An empirical Central Limit Theorem in L 1 for stationary sequences

Transcription

1 Stochastic rocesses and their Applications 9 (9) An empirical Central Limit heorem in L for stationary sequences Sophie Dede LMA, UMC Université aris 6, Case courrier 88, 4, lace Jussieu, 755 aris Cedex 5, France Received 5 December 8; received in revised form 6 June 9; accepted June 9 Available online 7 June 9 Abstract In this paper, we derive asymptotic results for the L -Wasserstein distance between the distribution function and the corresponding empirical distribution function of a stationary sequence. Next, we give some applications to dynamical systems and causal linear processes. o prove our main result, we give a Central Limit heorem for ergodic stationary sequences of random variables with values in L. he conditions obtained are expressed in terms of projective-type conditions. he main tools are martingale approximations. c 9 Elsevier B.V. All rights reserved. MSC: 6F7; 6G; 6G3 Keywords: Empirical distribution function; Central Limit heorem; Stationary sequences; Wasserstein distance. Introduction he Kantorovich or L -Wasserstein distance between two probability measures and on R with finite mean, is defined by { } d (, ) : inf x y dν(x, y) : ν (R ) with marginals,, where (R ) is the space of probability measures on R. address: sophie.dede@upmc.fr /$ - see front matter c 9 Elsevier B.V. All rights reserved. doi:.6/j.spa.9.6.6

2 S. Dede / Stochastic rocesses and their Applications 9 (9) Let Λ be the space of -Lipschitz functions. It is well known that d can also be written as follows: d (, ) F (t) F (t) dt sup f d f d, f Λ where F (respectively F ) is the distribution function of (respectively of ). Let (X i ) i Z be a stationary sequence of real-valued random variables. In this paper, we are concerned with the Central Limit heorem (CL) for the L -Wasserstein distance, defined by F n (t) F X (t) dt, (.) R where F X is the common distribution function of the variables X i, and F n is the corresponding empirical distribution function (see Section 3). In the literature, several previous works on the Kantorovich or L -Wasserstein distance, have already been done, for a sequence of i.i.d random variables X (X i ) i Z (see for instance del Barrio, Giné and Matrán []). Recall that if X has the distribution function F X, then the condition FX (t)( F X (t))dt <, is equivalent to Λ, (X) : ( X > t)dt <. In their heorem., del Barrio, Giné and Matrán [] prove that if (X i ) i Z is i.i.d, then the processes n(f n F X ) converge in law in L to the process {B(F X (t)), t R}, where B is a Brownian bridge, if and only if Λ, (X) <. Our main result extends heorem. in del Barrio, Giné and Matrán [] to the case of stationary sequences, satisfying some appropriate dependence conditions. Before giving the idea of the proof, let us introduce L (µ) L (, µ), where µ is a σ -finite measure and is a real interval, the Banach space of µ-integrable real functions on, with the norm.,µ, defined by x,µ x(t). Let L (µ) be its dual space. First, we give the Central Limit heorem (CL) for ergodic stationary sequences of martingale differences in L (µ) (see Section 4.). hen, by martingale approximation (see for instance Volný []), we derive a Central Limit heorem for some ergodic stationary sequences of L (µ)- valued random variables satisfying some projective criteria. his result allows us to get sufficient conditions to derive the asymptotic behavior of (.). he paper is organized as follows. In Section, we state our main result. In Section 3, we derive the empirical Central Limit heorem for statistics of the type (.) for a large class of dependent sequences. In particular, the results apply to unbounded functions of expanding maps of the interval, and to causal linear processes.. Central Limit heorem for stationary sequences in L (µ) From now, we assume that the ergodic stationary sequence (X i ) i Z of centered random variables with values in L (µ), is given by X i X i, where : Ω Ω is a bijective bimeasurable transformation preserving the probability on (Ω, A). Let S n n j X j, be

