A Duclos

D Lafarge

V Pagneux

Transmission of acoustic waves through 2D phononic crystal : visco-thermal and multiple scattering effects

Keywords: numbers: 43, 20, +g 43, 35, +d 63, 20, -e

In this paper, we are interested in the transition between regimes where either visco-thermal or multiple scattering effects dominate for the propagation of acoustic waves through a 2D regular square array of rigid cylinders embedded in air. An extension of the numerical method using Schlömilch series is performed in order to account for visco-thermal losses. Comparison with experimental data and results from classical homogenization theory allows to study the transition between a low frequency limit (where viscous and thermal effects dominate) and a high frequency regime (where multiple scattering effects become predominant). For this particular geometry, a large frequency domain where visco-thermal and multiple scattering effects coexist is found.

I. INTRODUCTION

A precise description of sound propagation and absorption in phononic crystal for a large frequency domain is of great interest in order to study transition between low and high frequency. At low frequencies, i.e. when the viscous and thermal skin depths are comparable to the crystal characteristic dimension, important dissipation effects occur. These effects are well described by the long wavelength homogenization theories [1][2][3][START_REF] Allard | Propagation of Sound in Porous Materials[END_REF][START_REF] Lafarge | [END_REF][6]. Simple fluid dynamics problems are to be solved: a Stokes unsteady fluid flow problem for the velocity, and a Fourier unsteady heat conduction problem for the temperature. Studies describing the propagation by the homogenization theory can be found in literature [7,8]. When the frequency increases enough, such separation no longer is possible and the problem transforms into a full wave propagation problem with multiple scattering effects becoming predominent. These effects may lead to band gaps (frequency regions where the propagation is forbidden). Different existing numerical methods allowing to determine these band gaps can be found in the literature [9][10][11][12][13][14]. For the purpose of studying the transition between low and high frequency regime we first extend the numerical method using Schlömilch series [15][16][17] to include visco-thermal effects. As a result, a numerical method is obtained which takes into account viscous, thermal and multiple scattering effects. Then, a comparison with experimental data (high frequencies) on one hand and with homogenization theory results (low frequencies) on the other hand is performed. Finally the transition between low and high frequencies is discussed.

II. THEORY

We consider sound propagation and absorption through a 2D periodic square arrangement of rigid cylinders embedded in air for a large frequency domain. This domain includes important multiple scattering as well as visco-thermal effects. The three geometries represented on figure 1 are considered. The first problem is a row containing an infinite number of cylinders periodically spaced according to y axis and insonified by (right and left) ingoing plane waves. The second one is an arrangement of N rows of cylinders with the same periodicity according to x and y axis and subject to the same ingoing plane waves. In a last problem, we consider the case of an infinite periodic medium called "phononic crystal". For the cases (a) and (b), we are interested in the reflection and transmission properties of the samples, and for the infinite phononic crystal (c), we study the dispersion relation of the Floquet-Boch modes. The solution of the problems (b) and (c) is based on extensions of the solution of the problem (a). In order to account for visco-thermal and multiple scattering effects, two descriptions are introduced. Firstly, we consider an equivalent surface admittance surrounding each cylinder (method A). Using this concept, previously described by Tournat et al. [18], weak dissipation localized in the cylinder boundary layers can be considered. Secondly, a complete description allowing for losses (localized or not) in all the fluid is presented (method B). Analytically, the two problems are formulated in terms of the displacement u and the (excess) temperature τ fields. In the first case (method A), these fields derive from an acoustic scalar potential φ a ,

     u = -→ ∇ φ a τ = ω 2 C p φ a , (1)
which satisfies the 2D Helmholtz equation [START_REF] Morse | Theoretical Acoustics[END_REF] (convention e -iωt ),

(∆ + k 2 a )φ a = 0 , k 2 a = (ω/c 0 ) 2 (2)
where ω is the angular frequency and c 0 is the adiabatic sound speed. The visco-thermal effects occuring in the boundary layers are considered in the following boundary condition set at cylinder surfaces,

(∂ n + ik a β) φ a = 0 , (3)
where ∂ n is the normal derivative and β is the equivalent surface admittance operator which depends notably on fluid properties (kinematic viscosity and thermal diffusivity). Expression of this operator (in the limit of small boundary layers) can be found in [18] for the 2D cylindrical case.

