaker tha ay projective criterio derived from Gordi s theorem 969 about approximatig martigales. Moreover, our criterio is equivalet to the coditioal cetral it theorem, which implies stable covergece i the sese of Réyi to a mixture of ormal distributios. We also establish fuctioal ad triagular versios of this theorem. From these geeral results, we derive sufficiet coditios which are easier to verify ad may be compared to other results i the literature. To be complete, we preset a applicatio to kerel desity estimators for some classes of discrete time processes. Mathematics Subject Classificatios 99: 60 F 05, 60 F 7. Key words: Statioary processes, cetral it theorem, ivariace priciples, stable covergece, triagular arrays. Short title: Coditioal cetral it theorem

2 Itroductio I 963 Réyi itroduced the cocept of stable covergece of radom variables. This otio is more precise tha covergece i distributio ad may be useful i several cotexts, especially i coectio with radom ormalizatio this fact was first poited out by Smith 945 i the particular case of sums of Rademacher radom variables. Aldous ad Eagleso 978 made clear the equivalece betwee stability ad weak-l covergece of some fuctios of the variables, ad proposed some powerful tools to establish stability of it theorems. Further, adaptig a result of McLeish 974, they gave sufficiet coditios for a sequece of martigale differeces to coverge stably to a mixture of ormal distributios. Their results amog may others have bee used ad developed by Hall ad Heyde 980, Chapter 3 to provide a elegat ad rather complete cotributio to martigale cetral it theory. Some of these results have bee exteded to geeral sequeces by providig strog eough coditios to esure that the partial sums behave asymptotically like a martigale. I this cotext, McLeish 975b, 977 used the cocept of mixigale, while Peligrad 98 followed Gordi s approach. The commo feature of these works is the applicatio of Theorem 9.4 i Billiglsey 968: firstly they prove tightess of the partial sum process ad secodly they idetify the it by usig a suitable characterizatio of the Wieer process. They obtai mixig covergece of the partial sum process to a Browia motio, which coicides with stable covergece provided that the coditioal variace of the partial sums with respect to the past σ-algebra is asymptotically costat cf. Remark 5, Sectio 2 for the relatio betwee stability ad mixig. I this paper we focus o the cetral it questio for strictly statioary sequeces idexed by Z. We propose i Theorem a simple criterio which implies stable covergece of the ormalized partial sums to a mixture of ormal distributios. More precisely, we show that this criterio is ecessary ad sufficiet to obtai a stroger result tha stable covergece. We shall see that this ew type of covergece to a mixture of ormal distributios, close to the oe itroduced by Touati 993, Theorem 3, H-2, is satisfied for a wide class of statioary sequeces. Notatios. Let Ω, A, P be a probability space, ad T : Ω Ω be a bijective bimeasurable trasformatio preservig the probability P. A 2

3 elemet A is said to be ivariat if T A = A. We deote by I the σ-algebra of all ivariat sets. The probability P is ergodic if each elemet of I has measure 0 or. Fially, let H be the space of cotiuous real fuctios ϕ such that x + x 2 ϕx is bouded. Theorem Let M 0 be a σ-algebra of A satisfyig M 0 T M 0 ad defie the odecreasig filtratio M i i Z by M i = T i M 0. Let X 0 be a M 0 -measurable, square itegrable ad cetered radom variable. Defie the sequece X i i Z by X i = X 0 T i, ad S = X + + X. The followig statemets are equivalet: s There exists a oegative M 0 -measurable radom variable η such that, for ay ϕ i H ad ay positive iteger k, E ϕ /2 S ϕx ηgxdx where g is the distributio of a stadard ormal. M k = 0 s2 a the sequece S 2 >0 is uiformly itegrable. b the sequece E /2 S M 0 teds to 0 as teds to ifiity. c there exists a oegative M 0 -measurable radom variable η such that E S 2 η M 0 teds to 0 as teds to ifiity. Moreover the radom variable η satisfies η = η T almost surely. A statioary sequece X T i i Z of radom variables is said to satisfy the coditioal cetral it theorem CCLT for short if it verifies s. Before presetig the applicatios of Theorem, let us compare Coditio s2 with Theorem 9.5 i Jakubowski 993 takig B = therei, where ecessary ad sufficiet coditios for the usual CLT are give. There are two distict sets of coditios i this Theorem: o oe had Coditio B ad o the other had Coditios 9.5, 9.6 ad 9.7. Firstly, it is clear that s2a is stroger tha Coditios 9.5 ad 9.6. Secodly, s2a together with s2c imply 9.7 with σ 2 = Eη. Sice Jakubowski s result oly deals with pure Gaussia it, we ifer from the two precedig remarks that s2 implies Coditio B if ad oly if the radom variable η is costat. This meas that the two results are of a differet ature ad that Theorem may ot be derived by usig Jakubowski s result a other reaso is that the latter does ot ecessarily imply stable covergece. I fact, Coditio B 3

4 is a kid of mixig property ivolvig the whole past ad the whole future of the sequece, so that radom variaces are forbidde this is also the case whe cosiderig classical mixig coefficiets. Note also that to obtai Coditio B from s2 i the case where η is costat seems as difficult as to prove s directly. We ow give sufficiet coditios for the CCLT to hold. Note first that criterio s2 is satisfied for statioary sequeces of martigale differeces: ideed, i that case, s2a follows from Doob s maximal iequality, s2b is straightforward ad s2c is a cosequece of the L -ergodic theorem. Now, as first oticed by Gordi 969, it is ofte possible to approximate the partial sums of a statioary process by a aturally related martigale with statioary differeces. Such a approximatio provides a possible approach to obtai sufficiet coditios for the CCLT, as show by Propositio : Propositio Let M i i Z ad X i i Z be as i Theorem. Let H i be the Hilbert space of M i -measurable, cetered ad square itegrable fuctios. For all iteger j less tha i, deote by H i H j, the orthogoal of H j ito H i. Let Q be the set of all fuctios from H i H j where < j i <. If if sup f Q the s2 hece s holds. X 0 T i f T i 2 = 0,. i= Remark. Coditio. is due to Gordi 969. However, compared to Gordi s theorem, we have to restrict ourselves to adapted sequeces which meas that X i is M i -measurable. Note that, as i Eagleso 975, we do ot require P to be ergodic. It follows from Propositio that ay projective criterio derived from. leads to the CCLT. For istace, Coditios ad 2 of Theorem i Heyde 974 yield., ad so does Coditio 3.5 i Dürr ad Goldstei 986. From Theorem 2 i Heyde 974 we kow that. is satisfied as soo as i=0 P 0 X i coverges i L 2 to g ad /2 S 2 = g 2,.2 where P 0 is the projectio operator oto H 0 H. Volỳ 993, Theorems 5 ad 6, proposed sufficiets coditios based upo the sequece P 0 X i i 0 4