3 3496 S. Dede / Stochastic rocesses and their Applications 9 (9) the partial sums. For a σ -algebra F of A, satisfying F (F ), let F i i (F ), F n F n and F F k. Notation.. For any integer p and for any real-valued random variable Y, we denote. p, the L p -norm defined by Y p E( Y p ) /p, and. denotes the L -norm, that is the smallest u such that ( Y > u). Definition.. G is said to be an L (µ)-valued Gaussian random variable if for any f L (µ), f (G) is a real-valued Gaussian random variable, and its covariance operator, noted by Φ G, is defined by: for any ( f, g) in L (µ) L (µ), Φ G ( f, g) E ( f (G E(G))g(G E(G))) (see for instance Araujo and Giné []). Here is our main result: heorem.3. Assume that, for any real t, E(X (t) F ), E(X (t) F ) X (t) and X (t) <. (.) Let (X (t)) E(X (t) F ) E(X (t) F ) and assume that (X k (t)) <. (.) hen n / n X i n G in law in L (µ), (.3) where G is an L (µ)-valued centered Gaussian random variable with covariance operator: for any ( f, g) in L (µ) L (µ), ( ( ) ( )) Φ G ( f, g) E f (X k ) g (X k ) Cov( f (X ), g(x k )). (.4) As a consequence, we have Corollary.4. Assume that (.) holds. Moreover, suppose that n and that n n n E(X n (t) F ) <, (.5) X n E(X n (t) F ) <. (.6) hen, the conclusion of heorem.3 holds.

4 S. Dede / Stochastic rocesses and their Applications 9 (9) Applications to the empirical distribution function Let Y (Y i ) i Z be a sequence of real-valued random variables. We denote their common distribution function by F Y and by F n the corresponding empirical distribution function of Y : t R, F n (t) n Yi t. n Let λ be the Lebesgue measure on R. If E( Y ) <, the random variable X i (.) {t Yi t F Y (t), t R} may be viewed as a centered random variable with values in L (λ). Notation 3.. Let Z be a real-valued random variable and F a σ -algebra of A. Let F Z F be the conditional distribution function of Z given F. With these notations, the following equalities are valid: for every k in Z, (X k (t)) dt F Yk F (t) F Yk F (t) dt, (3.) and 3.. Dependent sequences E(X k (t) F ) dt F Yk F (t) F Y (t) dt. (3.) As we shall see in this section, applying Corollary.4, we can derive sufficient conditions for the convergence in L (λ) of the process n(f n F Y ), as soon as the sequence Y satisfies some weak dependence conditions. Set F σ (Y i, i ). We first recall the following dependence coefficients as defined in Dedecker and rieur [9]: for any integer k, and φ(k) sup (Y k t F ) (Y k t), t R α(k) sup (Y k t F ) (Y k t). t R When the sequence Y is φ-dependent, the following result holds: roposition 3.. Assume that φ(k) < and k k ( Y > t)dt <, (3.3) then {t n(f n (t) F Y (t)), t R} converges in L (λ), to an L (λ)-valued centered Gaussian random variable G, with covariance operator: for any ( f, g) in L (λ) L (λ), Φ G ( f, g) f (s)g(t)c(s, t)dtds (3.4) R with C(s, t) ((Y s, Y k t) F Y (s)f Y (t)).