In the second case (method B), the displacement and temperature fields can be related ( [START_REF] Pierce | Acoustics, An Introduction to Its Physical Principles and Applications[END_REF]) to three potentials : acoustic φ a , entropic φ h and vorticity ψ,

     u = -→ ∇ (φ a + φ h ) + -→ ∇ ∧ ψ τ = ω 2 C p φ a - iω β 0 ν h φ h , (4)
where β 0 is the coefficient of thermal expansion, T 0 the temperature, C p the specific heat at constant pressure and ν h the thermal diffusivity. All these parameters depend on the fluid nature. The three potentials (we set ψ = φ v e z ) obey to the 2D Helmholtz equations ( [START_REF] Pierce | Acoustics, An Introduction to Its Physical Principles and Applications[END_REF]),

         (∆ + k 2 a )φ a = 0 , k 2 a = (ω/c 0 ) 2 (∆ + k 2 h )φ h = 0 , k 2 h = iω/ν h (∆ + k 2 v )φ v = 0 , k 2 v = iω/ν v , (5)
where ν v is the kinematic viscosity. The real wavenumber k a is related to acoustic propagation while the complex wavenumbers k v and k h are related to viscous and thermal waves respectively. The three potentials are coupled at the cylinders boundaries by the following conditions,

   u = 0 τ = 0 , (6)
cancelling displacement and temperature fields.

Having in view the determination of the Floquet-Bloch modes (in the x direction of the phononic cristal), we will in methods A and B expand the potentials in terms of right and left going plane waves,

φ pot ( r) = ∞ n=-∞ A + pot,n e i kpot,n. r + A - pot,n e -i kpot,n. r , (7)
where the "pot" index is either a (for the acoustic potential), h (for the entropic potential) or v (for the vorticity potential), and the index n determines the possible wavevectors through ( k 2 pot,n = k 2 pot ) and the condition ( k pot,n ) y L = 2πn which must be satisfied in order to respect the y periodicity. The amplitudes of all potentials are merged into coefficients A in figure (1). To obtain the reflection and transmission properties of one row (case (a)), the scattering matrix, which relates the outgoing waves to the ingoing ones, is introduced,

  A - 0 A + L   =   [T] [R] [R] [T]     A - L A + 0   , (8)
where [R] and [T] are the reflection and transmission matrices. To determine theirs elements

r pot pot ′ nn ′ and t pot pot ′ nn ′
, the outgoing potentials are expressed in two different forms, on each side of the row for a given ingoing potential φ ext ( r, n ′ , pot ′ ) = e i k pot ′ ,n ′ . r . The outgoing potentials can be interpreted as the reflected and transmitted fields by the row,

           φ R pot,0 ( r) = ∞ n=-∞ r pot pot ′ nn ′ e -i kpot,n. r φ T pot,L ( r) = ∞ n=-∞ t pot pot ′ nn ′ e i kpot,n. r . ( 9
)
These can also be interpreted in term of the potential fields scattered by all cylinders belonging to the row,

           φ R pot,0 ( r) = ∞ n=-∞ ∞ j=-∞ C pot pot ′ nn ′ i m H m (k pot | r -r j |)e imΦ r-r j φ T pot,0 ( r) = ∞ n=-∞ ∞ j=-∞ D pot pot ′ nn ′ i m H m (k pot | r -r j |)e imΦ r-r j + φ ext ( r, n ′ , pot ′ ) , (10)
where the coefficients