5 for.2 to hold. From these coditios, we easily ifer cf. Sectio 6.2 that.2 is satisfied as soo as there exists a sequece L k k>0 of positive umbers such that i L k < ad L k EX k M <..3 i>0 k= Accordig to McLeish s defiitio 975a,.3 is a mixigale-type coditio. Note that it is close to optimality, sice the choice L k = is ot strog eough to imply weak covergece of /2 S see Propositio 7 Sectio 6. k>0 We ow tur to the fuctioal versio of Theorem. As usual, to prove tightess of the ormalized Dosker lie, we eed to cotrol the maximum of partial sums. More precisely, uder the coditio b ad c of s2 hold, ad a is replaced by s2 a 2 : max S i is uiformly itegrable, i we obtai the fuctioal versio of the CCLT described i Theorem 3, Sectio 2. Note that if η is costat, the usual fuctioal cetral it theorem follows from s2 by applyig Theorem 9.4 i Billigsley 968. Note also that if the covergece rate to zero i s2c is of order θ for some positive θ ad the variables have fiite 2 + δ-momets, the it follows from Theorem 2. i Serflig 968 that s2a is automatically satisfied. Assumig that the same rate holds for s2b, Eberlei 986a has obtaied strog ivariace priciples with order of approximatio Ot /2 κ for some positive κ. Oce agai, there exists a large class of statioary sequeces satisfyig s2, as show by the two propositios below. Propositio 2 Let M i i Z ad X i i Z be as i Theorem. Coditio.3 implies s2. The key for Propositio 2 is to establish a maximal iequality which together with.3 implies s2a. This ca be doe by followig McLeish s approach for ostatioary mixigales cf. McLeish 975b, Lemma 6.3. I the particular case of statioary ad adapted sequeces, Propositio 2 improves o McLeish s results i three ways: Firstly, Coditio.3 is realized if either Coditio 2.5 i McLeish 977 holds or X, M is a mixigale of size -/2 cf. McLeish 975b Defiitios 2. ad 2.5. Secodly, the extra coditio i Theorem 2.6 of the latter may be removed i fact it 5

6 may be replaced by the weaker coditio s2c, which follows from.3 as we have already oticed. Thirdly, we obtai a stroger result i terms of covergece, the fuctioal CCLT implyig mixig-covergece as soo as η is costat, which we do ot require here. To get a idea of the wide rage of applicatios of mixigales, we refer to McLeish 975a, 975b, 977 ad Hall ad Heyde 980 Sectio 2.3. See also Eberlei 986b for a survey of results cocerig mixigales ad other geeralizatios of martigales. The secod coditio Coditio.4 below has a differet structure ad is ot obtaied via martigale approximatios, although may results i that field may be derived from it. Propositio 3 Let M i i Z, X i i Z ad S be as i Theorem. Cosider the coditio: X 0 EX k M 0 coverges i L..4 k= If.4 is satisfied, the s2 holds ad the sequece EX0 I 2 + 2EX 0 S I coverges i L to η. Coditio.4 was itroduced by Dedecker ad Rio 2000 to obtai the usual fuctioal cetral it theorem. I the adapted case, Coditio.4 is weaker tha the L 2 -criterio of Gordi 969 EX k M 0 coverges i L 2,.5 k= which has bee exteded to ostatioary sequeces by Peligrad 98. If X i = gξ i is a fuctio of a statioary Markov chai ξ i i Z with trasitio kerel K ad margial distributio µ, coditio.4 becomes gk k g coverges i L µ,.6 k=0 ad improves o classical results based upo the Poisso equatio. Coditio.6 was simultaeously discovered by Che 999 i the particular case of positive Harris Chai. Fially the applicatio of.4 to strogly mixig sequeces leads to the coditioal ad oergodic versio of the ivariace priciple of Doukha, Massart ad Rio 994, whose optimality is discussed i Bradley 997. The paper is orgaized as follows. I Sectio 2, we state the coditioal cetral it theorem Theorem 2 ad its fuctioal versio Theorem 3 i 6

7 the more geeral cotext of triagular arrays. As a cosequece, we derive i Corollary the stable covergece of the ormalized partial sums to a mixture of ormal distributios. The proofs of these results are postpoed to Sectios 3, 4 ad 5 respectively. Sectio 6 is devoted to the applicatios of Theorem : we prove Propositios, 2 ad 3 i Sectios 6., 6.2 ad 6.3 respectively. I Sectio 7, we give the aalogous of Propositios 2 ad 3 for triagular arrays. I Sectio 8, we explai how to apply these results to kerel desity estimators. 2 Mai results Theorem preseted i the itroductio is a straightforward cosequece of the followig theorem for triagular arrays with statioary rows. Theorem 2 For each positive iteger, let M 0, be a σ-algebra of A satisfyig M 0, T M 0,. Defie the odecreasig filtratio M i, i Z by M i, = T i M 0, ad M i,if = σ = k= M i,k. Let X 0, be a M 0, -measurable ad square itegrable radom variable ad defie the sequece X i, i Z by X i, = X 0, T i. Fially, for ay t i [0, ], write S t = X, + + X [t],. Suppose that /2 X 0, coverges i probability to zero as teds ifiity. The followig statemets are equivalet: S There exists a oegative M 0,if -measurable radom variable η such that, for ay ϕ i H, ay t i [0, ] ad ay positive iteger k, Sϕ : ϕ E /2 S t ϕx tηgxdx where g is the distributio of a stadard ormal. M k, = 0 S2 a S 2 sup E t S t = 0. t 0 t b For ay t i [0, ], the sequece E /2 S t M 0, teds to 0 as teds to ifiity. c There exists a oegative M 0,if -measurable radom variable η such that, for ay t i [0, ], the sequece E S 2 t tη M 0, teds to 0 as teds to ifiity. Moreover the radom variable η satisfies η = η T almost surely. 7

8 Remark 2. A assumptio o the asymptotic egligibility of the variables /2 X 0, seems to be atural i the cotext of triagular arrays see for istace the discussio p. 53 i Hall ad Heyde 980. We ow tur to the fuctioal versio of Theorem 2. Deote by H the space of cotiuous fuctios ϕ from C[0, ],. to R such that x + x 2 ϕx is bouded. Theorem 3 Let X i,, M i, ad S t be as i Theorem 2. For ay t i [0, ], set U t = S t + t [t]x [t]+, ad defie S t = sup 0 s t S s. The followig statemets are equivalet: S There exists a oegative M 0,if -measurable radom variable η such that, for ay ϕ i H ad ay positive iteger k, S E ϕ : ϕ /2 U ϕx ηw dx where W is the distributio of a stadard Wieer process. M k, = 0 S2 b ad c of S2 hold, ad a is replaced by : a the sequece S 2 >0 is uiformly itegrable, ad S t 2 sup E S t = 0. t 0 t Moreover the radom variable η satisfies η = η T almost surely. Remark 3. If we omit the assumptio that S 2 >0 is uiformly itegrable, we still obtai S ϕ for ay bouded fuctio ϕ of H. Note that Theorem 3 remais valid if we replace a by the stroger coditio sup S t 2 sup E I M t [0,] t St M t = 0. Remark 4. Let D[0, ] be the space of caldlag fuctios equipped with the Skorohod distace d. Deote by H D the space of cotiuous fuctios ϕ from D[0, ], d to R such that the fuctio x + x 2 ϕx is bouded. Applyig Theorem 5.5 i Billigsley 968, we ca also obtai that S2 holds if ad oly if, for ay ϕ i H D ad ay positive iteger k, E ϕ /2 S ϕx ηw dx M k, = 0. The followig result is a importat cosequece of Theorems 2 ad 3: 8