5 3498 S. Dede / Stochastic rocesses and their Applications 9 (9) Remark 3.3. roposition 3. is also true with the φ-mixing coefficient of Ibragimov [3], defined by φ(k) : φ(f, σ (X k )) sup E( Xk A F ) E( Xk A). A B(R) φ is weaker than φ because we consider only the coarser set {], t], t R}, instead of B(R). Notice that this result contains the i.i.d case, developed in del Barrio, Giné and Matrán []. Before giving sufficient conditions when the sequence Y is α-dependent, we first recall the following definition: Definition 3.4. For any nonnegative and integrable random variable Y, define the quantile function Q Y of Y, that is the cadlag inverse of the tail function x (Y > x). roposition 3.5. Assume that k α(k) k Q Y (u) u du <, (3.5) then the conclusion of roposition 3. holds. Remark 3.6. Notice that roposition 3.5 is also true with strong α-mixing coefficients of Rosenblatt [], defined by α(k) : α(f, σ (X k )) sup E( Xk A F ) E( Xk A). A B(R) α is weaker than α because we consider only the coarser set {], t], t R}, instead of B(R). Notice also that (3.5) is equivalent to α(k) ( Y > t)dt <. (3.6) k k 3... Application to expanding maps Let be a map from [, ] to [, ] preserving a probability µ on [, ]. Recall that the erron Frobenius operator K from L (µ) to L (µ) is defined via the equality: for any h L (µ) and f L (µ), (K h)(x) f (x)µ(dx) h(x)( f )(x)µ(dx). Here we are interested by giving sufficient conditions for the convergence in L (λ) of the empirical distribution function associated to F Y, where λ is the Lebesgue measure, and the random variables (Y i ) i Z are defined as follows: for a given measurable function f, let Y k f k. In fact since on the probability space ([, ], µ), the random variable (,,..., n ) is distributed as (Z n, Z n,..., Z ), where (Z i ) i is a stationary Markov chain with invariant measure µ and transition Kernel K (see Lemma X I.3 in Hennion and Hervé []), the convergence in L (λ) of the empirical distribution function associated to F Y is reduced to the convergence of the empirical distribution function associated to F f (Z). (3.7)

6 S. Dede / Stochastic rocesses and their Applications 9 (9) In this section we consider two cases: first the case of a class of BV-contracting maps and secondly the case of a class of intermittent maps. (a) he case of BV-contracting maps. Let BV be the class of bounded variation functions from [, ] to R. For any h BV, denote by dh the variation norm of the measure dh. A Markov kernel K is said to be BV -contracting if there exist C > and ρ [, [ such that d K n (h) Cρ n dh. A map is then said to be BV-contracting if its erron Frobenius operator K is BV-contracting (see for instance Dedecker and rieur [9], for more details and examples of maps which are BV-contracting). In this case, the following result holds: Corollary 3.7. If is BV-contracting and f is piecewise monotonic (with a finite number of branches) from ], [ to R satisfying λ( f > t)dt <, then the conclusion of roposition 3. holds for the sequence (Y k ) where Y k is defined by (3.7). Remark 3.8. In the particular case when f is positive and non-increasing on ], [, with f (x) Dx a for some a > and D a constant, we get that λ( f > t)dt C dt, t/(a) where C is a constant. Consequently, Corollary 3.7 holds as soon as a < holds. (b) Application to intermittent maps. For γ in ], [, we consider the intermittent map γ from [, ] to [, ], studied for instance by Liverani, Saussol and Vaienti [6], which is a modification of the omeau Manneville map [8]: { x( + γ x γ ) if x [, /[ γ x if x [/, ]. We denote by ν γ the unique γ -invariant probability measure on [, ] and by K γ the erron Frobenius operator of γ with respect to ν γ. For these maps, we obtain the following result: Corollary 3.9. For γ in ], [, if γ is an intermittent map and f is piecewise monotonic (with a finite number of branches) from ], [ to R, satisfying ν γ ( f > t)dt <, (3.9) k k k γ γ then the conclusion of roposition 3. holds for the sequence (Y k ) where Y k is defined by (3.7). Remark 3.. In the particular case when f is positive and non-increasing on ], [, with f (x) Dx a for some a > and g νγ the density of ν γ such that g νγ (x) V γ x γ where V γ is a constant, we can prove that (3.9) holds as soon as a < γ does (see Section 4.8). In his comment after heorem 3, Gouëzel [] proved that if f (x) x a, then n / n k ( f γ i ν γ ( f )) converges to a normal law if a < / γ, and that there is a convergence to a stable law (with a different normalization) if a > / γ. his example shows that our condition is close to optimality. (3.8)