C pot pot ′ nn ′ and D pot pot ′ nn ′
are independent of the scatterer (their determination is performed numerically according to [START_REF] Chen | [END_REF] for instance). At this point, we must estimate Schlömilch series of the form ∞ j=-∞ i m H m (k pot | r -r j |)e imΦ r-r j . For the usual case k pot = k a defined in relation [START_REF] Lafarge | [END_REF] with k a ∈ R, and due to the slow convergence of Schlömilch series, these sums are developed in the form [15][16][17] which present a better convergence. For the more original cases k pot = k h and k pot = k v also defined in relation [START_REF] Lafarge | [END_REF] and having non zero Im(k h ) and Im(k v ), the Schlömilch series are naturally convergent. Then an identification between the two couples of equations ( 9) and (10)

- 0 A + N L   =   [T N ] [R N ] [R N ] [T N ]     A - N L A + 0   , (11)
where [R N ] and [T N ] are the reflection and transmission matrices for N rows. These two matrices can be related to [R] and [T] using the following recurrence relations,

   [T N ] = [T] [I d ] -[R N -1 ][R] -1 [T N -1 ] [R N ] = [R] + [T] [I d ] -[R N -1 ][R] -1 [R N -1 ][T] , (12)
where [I d ] is the identity matrix. Using these relations, the reflection and transmission properties of N rows are simply deduced from the two known matrices [R] and [T].

The phononic crystal (case (c)) is interpreted as a superposition of an infinite number of rows. Taking an arbitrary row, the scattering matrix concept (relation ( 8)) remains valid.

Moreover, the potentials obey the Bloch condition [START_REF] Brillouin | Wave Propagation in Periodic Structures[END_REF],

  A + L A - L   = e ik B L   A + 0 A - 0   , (13)
where the k B are the Bloch wavenumbers to determine. The simultaneous use of the two relations ( 8) and ( 13) leads to the following eigenvalue problem,

  [T] [R] [0] [I d ]     A + 0 A - L   = e ik B L   [I d ] [0] [R] [T]     A + 0 A - L   , (14)
where [0] is the zero matrix. As the reflection and transmission matrices [R] and [T] are already known, the dispersion relation giving the solution for the Bloch wavenumbers k B can be solved numerically.

III. EXPERIMENTAL AND THEORETICAL RESULTS

A first comparison is performed at high frequencies (acoustic wavelength comparable or less than the lattice parameter L, k a L/π ≥ 1) corresponding to a regime governed by important multiple scattering effects. The two models described above allow to compute the scattering matrix (8) of one row and then, using equation (12), to predict the transmission coefficient of a sample containing a finite number of rows at normal incidence. This transmission coefficient can be experimentally obtained. Here, two samples of aluminium rods embedded in air in square configuration (as presented in figure 1 (b)) are used. The radius of the cylinders is fixed at R=1 mm. The lattice parameter of the sample is L=3 mm. The two samples contain respectively 3 and 6 rows. Each of them is 100 cylinders wide. The experiment is carried out at ultrasonic frequencies using a couple of broadband transducers (centered on 100 kHz) with radius 20 mm. The experimental results are reported on figure 2.

When the number of rows increases, the transmission coefficient tends towards zero inside frequency domains corresponding to band gaps defined for infinite phononic crystals. We have also reported the predictions of the two models A and B, which are undistinguishable, and the prediction for the ideal fluid (i.e. β = 0 in (3)). The band gap locations are well described in all cases because they are essentially due to multiple scattering effects only.

However, outside these bands, an accurate prediction of the transmission coefficients needs taking into account the visco-thermal losses. The two visco-thermal descriptions lead to the same predictions because the dissipation effects occuring during the propagation are entirely localized in the small boundary layers. Discrepancies between the ideal and visco-thermal fluid models are solely due to these weak losses.