9 Corollary Let X i,, M i,, S t be as i Theorem 2 ad U t as i Theorem 3. Suppose that the sequece M 0, is odecreasig. If Coditio S2 resp. S2 is satisfied, the, for ay ϕ i H resp. H, the sequece ϕ /2 S t resp. ϕ /2 U coverges weakly i L to ϕx tηgxdx resp. ϕx ηw dx. Remark 5. Corollary implies that the sequece /2 S t coverges stably to a mixture of ormal distributios. We refer to Aldous ad Eagleso 978 for a complete expositio of the cocept of stability itroduced by Réyi 963 ad its coectio to weak L -covergece. Note that stable covergece is a useful tool to establish weak covergece of joit distributios see agai Aldous ad Eagleso, or Hall ad Heyde 980 Chapter 3, Sectio 3.2.vi. If furthermore η is costat, the the covergece is mixig. If P is ergodic, this result is a cosequece of Theorem 4 i Eagleso 976 see Applicatio 4.2 therei. For a review of mixig results see Csörgö ad Fischler Proof of Theorem 2 The fact that S implies S2 is obvious. I this sectio, we focus o the cosequeces of coditio S2. We start with some preiary results. 3. Defiitios ad preiary lemmas Defiitios. Let µ be a siged measure o a metric space S, BS. Deote by µ the total variatio measure of µ, ad by µ = µ S its orm. We say that a family Π of siged measures o S, BS is tight if for every positive ɛ there exists a compact set K such that µ K c < ɛ for ay µ i Π. Deote by CS the set of cotiuous ad bouded fuctios from S to R. We say that a sequece of siged measures µ >0 coverges weakly to a siged measure µ if for ay ϕ i CS, µ ϕ teds to µϕ as teds to ifiity. Lemma Let µ >0 be a sequece of siged measure o R d, BR d, ad set ˆµ t = µ expi < t,. >. Assume that the sequece µ >0 is tight ad that sup >0 µ <. The followig statemets are equivalet. the sequece µ >0 coverges weakly to the ull measure. 9

10 2. for ay t i R d, ˆµ t teds to zero as teds to ifiity. The proof of Lemma will be doe i Appedix. Defiitios 2. Defie the set RM k, of Rademacher M k, -measurable radom variables: RM k, = {2I A : A M k, }. Recall that g is the N 0, -distributio ad that W is the Wieer measure o C[0, ]. the radom variable η itroduced i Theorem 2 ad ay bouded radom variable Z, let. ν [Z] be the image measure of Z.P by the variable /2 S t. 2. ν [Z] be the image measure of Z.P by the process /2 U. 3. ν[z] be the image measure of g.λ Z.P by the variable φ from R Ω to R defied by φx, ω = x tηω. 4. ν [Z] be the image measure of W Z.P by the variable φ from C[0, ] Ω to C[0, ] defied by φx, ω = x ηω. Lemma 2 Let µ [Z ] = ν [Z ] ν[z ] ad µ [Z ] = ν [Z ] ν [Z ]. For ay ϕ i H resp H, the statemet Sϕ resp. S ϕ is equivalet to: S3ϕ resp. S3 ϕ: for ay Z i RM k, the sequece µ [Z ]ϕ resp. µ [Z ]ϕ teds to zero as teds to ifiity. Proof of Lemma 2. We prove that Sϕ S3ϕ, the *-case beig uchaged. For Z i RM k, ad ϕ i H, we have µ [Z ]ϕ = E Z ϕ /2 S t ϕx tηgxdx ϕ E /2 S t ϕx tηgxdx M k,. Cosequetly Sϕ implies S3ϕ. Now to prove that S3ϕ implies Sϕ, choose A, ϕ = { E ϕ /2 S t ϕx tηgxdx } M k, 0, For ad Z ϕ = 2I A,ϕ. Obviously µ [Z ϕ ]ϕ = ϕ E /2 S t ϕx tηgxdx M k,, ad S3ϕ implies Sϕ. 0

11 3.2 Ivariace of η We first prove that if S2 holds, the radom variables η satisfies η = η T almost surely or equivaletly that η is measurable with respect to the P- completio of I. From S2c ad both the facts that X i, i Z is strictly statioary ad M 0, M,, we have for ay t i ]0, ], E η T S2 t T M 0, = t O the other had, defiig ψx = x 2 x ad usig the fact that T preserves P, we have St 2 t S2 t T t 2 St 2 t S t + t ψ S t S t T ψ. 3.2 To cotrol the secod term o right had, ote that the fuctio ψ is 3- lipschitz ad bouded by. It follows that for each positive ɛ, ψ S t S t T ψ 3ɛ + 2P X 0, X [t], > ɛ. Usig that /2 X 0, coverges i probability to 0, we derive that ψ S t S t T ψ = 0, ad the secod term o right had i 3.2 teds to 0 as teds to ifiity. This fact together with iequality 3.2 ad Coditio S2a yield sup St 2 t 0 S2 t T t t = 0, which together with S2c imply that sup t 0 E η S2 t T M 0, = t Combiig 3. ad 3.3, it follows that Eη η T M 0, = 0, which implies that E η η T M 0,k = 0. k

12 Applyig the martigale covergece theorem, we obtai E η η T M 0,k = Eη η T M 0,if = k Accordig to S2c, the radom variable η is M 0,if -measurable. Therefore, 3.4 implies that Eη T M 0,if = η. The fact that η T = η almost surely is a direct cosequece of the followig elemetary result, whose proof will be doe i Appedix. Lemma 3 Let Ω, A, P be a probability space, X a itegrable radom variable, ad M a σ-algebra of A. If the radom variable EX M has the same law as X, the EX M = X almost surely. 3.3 S2 implies S We ow tur to the mai proof of the paper. First, ote that we ca restrict ourselves to bouded fuctios of H: if S2 implies Sh for ay cotiuous ad bouded fuctio h the we easily ifer from S2c that St 2 is uiformly itegrable for ay t i [0, ], which implies that S exteds to the whole space H. Notatios 2. Let BR 3 be the class of three-times cotiuously differetiable fuctios from R to R such that max h i, i {0,, 2, 3}. Suppose ow that Sh holds for ay h i BR. 3 Applyig Lemma 2, this is equivalet to say that S3h holds for ay h i BR, 3 which obviously implies that S3h holds for h t = expit.. Usig that the probability ν [] is tight sice it coverges weakly to ν[] ad that µ [Z ] ν [] + ν[], we ifer that µ [Z ] is tight, ad Lemma implies that S3h ad therefore Sh holds for ay cotiuous bouded fuctio h. O the other had, from the asymptotic egligibility of /2 X 0, we ifer that, for ay positive iteger k, /2 S t S t T k coverges i probability to zero. Cosequetly, sice ay fuctio h belogig to BR 3 is -lipschitz ad bouded, we have h /2 S t h /2 S t T k = 0, ad Sh is equivalet to h E /2 S t T k hx tηgxdx M k, = 0. 2

13 Now, sice both η ad P are ivariat by T, we ifer that Theorem 2 is a straightforward cosequece of Propositio 4 below: Propositio 4 Let X i, ad M i, be defied as i Theorem 2. If S2 holds, the, for ay h i BR 3 ad ay t i [0, ], h E /2 S t hx tηgxdx where g is the distributio of a stadard ormal. M 0, = 0 Proof of Propositio 4. We prove the result for S, the proof of the geeral case beig uchaged. Without loss of geerality, suppose that there exists a sequece ε i i Z of N 0, -distributed ad idepedet radom variables, idepedet of M, = σ k, M k,. Notatios 3. Let i, p ad be three itegers such that i p. Set q = [/p] ad defie U i, = X iq q+, + + X iq,, V i, = U, + U 2, + + U i, η i = ε iq q+ + + ε iq, Γ i = i + i+ + + p. Notatios 4. Let g be ay fuctio from R to R. For k ad l i [, p] ad ay positive iteger p, set g k,l; = gv k, + Γ l, with the covetios g k,p+; = gv k, ad g 0,l; = gγ l. Afterwards, we shall apply this otatio to the successive derivatives of the fuctio h. For brevity we shall omit the idex. Let s = ηε + + ε. Sice ε i i Z is idepedet of M,, we have, itegratig with respect to ε i i Z, E h /2 S hx ηgxdx M 0, = Eh /2 S hv p, M 0, + EhV p, hγ M 0, + EhΓ h /2 s M 0,. 3.5 Here, ote that /2 S V p, /2 X p+2, + + X,. Usig the asymptotic egligibility of /2 X 0,, we ifer that /2 S V p, coverges i probability to zero. Sice furthermore h is -lipschitz ad bouded, we coclude that h /2 S hv p, = 0, 3.6 3