7 35 S. Dede / Stochastic rocesses and their Applications 9 (9) Causal linear processes We focus here on the stationary sequence Y k j a j ε k j, (3.) where (ε i ) i Z is a sequence of real-valued i.i.d random variables in L and j a j < +. Corollary 3.. Assume that, ε has a density bounded by K and that a. Moreover, assume that k ak Q Y (u) u du <. (3.) hen the conclusion of roposition 3. holds for the sequence (Y k ) where Y k is defined by (3.). Remark 3.. Since j a j <, (3.) is true provided that k ak Q ε (u) u du <. (3.) roof of Remark 3.. By using Lemma. in Rio [9], page 35, we have that ( ) ak Q Y (u) ak Q ε (u) du a j du. u j u Consequently since j a j <, (3.) is true provided that ak Q ε (u) u du <. (3.3) As a consequence, we get the following result Corollary 3.3. Assume either Item or below:. For some r >, the i.i.d random variables (ε i ) i Z are in L r and ε has a density bounded by K. In addition a and (k + ) /(r ) a k (r )/(r ) <. (3.4) k. For some r >, ( c ) r x >, ( ε > x) where c is a positive constant, x and ε has a density bounded by K. In addition a and a k /r <. k (3.5) hen the conclusion of roposition 3. holds for the sequence (Y k ) where Y k is defined by (3.).

8 S. Dede / Stochastic rocesses and their Applications 9 (9) roofs 4.. Central Limit heorem for L (µ)-valued martingale differences We extend to L (µ)-valued martingale differences, a result of Jain [4], which is, for a sequence of i.i.d centered L (µ)-valued random variables X (X i ) i Z, the Central Limit heorem holds if and only if X (t) <. heorem 4.. Let (m i ) i Z be a sequence of stationary ergodic martingale differences with values in L (µ) such that, for any i Z, m i m i. We note M n n m i. Assume that m (t) <. (4.) hen M n n n G in law in L (µ), (4.) where G is an L (µ)-valued centered Gaussian random variable with covariance operator: for any f in L (µ), Φ G ( f, f ) E( f (m )). roof of heorem 4.. Before giving the proof, we recall some notations and definitions used in the proof. Notation 4.. Using the notations of Jain [4], we consider for any real separable Banach space B and its dual B, } W M {ν probability measures on B: f dν <, f dν, f B. Definition 4.3. If ν W M, its covariance kernel Φ ν is given by: for any ( f, g) in B B, Φ ν ( f, g) f gdν. Definition 4.4. µ W M is pregaussian if there is a Gaussian measure ν, such that Φ ν Φ µ. By the classical Linderberg s theorem for stationary ergodic martingale differences in Billingsley [3], n / f (M n ) converges in law to the centered Gaussian random variable Z in R, with variance E( f (m )), for each f L (µ). Now, we have to prove that the sequence of the distributions of n / M n is relatively compact. As L (µ) is of cotype (for a definition see for instance Jain [4] or Ledoux and alagrand p 45 [5]), we use the same approach as in the proof of heorem 6.4 in de Acosta, Araujo and Giné []. By stationarity, it follows that ( [ ( )] ) Mn E f n E([ f (m i )] ) E([ f (m )] ). n n By heorem in Jain [4], if (4.) holds, then m is pregaussian, so there exists an L (µ)- valued centered Gaussian random variable G (or a Gaussian measure γ on L (µ)), with

9 35 S. Dede / Stochastic rocesses and their Applications 9 (9) covariance operator Φ G, such that, for any f in L (µ), E([ f (G)] ) Φ G ( f, f ) E([ f (m )] ). By heorem 5.6 in de Acosta, Araujo and Giné [], every centered Gaussian measure on L (µ), is strongly Gaussian which means that there exist a Hilbert space H, a continuous linear map M : H L (µ) and a tight centered Gaussian measure ν on H such that γ ν M. herefore, we can apply heorem 6. [], with K {ξ a probability measure on L (µ) such that Φ ξ ( f, f ) Φ γ ( f, f ), for all f L (µ)}, so K is relatively compact. We have proved that the sequence of the distributions of n / M n is relatively compact. 4.. roof of heorem.3 Before giving the proof, we recall a notation used in the proof. Notation 4.5. i is the projection operator from L to J i where J i is the orthogonal of H i in H i, where H i is the space of F i -measurable and square integrable random variables. We construct the martingale n M n m i, where m (X k ). Notice that (m i ) i Z is a sequence of stationary ergodic martingale differences. By triangle inequality and by assumption (.), m (t) (X k (t)) (X k (t)) <. Applying heorem 4., we infer that n M n n G in law in L (µ), where G is an L (µ)-valued centered Gaussian random variable such that Φ G ( f, f ) E([ f (m )] ), for any f L (µ). o conclude the proof, it suffices to prove that lim n S n (t) n n n m (t) i. (4.3) he proof is inspired by the proof of heorem in Dedecker, Merlevède and Volný [8]. By triangle inequality, S n (t) n m (t) i n n S n (t) E(S n(t) F n ) + E(S n(t) F n ) + E(S n(t) F ) n n n n