A comparison between our computations and the low frequency (i.e. long wavelength)

model is now performed when the dissipation effects become more important. Previous numerical studies based on homogenization theory (limited to leading order terms) and using finite element methods to solve the problem of long wavelength sound absorption in 2D phononic crystal (figure 1 case (c)) have been performed by Cortis [7]. The principle of these long wavelength simulations is to solve the unsteady Stokes equation for velocity and the unsteady Fourier equation for temperature. It exploits in consistent manner the fact that (to leading order) the velocity field becomes divergence-free at the pore scale (for the purpose of evaluating viscous effects) and the pressure field becomes uniform (for the purpose of evaluating thermal effects). In order to compare these existing homogenization results to those given by the two visco-thermal models including multiple scattering, we compare the real part of our complex velocity V associated to the phononic crystal with that given by Cortis results. This quantity can be expressed from the real part of the Bloch wavenumber radius is R=1 mm and lattice periodicity is L=3 mm (thus porosity is Φ = 0.65). We have reported the theoretical predictions with (-) and without (--) dissipation effects and the experimental data (+). frequency limit, the two visco-thermal models tends towards Cortis results. Consequently and for the geometrical dimension considered here, our two models successfully fulfill the two asymptotic limits. This is because in the low frequency regime considered here, the dissipation effects are still localized in small boundary layers. When the frequency increases, multiple scattering effects emerge and, as expected, the (leading order terms) classical homogenization results are no more able to predict the correct velocity. Here, the real part of the velocity decreases and tends towards zero quickly due to the first band gap where the propagation is forbidden. At this point, we can precisely study the transition between a long wavelentgh visco-thermal description of sound propagation and the emergence of multiple scattering effects. The limit of the classical homogenization theory for a simple 2D phononic crystal of 0.90 porosity case appears near k a L/π = 0.2. The same limit was found for other porosities decreasing from 0.90 to 0.50. Description of sound absorption using classical homogenization theory is not accurate beyond this frequency limit. In the presented case, the dissipation effects are very weak (1% modification on phase velocity) and localized in very small boundary layers. Then, the transition description can be performed by the equivalent surface admittance concept (method A) as well as the complete description (method B). Now, we translate the frequency values and we consider a 1 µm radius cylinders configuration. This scale law allows us to consider the same multiple scattering effects but more important dissipation. Associated results are reported on figure 4 in the transition domain where both visco-thermal and multiple scattering effects are significant. As expected, the observed limit for classical homogenization theory, due to multiple scattering emergence, remains unchanged. In this second case, our equivalent surface admittance concept cannot describe the transition due to important visco-thermal effects (about 20% modification on phase velocity). For this geometry, the transition shall be described by considering the complete multiple scattering calculations. Besides, we know that in the high frequency limit, the velocity V (ω) given by the homogenization theory reaches an asymptotic value which is related to the tortuosity α ∞ [1] (geometrical characteristic of porous media) with the following relation,

lim ω→∞ V (ω) = c 0 √ α ∞ . (15)
This high frequency limit in view of long wavelength theory is just that matched by the ideal fluid results, in view of multiple scattering theory, in the low frequency limit (absence of multiple scattering effects). Corresponding values are indicated on figures 3 and 4 by the top horizontal line. The tortuosity values obtained with the ideal fluid model (at the low frequency limit) are in very good agreement with those from literature [7,[START_REF] Perrins | Proc. R. Soc. Lond. A[END_REF]. However, these values are never reached when considering both visco-thermal and multiple scattering effects as illustrated on figures 3 and 4. This observation confirms the existence of a large frequency domain (from k a L/π = 0.2 for this example) where viscous, thermal and multiple scattering effects coexist. In this region, sound propagation and absorption can only be described by taking into account all these effects together.

IV. CONCLUSION

In summary, we have extended an existing model describing the multiple scattering effects on 2D phononic crystal (using Schlömilch series) by including viscous and thermal effects.