14 ad the same argumets yield hγ h /2 s = I view of 3.6 ad 3.7, it remais to cotrol the secod term i the right had side of 3.5. To this ed, we use Lideberg s decompositio, as doe i Dedecker ad Rio hv p, hγ = p h i,i+ h i,i+ + i= p h i,i+ h i,i. 3.8 i= Now, applyig Taylor s itegral formula we get that: h i,i+ h i,i+ = U i, h i,i+ + 2 U i,h 2 i,i+ + R i η h i,i+ h i,i = ih i,i+ η 2 2 i h i,i+ + r i where R i U 2 i, U i, ad r i η 2 i η i. 3.9 Sice i is cetered ad idepedet of σ M, σh i,i+, we have E η i h i,i+ M 0, = E i E ηh i,i+ M 0, = 0. It follows that EhV p hγ M 0, = D + D 2 + D 3, 3.0 where D = p E /2 U i, h i,i+ M 0,, i= D 2 = p E Ui, 2 η 2 i h 2 i,i+ M 0,, i= p D 3 = ER i + r i M 0,. i= Cotrol of D 3. From 3.9 ad the fact that T preserves P, we get p S 2 R i E /p S /p, /p i= 4

15 ad S2a implies that sup p p R i = i= Moreover, sice for t [0, ], the sequece η/ /2 ε + +ε [t] obviously satisfies S2a, the same argumet applies to p i= r i. Fially p sup D 3 = Cotrol of D. To prove that D teds to zero as teds to ifiity, it suffices to show that, for ay positive iteger i less tha p, E /2 U i, h i,i+ M 0, = Deote by E ε the itegratio with respect to the sequece ε i i Z. Set li, = i [/p]. Bearig i mid the defiitio of h i,i+ ad itegratig with respect to ε i i Z we deduce that the radom variable E ε h i,i+ is M li,, -measurable ad bouded by oe. Now, sice the σ-algebra M 0, is icluded ito M li,,, we obtai E /2 U i, h i,i+ M 0, E /2 U i, M li,,. Usig that T preserves P, the latter equals E /2 S /p M 0, ad S2b implies that 3.3 holds. Cotrol of D 2. To prove that D 2 teds to zero as teds to ifiity, it suffices to show that, for ay positive iteger i less tha p, E Ui, 2 η 2 i h i,i+ M 0, = Itegratig with respect to ε i i Z, we have EU 2 i, η 2 i h i,i+ M 0, = EU 2 i, η[p ]E ε h i,i+ M 0,. Sice [p ] coverges to p, 3.4 will be proved if for ay positive iteger i less tha p E Ui, 2 ηp E ε h i,i+ M 0, = 0. 5

16 Arguig as for the cotrol of D, we have E U 2 i, ηp E ε h i,i+ M 0, E U 2 i, ηp M li,,. Sice both η ad P are ivariat by the trasformatio T, the latter equals E S 2 /p ηp M 0, ad S2c implies that 3.4 holds. Ed of the proof of Propositio 4. From 3.2, 3.3 ad 3.4 we ifer that, for ay h i B 3 R, p sup D + D 2 + D 3 = 0. This fact together with 3.5, 3.6, 3.7 ad 3.0 imply Propositio 4. 4 Proof of Theorem 3 Oce agai, it suffices to prove that S2 implies S. Suppose that S ϕ holds for ay bouded fuctio ϕ of H. Sice S 2 >0 is uiformly itegrable, S ϕ obviously exteds to the whole space H. Cosequetly, we ca restrict ourselves to the space of cotiuous bouded fuctios from C[0, ] to R. Accordig to Lemma 2, the proof of Theorem 2 will be complete if we show that, for ay Z i RM k,, the sequece µ [Z ] coverges weakly to the ull measure as teds to ifiity. Defiitios 3. For 0 t < < t d, defie the fuctios π t...t d ad Q t...t d from C[0, ] to R d by the equalities π t...t d x = xt,..., xt d ad Q t...t d x = xt, xt 2 xt,..., xt d xt d. For ay siged measure µ o C[0, ], BC[0, ] ad ay fuctio f from C[0, ] to R d, deote by µf the image measure of µ by f. Let µ ad ν be two siged measures o C[0, ], BC[0, ]. Recall that if µπt...t d = νπt...t d for ay positive iteger d ad ay d-tuple such that 0 t < < t d, the µ = ν. Cosequetly Theorem 2 is a straightforward cosequece of the two followig items. relative compactess: for ay Z i RM k,, the family µ [Z ] >0 is relatively compact with respect to the topology of weak covergece. 2. fiite dimesioal covergece: for ay positive iteger d, ay d-tuple 0 t < < t d ad ay Z i RM k, the sequece µ [Z ]π t...t d coverges weakly to the ull measure as teds to ifiity. 6

17 4. Fiite dimesioal covergece Clearly it is equivalet to take Q t...t d istead of π t...t d i item 2. The followig lemma shows that fiite dimesioal covergece is a cosequece of Coditio S2. The stroger coditio S2 is oly required for tightess. Lemma 4 For ay a i R d defie f a from R d to R by f a x =< a, x >. If S2 holds the, for ay a i R d, ay d-tuple 0 t < < t d ad ay Z i RM k,, the sequece µ [Z ]f a Q t...t d coverges weakly to the ull measure. Write µ [Z ]f a Q t...t d expi. = µ [Z ]Q t...t d expi < a,. >. Accordig to Lemma 4, the latter coverges to zero as teds to ifiity. Takig Z =, we ifer that the probability measure ν[]q t...t d coverges weakly to the probability measure ν []Q t...t d ad hece is tight. Sice µ [Z ]Q t...t d ν []Q t...t d + ν []Q t...t d, the sequece µ [Z ]Q t...t d >0 is tight. Cosequetly we ca apply Lemma to coclude that µ [Z ]Q t...t d coverges weakly to the ull measure. Proof of Lemma 4. Accordig to Lemma 2, we have to prove the property S ϕ f a Q t...t d for ay cotiuous bouded fuctio ϕ. Arguig as i Sectio 3.3, we ca restrict ourselves to the class of fuctio B 3 R. Let h be ay elemet of B 3 R ad write h f a Q t...t d /2 U = d l= U t l U t l h l a l where the radom variable h l x is equal to l U t i U t i h ai + x + i= h f a Q t...t d x ηw dx d i=l+ h l a l x t l t l η gx dx, d a i x i ti t i η i=l+ gx i dx i. Note that for ay ω i Ω, the radom fuctio h l belogs to B 3 R. complete the proof of Lemma 4, it suffices to see that, for ay positive itegers k ad l, the sequece E U t l U t l h l a l To h l a l x t l t l η gx dx M k, 7 4.