10 S. Dede / Stochastic rocesses and their Applications 9 (9) E(S n(t) F ) n n n m (t) i S n (t) E(S n(t) F n ) n n E(S n (t) F n ) + E(S n(t) F ) n m (t) i n n n + E(S n (t) F ). (4.4) n It suffices to prove that each term on the right-hand side of Inequality (4.4) tends to, as n tends to infinity. Let us first control the second term. Since E(S n (t) F n ) E(S n (t) F ) n k n i (X k (t)), it follows that, by stationarity and orthogonality, E(S n (t) F n ) E(S n (t) F ) n m (t) i n n n n n (X k i (t)) m (t) k n n i n (X j (t)) m (t) + n n j i n (X j (t)) n j i j n i+ (X j (t)). (4.5) Splitting the sum on i of the first term on the right-hand side of Inequality (4.5), we get that n n (X j (t)) n j i N (X j (t)) + n j i n in+ (X j (t)) j i

11 354 S. Dede / Stochastic rocesses and their Applications 9 (9) N n (X j (t)) + n j i n (X j (t)). in+ j i Fubini entails that N n (X j (t)) Moreover, since n n in+ we infer by (.) that j i lim lim sup N n Whence n j i n n (X j (t)) j i n n in+ j i N (X j (t)) N N n j Z n (X j (t)) j j i (X j (t)) (X j (t)) n. ( ) n (n N) (X j (t)) j N ( j N (X j (t)). n. j N (X j (t)) ) (X j (t)), In the same way, splitting the sum on i of the second term on the right-hand side of Inequality (4.5), we derive that n n (X j (t)) n j n i+ n N j n i+ (X j (t)) + n n in N+ j n i+ (X j (t))

12 Since S. Dede / Stochastic rocesses and their Applications 9 (9) n N n (X j (t)) j n i+ + n n (X j (t)). in N+ j n i+ n n N j n i+ ( ) (n N) (X j (t)) (X j (t)) n j N+ (X j (t)), j N+ and n n in N+ n N n n in N+ j Z j n i+ (X j (t)) j n i+ (X j (t)), (X j (t)) we deduce by (.), that lim n n n Consequently, we derive that E j n i+ (X j (t)). ( ) ( ) Sn (t) Sn (t) F n E F n n n n m (t) i n. (4.6) o prove that the last term of Inequality (4.4) tends to as n tends to infinity, we first write that ( ) E Sn (t) F N E(X n n k (t) F ) k + n E(X n k (t) F ). kn+

13 356 S. Dede / Stochastic rocesses and their Applications 9 (9) By orthogonality, n E(X k (t) F ) kn+ n n kn+ ln+ n n kn+ ln+ E(E(X k (t) F )E(X l (t) F )) ( ) E m (X k (t)) m (X l (t)). Using Cauchy Schwarz inequality and stationarity, it follows that n E(X k (t) F n ) n+m n+m (X k (t)) (X l (t)) n kn+ m kn+m+ ln++m ( ) (X k (t)). kn+ Consequently n n kn+ and by (.), it follows that lim lim sup N n E(X k (t) F ) n n kn+ m kn+ On the other hand, by stationarity, N N E(X k (t) F ) E( i (X k (t)) F ) k k i Z ( ) N i (X k (t)) k i Z (X k (t)), E(X k (t) F ). (4.7) N i Z (X i (t)). Hence by (.), we get that N lim E(X k (t) F ). (4.8) n n k herefore, (4.7) and (4.8) imply that ( ) E Sn (t) F. (4.9) n lim n o prove that the first term of Inequality (4.4) tends to as n tends to infinity, we write that ( ) S n (t) Sn (t) E F n n n