18 teds to zero as teds to ifiity. Sice h l is -lipshitz ad bouded, we ifer from the asymptotic egligibility of /2 X 0, that h U t l U t l S t l a l l t l T [t l ]+ h l a l = Deote by g l the radom fuctio g l = h l T [t l ]. Combiig 4., 4.2 ad the fact that M k [tl ], M k,, we ifer that it suffices to prove that g E l a l /2 S u g l a l x uη gx dx M k, = Sice the radom fuctios g l is M 0, -measurable 4.3 ca be prove exactly as property S see Sectio 3.3. This completes the proof of Lemma Relative compactess I this sectio, we shall prove that the sequece µ [Z ] >0 is relatively compact with respect to the topology of weak covergece. That is, for ay icreasig fuctio f from N to N, there exists a icreasig fuctio g with value i fn ad a siged mesure µ o C[0, ], BC[0, ] such that µ g [Z g] >0 coverges weakly to µ. Let Z + resp. Z be the positive resp. egative part of Z, ad write µ [Z ] = µ [Z + ] µ [Z ] = ν [Z + ] ν [Z ] ν [Z + ] + ν [Z ], where ν [Z] ad ν [Z] are defied i 2. ad 4. of Defiitios 2. Obviously, it is eough to prove that each sequece of fiite positive measures ν [Z + ] >0, ν [Z ] >0, ν [Z + ] >0 ad ν [Z ] >0 is relatively compact. We prove the result for the sequece ν [Z + ] >0, the other cases beig similar. Let f be ay icreasig fuctio from N to N. fuctio l with value i fn such that We must sort out two cases: EZ+ l = if EZ+ f. Choose a icreasig. If EZ + l coverges to zero as teds to ifiity, the, takig g = l, the sequece νg [Z+ g ] >0 coverges weakly to the ull measure. 2. If EZ + l coverges to a positive real umber as teds to ifiity, we itroduce, for large eough, the probability measure p defied by 8

19 p = EZ + l ν l [Z+ l ]. Obviously if p >0 is relatively compact with respect to the topology of weak covergece, the there exists a icreasig fuctio g with value i ln ad hece i fn ad a measure ν such that ν g [Z+ g ] >0 coverges weakly to ν. Sice p >0 is a family of probability measures, relative compactess is equivalet to tightess. Here we apply Theorem 8.2 i Billigsley 968: to derive the tightess of the sequece p >0 it is eough to show that, for each positive ɛ, δ 0 sup p x : wx, δ ɛ = 0, 4.4 where wx, δ is the modulus of cotiuity of the fuctio x. Accordig to the defiitio of p, we have p x : wx, δ ɛ = EZ + l Z+ l.p Ul w, δ ɛ. l Sice both EZ + l coverges to a positive umber ad Z+ l is bouded by oe, we ifer that 4.4 holds if Ul sup P w, δ ɛ = δ 0 l From Theorem 8.3 ad iequality 8.6 i Billigsley 968, it suffices to prove that, for ay positive ɛ, sup δ 0 δ P Sl δ lδ ɛ δ = We coclude by otig that 4.6 follows straightforwardly from S2a ad Markov s iequality. Coclusio: I both cases there exists a icreasig fuctio g with value i fn ad a measure ν such that ν g [Z+ g ] >0 coverges weakly to ν. Sice this is true for ay icreasig fuctio f with value i N, we coclude that the sequece ν [Z + ] >0 is relatively compact with respect to the topology of weak covergece. Of course, the same argumets apply to the sequeces ν [Z ] >0, ν [Z + ] >0 ad ν [Z ] >0, which implies the relative compactess of the sequece µ [Z ] >0. 9

20 5 Proof of Corollary We have to prove that if S2 holds, the, for ay bouded radom variable Z, ay t i [0, ] ad ay ϕ i H, E Zϕ /2 S t = E Z ϕx tηgxdx. 5. Sice S 2 t is uiformly itegrable, we eed oly prove 5. for cotiuous bouded fuctios. Recall that M, = σ k, M k,. Sice both S t ad η are M, -measurable, we ca ad do suppose that so is Z. Set Z k, = EZ M k,, ad use the decompositio E Zϕ /2 S t E Z ϕx tηgxdx = T + T 2 + T 3, where T = E Z Z k, ϕ /2 S t T 2 = E Z k, ϕ /2 S t ϕx tηgxdx T 3 = E Z k, Z ϕx tηgxdx. By assumptio, the array M k, is odecreasig i k ad. Sice the radom variable Z is M, -measurable, the martigale covergece theorem implies that k Z k, Z = 0. Cosequetly, k sup T = sup T 3 = 0. k O the other had, Theorem 2 implies that T 2 teds to zero as teds to ifiity, which completes the proof of Corollary. 6 Applicatios of Theorem 6. Proof of Propositio From Theorem i Volý 993, we kow that. is equivalet to the existece of a radom variable m i H 0 H such that X 0 T i m T i 2 = i= 20

21 Let W = m T + + m T. Sice m T i i Z is a statioary sequece of martigale differeces with respect to the filtratio M i i Z, it satisfies s2. More precisely, EW 2 M 0 coverges to η = Em 2 I i L. Now, we shall use 6. to see that the sequece X i i Z also satisfies s2. Proof of s2b. From 6. it is clear that /2 ES M 0 2 teds to zero as teds to ifiity. Proof of s2c. To see that ES 2 M 0 coverges to η i L, write ES 2 W M 2 0 S 2 W 2 S + W 2 S W From 6. the latter teds to zero as teds to ifiity ad therefore X i i Z satisfies s2c with η = Em 2 I. Proof of s2a. Usig both that W 2 is uiformly itegrable ad that the fuctio x x is -lipschitz, we have, for ay positive real M, W 2 S M W M = Sice x 2 y z 2 t x 2 z 2 + z 2 y t, we ifer from 6.2 ad 6.3 that S 2 S M W 2 Now, the uiform itegrability of W 2 yields M sup E W M = 0. S 2 S M = 0, which meas exactly that S 2 is uiformly itegrable. This completes the proof of Propositio. 6.2 Proof of Propositio 2 Let P i be the projectio operator oto H i H i : for ay fuctio f i L 2 P, P i f = Ef M i Ef M i. We first recall a result due to Volý 993, Theorem 6 see Theorem 5 of the same paper or Aexe A Corollary 2 i Dedecker 998 for weaker coditios. 2

22 Propositio 5 Let M i i Z ad X i i Z be as i Theorem. Defie the σ-algebra M = i Z M i ad cosider the coditio EX 0 M = 0 ad Coditio 6.4 implies.2. P 0 X i 2 <. 6.4 We ow prove that.3 implies 6.4 ad hece.2. First, we have the orthogoal decompositio i>0 X k = EX k M + P k i X k. 6.5 i=0 Sice.3 implies that EX k M = 0, we ifer from 6.5 ad the statioarity of X i i Z that k>0 L k EX k M = k>0 L k P i X k 2 2 = i L k P 0 X i 2 2. i>0 Settig a i = L + + L i, we ifer that.3 is equivalet to EX 0 M = 0, i 0 Now, Hölder s iequality i l 2 gives k= a i P 0 X i 2 2 < ad i>0 i>0 /2 /2 P 0 X i 2 a i P 0 X i 2 a 2 <, i i>0 i>0 ad 6.4 hece.2 follows from.3. i>0 a i <. 6.6 Now, to complete the proof of Propositio 2, it remais to show that.3 implies s2a. This is a direct cosequece of Propositio 8 Sectio 7, whose proof will be doe by applyig the followig maximal iequality: Propositio 6 Let X i i Z be a sequece of square-itegrable ad cetered radom variables, adapted to a odecreasig filtratio M i i Z. Defie the σ-algebra M = i Z M i ad the radom variables S = X + + X ad S = max{0, S,..., S }. For ay positive iteger i, let Y i,j j be the martigale Y i,j = j k= P k ix k ad Yi, = max{0, Y i,,..., Y i, }. Let λ be ay oegative real umber ad Γi, k, λ = {Yi,k > λ}. Assume that the sequece is regular: for ay iteger k, EX k M = 0. For ay two 22