14 S. Dede / Stochastic rocesses and their Applications 9 (9) n N [X n k (t) E(X k (t) F n )] k + n [X n k (t) E(X k (t) F n )]. kn N+ By orthogonality, n N [X k (t) E(X k (t) F n )] k n N k n N k n N l n N l E([X k (t) E(X k (t) F n )][X l (t) E(X l (t) F n )]) E ( mn+ m (X k (t)) m (X l (t)) Consequently by using Cauchy Schwarz inequality and stationarity, n N [X n k (t) E(X k (t) F n )] n N m n N m (X k (t)) (X l (t)) n k mn+ k m l m ( ) (N+) (X k (t)). k Hence we get n N [X k (t) E(X k (t) F n )] n k and by (.), it follows that lim N lim sup n n ). (N+) k (X k (t)), n N [X k (t) E(X k (t) F n )]. (4.) k Also by stationarity, n [X k (t) E(X k (t) F n )] kn N+ ( ) n m (X k (t)) kn N+ m Z N i Z (X i (t)). n kn N+ mn+ m (X k (t)) Hence n n kn N+ [X k (t) E(X k (t) F n )]

15 358 S. Dede / Stochastic rocesses and their Applications 9 (9) N (X i (t)). (4.) n i Z herefore, (4.) and (4.) imply that S n (t) E n lim n ( Sn (t) n F n ). (4.) By (4.6), (4.9) and (4.), it follows that (4.3) holds. Consequently, n / S n converges in law in L (µ) to an L (µ)-valued centered Gaussian random variable with covariance operator, for any f L (µ), ( ) Φ G ( f, f ) E f (m ) ( E ( )) f (X k ). o conclude, we prove that, for any ( f, g) in L (µ) L (µ), Cov( f (X ), g(x k )) <, (4.3) and Φ G ( f, g) Cov( f (X ), g(x k )). (4.4) Clearly, it suffices to prove it for f g. In that case, by using Corollary in Dedecker, Merlevède and Volný [8], we know that (4.3) and (4.4) hold as soon as ( f (X k )) <. (4.5) As f is a linear form on L (µ) then f belongs to L (µ). It follows that ( f (X k )) f ( (X k )) C( f ) (X k )(t) C( f ) (X k )(t), where C( f ) is a constant depending on f. Consequently, (4.5) holds as soon as (.) holds. his entails that (4.4) follows roof of Corollary.4 We prove Corollary.4 with the same arguments as in the end of the proof of Corollary in eligrad and Utev [7]. By stationarity and orthogonality, for all k in Z, we have E(X k (t) F ) j (X k (t)) j (X (t)) j jk

16 S. Dede / Stochastic rocesses and their Applications 9 (9) and X k (t) E(X k (t) F ) j (X k (t)) j j (X k (t)) j jk+ j (X (t)). Now, applying Lemma A. in eligrad and Utev [7], to a i : i (X (t)), it follows ( i (X (t)) 3 n / i (X (t)) n in 3 n / E(X n (t) F ), (4.6) n and then to a i : i (X (t)), ( i (X (t)) 3 n / i (X (t)) n in 3 n / X (n ) (t) E(X (n ) (t) F ) 3 n (n + ) / X n (t) E(X n (t) F ) n 3 X (t) E(X (t) F ) + 3 ) / ) / n / X n (t) E(X n (t) F ) n 3 n / X n (t) E(X n (t) F ) + 6 X (t). (4.7) n herefore, by (4.6) and (4.7), we deduce that (X k (t)) k (X (t)) + k (X (t)) + (X (t)) k k ) / ( 3 n / i (X (t)) n in ( / + 3 n / i (X (t)) ) + (X (t)) n in [ ] 3 n / E(X n (t) F ) + n / X n (t) E(X n (t) F ) n n + 8 X (t). Consequently, (.5) and (.6) imply (.).