23 sequeces of oegative umbers a i i 0 ad b i i 0 such that K = a i is fiite ad b i = we have E S λ 2 + 4K a i i=0 k= EPk ix 2 k I Γi,k,bi λ. Proof. The proof is adapted from McLeish 975b. From decompositio 6.5 with EX k M = 0, we have S j = i 0 Y i,j ad therefore S j λ + i 0 Y i,j b i λ +. Applyig Hölder s iequality ad takig the maximum o both side, we get S λ 2 + K i 0 a i Y i, b i λ 2 +. Takig the expectatio ad applyig Propositio a of Dedecker ad Rio 2000 to the martigale Y i,, we obtai Propositio 6. To be complete, we would like to metio that coditio.3 is close to optimality, as show by Propositio 7 below. Propositio 7 There exists a sequece X i i Z satisfyig the assumptios of Theorem, such that ES 2 coverges to σ 2 ad EX k M <, but the radom variables /2 S do ot coverges i distributio. See Dedecker 998 Aexe A.3 for a proof. k>0 6.3 Proof of Propositio 3 Proof of s2a. Let S = max{ S,..., S }. From Propositio i Dedecker ad Rio 2000, we ifer that S 2 >0 is uiformly itegrable as soo as.4 holds. Proof of s2c. The fact that E S M 2 0 coverges i L to η has bee already proved i Dedecker ad Rio 2000, Sectio 4, Cotrol of D 2. Proof of s2b. Usig first the statioarity of X i i Z ad ext the orthogoal decompositio 6.5, we obtai P 0 X i 2 2 = i=0 P i X X i=0 23

24 Now, from the decompositio we ifer that X ES M 0 = X ES M + X X ES M 0 X 0 ES T M 0 + X 0 2 P 0 X i, i= P 0 X i 2. i= 6.8 By.4, the first term o right had teds to zero as teds to ifiity. O the other had, we ifer from 6.7 ad Cauchy-Shwarz s iequality that /2 i= P 0X i 2 vaishes as goes to ifiity, ad so does the left had term i 6.8. By iductio, we ca prove that for ay positive iteger k, Now X k ES M 0 = E X 0 IES M 0 ES M 0 E X 0 I k + ES M 0 k k i= X i k X i. 6.0 From 6.9, the secod term o right had teds to zero as teds to ifiity. Applyig first Cauchy-Schwarz s iequality ad ext the L 2 -ergodic theorem, we easily deduce that the first term o right had is as small as we wish by choosig k large eough. Therefore i= E X 0 IES M 0 = Set A = {I E X0 I>0} ad B = A c = {I E X0 I=0}. For ay positive real m, we have I A ES M 0 m E X 0 IES M 0 + I 0<E X0 I<mES M From 6., the first term o right had teds to zero as teds to ifiity. Lettig m goes to zero we ifer that the secod term o right had of 6.2 is as small as we wish. Cosequetly I A ES M 0 =

25 O the other had, otig that E X 0 I B = 0, we ifer that X 0 is zero o the set B. Sice B is ivariat by T, X k is zero o B for ay k i Z. Now arguig as i Claim b i Dedecker ad Rio 2000, we obtai EE S M 0 I = E S I. These two facts lead to I B ES M 0 EI B EE S M 0 I E S I B Collectig 6.3 ad 6.4, we coclude that /2 ES M 0 teds to zero as teds to ifiity. This completes the proof. 7 Applicatios of Theorem 3 I this sectio we exted Propositios 2 ad 3 to the case of triagular arrays. I the ext sectio, we shall see how to apply these results to Kerel desity estimators. Cosider first the followig coditio, close to.3: there exists a sequece L k k>0 of positive umbers such that i i>0 k= L k < ad N sup L k EX k, M 0, 2 2 = 0. k=n If 7. is satisfied, defie QN, X ad N X as follows: QN, X = sup L k EX k, M 0, 2 2 k=n 7. ad N X = if{n > 0 : QN = 0}. If N X is fiite, we say that the array X i, is asymptotically N X -coditioally cetered of type as usual, it is m-coditioally cetered if EX m+, M 0, = 0. Propositio 8 Let X i, ad M i, be as i Theorem 2. Defie the radom variables V t = X, X,[t] 2. Assume that 7. is satisfied, ad that Lideberg s coditio holds: for ay positive ɛ, EX 2 0,I X0, >ɛ = 0. Assume furthermore that The S2a is satisfied. sup V t sup E M t [0,] t I Vt Mt =

26 Remark 6. Let us compare Propositio 8, with Theorem 2.4 i McLeish 977. I the particular case of triagular arrays, coditio 2.2b i McLeish with σ 2,i = which seems to be the usual case is equivalet with our otatio to the assumptio that X 0, is uiformly itegrable. It is easy to verify that this assumptio esures that both Lideberg s coditio ad 7.2 hold. However, i may iterestig cases, the sequece X 0, is ot uiformly itegrable while both Lideberg s coditio ad 7.2 are satisfied it is the case, for istace, whe cosiderig Kerel estimators. As a cosequece of Propositio 8, we obtai the ivariace priciple: Corollary 2 Let X i, ad M i, be as i Theorem 2. Assume that both 7. ad Lideberg s coditio are satisfied. Assume furhtermore that, for each 0 k < N X, there exists a M 0,if -measurable radom variable λ k such that for ay t i ]0, ], t [t] i= X i+k, X i, coverges i L to λ k. 7.3 N X The Coditio S2 holds with η = λ λ k. Remark 7. Let us discuss the measurability assumptio o λ k. Startig from 7.3, oe ca easily show first that λ k is ivariat by T, ad ext that k= it is a it of M 0, -measurables radom variables. Suppose furthermore that the sequece M 0, is odecreasig, the λ k is a it of M 0,if - measurable radom variables. I that case, the assumptio that λ k is M 0,if - measurable is useless, beig automatically satisfied. Remark 8. Note that if X i, = X i the both Lideberg s coditio ad assumptio 7.3 are satisfied, so that Corollary 2 exteds Propositio 2 to the case of triagular arrays. The ext coditio is the atural extesio of Coditio.4 N sup sup m X 0, N m k=n If 7.4 is satisfied, defie RN, X ad N 2 X as follows: RN, X = sup sup X 0, N m k=n 26 EX k, M 0, = m EX k, M 0,,