17 35 S. Dede / Stochastic rocesses and their Applications 9 (9) roof of roposition 3. We apply Corollary.4 to the random variables X i (.) {t Yi t F Y (t), t R}. Let Z(t) Obviously, F Y k F (t) F Y (t) F Yk F (t) F Y (t). t R, E(X k (t) F ) X k (t) F Y (t)( F Y (t)). (4.8) By roposition. in Dedecker [4] and by (4.8), for any k in Z, we derive ( ) FYk F F Yk F (t) F Y (t) E (t) F Y (t) (F Yk F F Yk F (t) F Y (t) (t) F Y (t)) Cov(Z(t), X k (t)) Z(t) X k (t) φ(k) F Y (t)( F Y (t)) φ(k). Consequently, we deduce that (.5) holds as soon as (3.3) holds. o end the proof, it remains to prove (3.4). Clearly, as ( f, g) in L (λ) L (λ), and R Y s F Y (s) ds < (deduced by (3.3)), Fubini entails that for any k Z, Cov( f (X ), g(x k )) f (t)g(s)cov( Y t F Y (t), Yk s F Y (s))dtds. R Via Cauchy Schwarz inequality and Fubini, f (t) g(s) Cov( Y t F Y (t), Yk s F Y (s)) dtds R C( f )C(g) E( R i ( Y t) i ( Yk s)) dtds i C( f )C(g) ( Yi t) ( Yk+i s) dtds R i Z ( ) ( ) C( f )C(g) ( Yk s) ds ( Yk t) dt R R <, where C( f ) (respectively C(g)) is a constant depending on f (respectively g). Indeed, by using the proof of Corollary.4 and (3.3), it follows that ( Yk t) dt <. R herefore, we can still apply Fubini, for any ( f, g) in L (λ) L (λ), Cov( f (X ), g(x k ))

18 S. Dede / Stochastic rocesses and their Applications 9 (9) f (t)g(s)cov( Y t F Y (t), Yk s F Y (s))dtds R f (t)g(s) Cov( Y t F Y (t), Yk s F Y (s))dtds R f (t)g(s) ((Y t, Y k s) F Y (t)f Y (s))dtds. R (.4) entails that (3.4) follows roof of roposition 3.5 We apply Corollary.4 to the random variables X i (.) {t Yi t F Y (t), t R}. By Hölder s inequality for any k, we get E(X k (t) F ) E( Yk t F ) (Y k t) E( Yk t F ) (Y k t) E(Yk t F ) (Y k t) α(k). Using (4.8) and (4.9), E(X k (t) F ) dt α(k) FY (t)( F Y (t))dt R α(k) FY (t)dt + Notice that α(k) ( Y > t)dt. α(k) FY (t)dt α(k) ( Y > t) u ( Y >t) u α(k) du α(k) u ( Y >t) du α(k) Q Y (u ) t du. By Fubini and by a change of variable, we derive + ( ) α(k) ( Y > t)dt u ( Y >t) u α(k) du dt α(k) ( ) Q Y (u ) t dt du (4.9)

19 35 S. Dede / Stochastic rocesses and their Applications 9 (9) α(k) Q Y (u )du α(k) Q Y (u) du. u We deduce that (3.5) implies (.5). he CL holds, by applying Corollary roof of Corollary 3.7 Recall that to prove the convergence for the empirical distribution function of Y where (Y k ) is defined by (3.7), it suffices to show the convergence in L (λ) of the empirical distribution function of f (Z) where (Z i ) i Z is the stationary Markov chain with transition Kernel K. Hence we shall prove that f (Z) satisfies the conditions of roposition 3.. Since f is piecewise monotonic with a finite number of branches, say N, then f (], t]) is the union of N disjoints sets, which are either an interval or the complement of an interval. Consequently, φ(f, f (Z k )) N φ(f, Z k ). In addition, Dedecker and rieur proved that if (3.8) holds then φ(f, Z k ) C ρ k, with C a positive constant (see [9]). his entails that (3.3) holds roof of Corollary 3.9 roceeding as the beginning of the proof of Corollary 3.7, we have this time to bound α(f, f (Z k )). Once again, since f is piecewise monotonic with a finite number of branches, say N, then f (], t]) is the union of N disjoints sets, which are either an interval or the complement of an interval implying that α(f, f (Z k )) N α(f, Z k ). Recently, Dedecker, Gouëzel and Merlevède [6] proved in roposition., that there exists a constant C γ, such that, for any positive integer k, α(f, Z k ) It follows that C γ (k + ) γ γ α(k) ν γ ( f > t)dt N. (4.) Consequently, (3.6) holds as soon as (3.9) holds roof of Remark 3. Cγ (k + ) γ γ ν γ ( f > t)dt. Since the density g νγ of ν γ is such that g νγ (x) V γ x γ, we infer that V γ ν γ ( f > t) D γ a γ t γ a, (4.)