27 ad N 2 X = if{n > 0 : RN, X = 0}. If N 2 X is fiite, we say that the array X i, is asymptotically N 2 X -coditioally cetered of type 2. Propositio 9 Let X i,, M i, be as i Theorem 2 ad V t as i Propositio 8. If coditios 7.2 ad 7.4 are satisfied the S2a holds. As a cosequece we obtai the followig ivariace priciple: Corollary 3 Let X i, ad M i, be as i Theorem 2. Assume that 7.4 ad S2b are satisfied. Assume furhtermore that, for each 0 k < N 2 X, there exists a M 0,if -measurable radom variable λ k such that 7.3 holds. The Coditio S2 holds with N 2 X η = λ λ k. Remark 9. Let us have a look to a particular case, for which N X = resp. N 2 X =. Coditios 7. resp. 7.4 ad 7.3 are satisfied if k= coditio R. resp. R ad R2. below are fulfilled R. k= L k EX k, M 0, 2 2 = 0. R k= X 0,EX k, M 0, = 0. R2. For ay t i ]0, ], t [t] i= X2 i, coverges i L to λ. I the statioary case, these results exted o classical results for triagular arrays of martigale differeces see for istace Hall ad Heyde 980, Theorem 3.2, for which Coditio R. resp. R is automatically satisfied. We shall see Sectio 8.3 that this particular case is sufficiet to improve o may results i the cotext of Kerel estimators. I the same way, Corollary 2 or 3 provides sufficiet coditios for asymptotically m-coditioally cetered arrays of type or 2 to satisfy the fuctioal CCLT. 7. Proofs of Propositio 8 ad Corollary 2 Proof of Propositio 8. With the same otatios as i Propositio 6, defie, for b i = 2 i, [t] T i, t = Pk ix 2 Ti, t k, ad c i, t, M = E t k= 27 I Γi,[t],bi M t.

28 Defie S t = sup 0 s t {0, S s}. From Propositio 6, we have t E St M t 2 + 4K a i c i, t, M. 7.5 Note that, by statioarity, c i, t, M c i, t, 0 P 0 X i, 2 2. Now, takig a i = L + + L i, we have a i P 0 X i, 2 2 = i=n i=0 L k EX k, M 0, k=n N k= ad from 7. ad the defiitio of N X, we ifer that N N X sup L k EX N, M 0, 2 2, 7.6 a i c i, t, 0 = i=n Suppose we ca prove that, for ay 0 i < N X, sup Ti, t sup E M t [0,] t I Ti, t Mt = Sice by statioarity ET i, t = [t] P 0 X i, 2 2, 7.8 implies first that sup >0 P 0 X i, 2 2 B i <, which together with Doob s iequality yield the upper bouds for 0 i < N X, sup t [0,] sup P Γi, [t], b i M t 4B i >0 b 2 i M Accordig to 7.7, we ca fid a fiite iteger Nɛ N X such that sup 4K i=nɛ a i c i, t, 0 ɛ. Now, sice c i, t, M c i, t, 0 we obtai from 7.5 that sup t E St M t 2 + Here ote that 7.8 together with 7.9 yield Nɛ ɛ + sup i=0 a i c i, t, M. 7.0 for 0 i < N X, sup sup c i, t, M = 0 M t [0,] so that, from 7.0 sup sup M t [0,] t E St M t 2 + ɛ

29 Of course the same argumets apply to the sequece X i i Z so that 7. holds for S t. This beig true for ay positive ɛ, S2a follows. To complete the proof, it remais to show that Lideberg s coditio together with assumptio 7.2 imply 7.8. Sice P 2 k ix k, 2 E 2 X k, M k i, + E 2 X k, M k i, 2 EX 2 k, M k i, + EX 2 k, M k i,, we ifer that 7.8 holds if, settig U i, t = [t] k= EX2 k, M k i,, Ui, t for ay i 0, sup M t [0,] sup E t I Ui, t Mt Now if Lideberg s coditio holds, classical argumets esure that t U i,t U i+, t = 0, = so that if 7.2 holds for i, it holds for i +. For i = 0, ote that 7.2 is exactly 7.2 sice U 0, t = V t. This eds the proof of Propositio 8. Proof of Corollary 2. Sice Coditio 7.3 implies 7.2, S2a follows from Propositio 8. It remais to prove S2b ad c. From 7.6 we ifer that 7. is equivalet to sup a i P 0 X i, 2 2 = N N X i=n Sice the sequece a i is oicreasig ad such that a i <, it follows that a i = oi, which implies that i = Oa i. Cosequetly 7.3 holds for a i = i ad from 7.6 agai we obtai N N X sup EX k, M 0, 2 2 = k=n Startig from 7.4, we first prove S2b. From 7.3 with k = 0 we easily ifer that, for each positive N, /2 EX, + + X N, M 0, 2 teds to zero as. Therefore, to prove S2b it suffices to see that sup N N X t EX k, M 0, 2 = k=n Sice 2EX i, M 0, EX j, M 0, E 2 X i, M 0, + E 2 X j, M 0,, EX k, M 0, 2 2 k=n 29 t k=n EX k, M 0, 2 2,

30 ad 7.5 follows from 7.4. We ow prove S2c. For ay fiite iteger 0 N N X, defie the variable η N = λ 0 + 2λ + + λ N ad the two sets Λ N = [, [t]] 2 {i, j Z 2 : i j < N} ad Λ N = [, [t]] 2 {i, j Z 2 : j i N}, so that, S 2 E t M0, η N η N X i, X j, t t Λ N + 2 EX i, X j, M 0,. 7.6 t Λ N From 7.3, we ca easily prove that the first term o right had goes to zero as teds to ifiity. It remais to cotrol the secod term o right had. Set Y i, = X i, EX i, M 0, ad write EX i, X j, M 0, EY i, Y j, M 0, t t ΛN ΛN + EX i, M 0, EX j, M 0,. 7.7 t Λ N Arguig as for 7.5, we ifer from 7.4 that the secod term o right had i 7.7 is as small as we wish by choosig N large eough. Next, by usig the operators P l, we have: EY i, Y j, M 0, t t Λ N t t i= k=n t EY i, Y i+k, M 0, i= k=n l= i P l X i, P l X i+k, Usig Cauchy-Schwarz s iequality, we obtai that the last term is less tha t t i i= l= P l X i, 2 ad by statioarity, we coclude that k=n P l X i+k, 2, EY i, Y j, M 0, P 0 X i, 2 t Λ N i=0 k=n 30 P 0 X k, 2.

31 This last iequality together with 7.3 ad Cauchy-Schwarz s iequality yield sup N N X EY i, Y j, M 0, = 0, t Λ N ad S2c follows. This completes the proof of Corollary Proofs of Propositio 9 ad Corollary 3 Proof of Propositio 9. Let S t be as i 7.5 ad defie the set Gt, M, by Gt, M, = {S t > M t}. From Propositio i Dedecker ad Rio 2000, we have, for ay positive iteger N, t E St M t 2 + 8E I Gt,M, t + 8 sup t N k= i=0 m X 0, N m t k=n+ X k, X k+i, EX k, M 0, 7.8 From 7.4 we ca choose Nɛ large eough so that the secod term o right had is less tha ɛ. Takig M = 0 i 7.8 ad usig Assumptio 7.2, we ifer that there exists a fiite real B such that sup t E St 2 B so that sup PGt, M, B M. 2 This last boud together with 7.2 imply that sup sup E I Gt,M, M t [0,] t t k= N i=0 X k, X k+i, = 0, so that, from 7.8, sup sup M t [0,] t E St M t 2 + ɛ. 7.9 Of course the same argumets apply to the sequece X i i Z so that 7.9 holds for S t. This beig true for ay positive ɛ, S2a follows. Proof of Corollary 3. Sice Coditio 7.3 implies 7.2, S2a follows from Propositio 9. It remais to prove S2c. Startig from iequality 7.6 for ay fiite iteger N N 2 X, we have to show that sup EX i, X j, M 0, = N N 2 X t Λ N 3