20 S. Dede / Stochastic rocesses and their Applications 9 (9) where D and V γ are positive constants. By Fubini and (4.), we then get that + ( k γ ) γ ν γ ( f > t)dt u νγ ( f >t) u k γ du dt γ ( K (γ, a)k ( γ ) γ where K (γ, a) is a constant. Consequently, (3.9) holds as soon as a < γ does roof of Corollary 3. )( a γ + ), We apply heorem.3 to the random variables X k (t) {t Yk t F Y (t)}. Let M i σ (ε k, k i). By the proof of Lemma 6 in Dedecker and Merlevède [7], F Yk M (t) F Yk M (t) K a a k ε ε. (4.) Moreover, we have F Yk M (t) F Yk M (t) F Yk M (t) F Y (t) + F Yk M (t) F Y (t) F Y (t)( F Y (t)). We deduce that (.) holds as soon as (K a a k ε ε ) ( ( Y k > t))dt <, and it may be reduced to a k ( Y k > t)dt < ak Q Yk (u) u du <. Now, from heorem.3, we infer that n(f n F Y ) converges in law to an L (λ)-valued centered Gaussian random variable G, with covariance operator Φ G defined by (3.4). 4.. roof of Corollary roof of Item of Corollary 3.3 o apply Corollary 3., it suffices to prove (3.). Firstly, recall that, if U is a uniform random variable on [, ], Q ε (U) and ε have the same law. We proceed as in Rio [9] p. 5. By Hölder s inequality on [, ], it follows that k ak Q ε (u) du u ( ) /r Q ε (u) r du Q ε (u) {u ak } k u {u ak } k u r/(r ) du (r )/r.

21 354 S. Dede / Stochastic rocesses and their Applications 9 (9) Using the same notations as in Dedecker and Doukhan [5], let δ (u) k {u ak } and f (x) x r/(r ). We infer that f (δ ) ( f ( j + ) f ( j)) {u a j } j (( j + ) r/(r ) j r/(r ) ) {u a j }. j Set C r ( r r ). Since ( j + )r/(r ) j r/(r ) C r ( j + ) /(r ), (3.) holds as soon as ( j + ) /(r ) {u a j } u (r ) r du <, j which is true provided that ( j + ) /(r ) a j r r <. j 4... roof of Item of Corollary 3.3 We apply Corollary 3., so it suffices to prove (3.). Notice that, the quantile function Q ε, here, is dominated by cu /r. hus, we derive ak Q ak ε (u) c du u u c r r a k /r. /+/r du Consequently, (3.) holds as soon as (3.5) does. Acknowledgments I would like to thank Jérôme Dedecker and Florence Merlevède for useful discussions. I also wish to thank the referee for his valuable advices and suggestions which helped me greatly in the writing of the revised revision. References [] A. de Acosta, A. Araujo, E. Giné, On poisson measures, gaussian measures and the central limit theorem in Banach spaces, in: robability on Banach Spaces, Adv. robab. Related opics. 4 (978) 68. [] A. Araujo, E. Giné, he central limit theorem for real and Banach valued random variables, in: Wiley Series in robability and Mathematical Statistics, John Wiley and Sons, New-York, Chichester, Brisbane, 98. [3]. Billingsley, he Lindeberg Lévy theorem for martingales, roc. Amer. Math. Soc (5) (96)

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