32 Usig first the iclusio M 0, M i, for ay positive i ad secod the statioarity of the sequece, we obtai succesively EX i, X j, M 0, X i, EX j, M i, t t Λ N Λ N sup m X 0, N m k=n EX k, M 0,. ad 7.20 follows from 7.4. This completes the proof of Corollary 3. 8 Kerel estimators Let Y be a real-valued radom variable with ukow desity f, ad defie the statioary sequece Y i i Z = Y T i i Z. We wish to estimate f at poit x from the data Y,..., Y. To this aim, we shall cosider as usual cf. Roseblatt 956b the kerel-type estimator of f f x = h i= x Yi K. h It is well-kow that the study of the bias Ef x fx does ot deped o the depedece properties of the process Y i i Z, but o the regularity of f. Cosequetly, we oly deal with the asymptotic behavior of f x Ef x. 8. A geeral result Defiitios 4. a kerel if: K <, We say that a Borel measurable fuctio K from R to R is u Ku = 0, ad u Ku du =. 8. Defiitios 5. For ay x i R d ad ay positive real M, deote by L M,x the space of itegrable fuctio from Rd to R with support i the cube [x M, x + M] [x d M, x d + M]. If f belogs to L M,x, deote by f,m x its orm. For ay positive measure µ o R d, BR d, defie the two quatities µ,m x = sup{µ f : f,m x } ad µ x = M 0 µ,m x. If µ x <, we say that µ belogs to L x. For each i i Z, deote by µ i the law of Y 0, Y i. We make the followig assumptios: 32

33 A Y has desity f which is cotiuous at x. A2 for each i i Z, µ i belogs to L x,x. Propositio 0 Let K be a kerel, ad h be a sequece of positive umbers such that h teds to zero ad h teds to ifiity as teds to ifiity. Let Y i be a strictly statioary sequece satisfyig A ad A2. Defie succesively M i, = M i = σy j, j i, X i, = x Yi {K h h x Yi } E K, 8.2 h ad U as i Theorem 3. Assume that either 7. or 7.4 ad S2b holds. If furthermore sup k k CovX0,, 2 Xi, 2 = 0, 8.3 i= the the process U satisfies S with η = fx K 2 2. Proof. It is a cosequece of either Corollary 2 or Corollary 3. I order to apply Corollary 2, we eed to prove that X 0, satisfies Lideberg s coditio. Defie K x = h Kh x. Sice A holds, classical argumets esure that EK x Y coverges to fx, ad cosequetly h EK x Y teds to 0 as teds to ifiity. Now, recall that X 0, = h K x Y EK x Y. Therefore it suffices to show that the sequece h K x Y satisfies Lideberg s coditio. The desity f beig cotiuous at x, there exist two positive reals M ad C such that for ay y i [x M, x + M], fy is less tha C. Settig K[M] = KI [ M,M] c, we have x Y E K 2 I h h h K x Y >ɛ K[M/h ] 2 h + C K 2 zi Kz >ɛ h dz. Note that uk 2 u teds to zero as u teds to ifiity, which implies that the first term o right had vaishes as h teds to zero. O the other had, sice K 2 is fiite, the secod term teds to zero as h teds to ifiity. To complete the proof, it remais to see that 8.3 implies 7.3 with λ 0 = fx K 2 2 ad λ k = 0 for ay positive iteger k. To prove the secod poit, ote that for ay k > 0 deotig the covolutio, X 0, X k, h µ k K K x, x + 3f K x

34 Agai, choose M such that for ay y i [x M, x + M], fy is less tha C. Splittig the itegral i four parts over the four sets [x M, x + M] 2, [x M, x + M] [x M, x + M] c, [x M, x + M] c [x M, x + M] ad [x M, x + M] c 2, we get the upper boud µ k K K x, x K 2 µ k,m x + 2C K[M/h ] K + K[M/h h h 2 ] Now assumptio A2 esures that for M small eough, the first term o right had is fiite. The two other terms o right had tedig to zero as teds to ifiity, we ifer that sup >0 µ k K K x, x <. Both this fact ad 8.4 imply that, for ay positive k, X 0, X k, teds to 0 as teds to ifiity, so that 7.3 holds with λ k = 0. It remais to see that 7.3 holds for k = 0 with λ 0 = fx K 2 2. Sice the fuctio K 2 2 K 2 is a kerel, classical argumets esure that EX 2 0, coverges to fx K 2 2. Now [t] X t 2 i, EX 0, 2 = [t] 2 t 2 VarX2 0,+ 2 t i= [t] k= k t i= CovX 2 0,, X 2 i,. Usig that K 4 4 K 4 is a kerel, we ifer that VarX0, 2 = Oh, so that VarX0, 2 teds to zero as h teds to ifiity. Sice furthermore 8.3 implies that the secod term o right had teds to zero as teds to ifiity, the result follows. 8.2 Applicatio to mixig sequeces I this sectio, we give three differets applicatios of Propositio 0 to the case of mixig sequeces. Defiitios 6. Let U ad V be two σ-algebras of A. The strog mixig coefficiet of Roseblatt 956a is defied by αu, V = sup{ PUPV PU V : U U, V V}. 8.6 The φ-mixig coefficiet itroduced by Ibragimov 962 ca be defied by φu, V = sup{ PV U PV, V V}. 8.7 Betwee those coefficiets, the followig relatio holds : 2αU, V φu, V. 34

35 Defiitios 7. Let Y ad ε be two real valued radom variables, ad defie Y i, ε i i Z = Y T i, ε T i i Z. We say that Y i, ε i i Z is a fuctioal autoregressive process if there exists a Borel-measurable fuctio ψ such that Y i = ψy i +ε i, where ε is idepedet of the σ-algebra M 0 = σy i, i 0. Cosider the two followig assumptios, which are stroger tha A2: A3 assumptio A2 holds ad moreover sup M 0 i Z µ i,m x, x <. A4 Y i, ε i i Z is a fuctioal autoregressive process ad ε has desity h which is cotiuous ad bouded. Corollary 4 Let Y i i Z be a strictly statioary sequece. Let h, X i, i Z, ad M i i Z be as i Propositio 0, ad defie the process U as i Theorem 3. Write φ, k = φm 0, σy k ad α, k = αm 0, σy k. Cosider the three followig coditios: i k>0 φ, k <, ii α, k = o, ad iii α, k <. k> k>0 If either A, A2, i, A, A3, ii or A4, iii holds, the the process U satisfies S with η = fx K 2 2. Remark 0. The mixig rate ii was first required by Robiso 983 for the more striget coefficiets α, k = αm 0, σy i, i k. To uderstad the differece betwee α, ad α,, ote that the covergece of α, to zero implies that M is idepedet of I, while the same property for α, does ot eve implies that σy 0, Y is idepedet of I. This meas i particular that a large class of oergodic processes are cocered by Corollary 4. For such processes, it may be surprisig that the variace term η is degeerate. I fact, this is due to Assumptio A2 which implies that, for ay positive iteger i, the covariace betwee X 0, ad X i, teds to zero as teds to ifiity. Remark. Let us recall some recet results about mixig rates for autoregressive processes cf. Tuomie ad Tweedie 994 ad Ago Nzé 998. Suppose that Y i, ε i i Z is a fuctioal autoregressive process ad that. the sequece ε i i Z is i.i.d, the desity h of ε satisfies h 0 > 0, ad there exists S such that E ε S <. 35