The main results consists in the existence of a large frequency domain where both multiple scattering and visco-thermal effects shall be taken into account. For an usual millimetric geometry surrounded by air, dissipation effects are very weak in the transition domain. Thus the description can be performed using the simple equivalent surface admittance concept considered in [18] (including the visco-thermal effects in the small boundary layers near the cylinders). However, for a micrometric geometry, this simple description becomes inaccurate due to the fact that the boundary layers are not small. In this paper, this case was successfully described by a general complete multiple scattering calculation. However, the studied transition will also be reproduced by an extended addmittance approach (removing the assumption of small boundary layers) as long as the viscous and thermal boundary layers are not overlapping between the different cylinders. Furthermore, we have shown that we can precisely recover the high frequency limit of the classical homogenization results with our complete multiple scattering calculation in the given 2D phononic crystal.

FIG. 1 :

 1 FIG. 1: Geometries under consideration. (a) one row containing an infinite number of cylinders, (b) n-rows containing an infinite number of cylinders, and (c) infinite phononic crystal.

  lead to the required reflection and transmission coefficients. Consequently the reflection and transmission properties of one row (case (a)) are entirely determined. Now we consider the case (b) of N rows (a) separated by the distance L. The outgoing and incoming wave amplitudes are related by a new scattering matrix,   A

kFIG. 2 :

 2 FIG. 2: Transmission coefficients of two samples containing 3 rows (a) and 6 rows (b) versus reduce frequency k a L/π (corresponding to a frequency range going from 20 kHz to 200 kHz). Cylinder

FIG. 3 :

 3 FIG.3: Real part of phase velocity versus reduce frequency k a L/π. Phononic crystal made up of 1 mm radius cylinders (porosity is Φ = 0.90). We have reported the theoretical predictions considering visco-thermal effects (+) obtained by Cortis, multiple scattering (--), equivalent surface admittance concept (∆) and visco-thermal as well as multiple scattering effects (-). The two visco-thermal models considering multiple scattering effects can describe the transition between low and high frequency regions. Indeed, in this present case, weak dissipation effects are localized at cylinder boundary layers.

FIG. 4 :

 4 FIG.4: Real part of phase velocity versus reduce frequency k a L/π. Phononic crystal made up of 1 µm radius cylinders. In this second geometry, the transition can only be described considering important visco-thermal effects not localized at cylinder boundary layers.


Marie Sauvé
email: marie.sauve@math.u-psud.fr

HISTOGRAM SELECTION IN NON GAUSSIAN REGRESSION

Keywords: 1991 Mathematics Subject Classification. 62G08, 62G05. CART, change-points detection, deviation inequalities, model selection, oracle inequalities, regression

We deal with the problem of choosing a piecewise constant estimator of a regression function s mapping X into R. We consider a non Gaussian regression framework with deterministic design points, and we adopt the non asymptotic approach of model selection via penalization developed by Birgé and Massart. Given a collection of partitions of X , with possibly exponential complexity, and the corresponding collection of piecewise constant estimators, we propose a penalized least squares criterion which selects a partition whose associated estimator performs approximately as well as the best one, in the sense that its quadratic risk is close to the infimum of the risks. The risk bound we provide is non asymptotic.

Introduction

We consider the fixed design regression framework. We observe n pairs (x i , Y i ) 1≤i≤n , where the x i 's are fixed points belonging to some set X and the Y i 's are real valued random variables. We suppose that:

Y i = s(x i ) + ε i , 1 ≤ i ≤ n, (1)
where s is an unknown function mapping X into R, and (ε i ) 1≤i≤n are centered, independent and identically distributed random perturbations. Our aim is to get informations on s from the observations (x i , Y i ) 1≤i≤n .

In order to get a simple estimator of s, we consider a partition M 0 of X with a large number of small cells containing at least one point x i , and we minimize the least squares contrast over the class S M0 of piecewise constant functions defined on the partition M 0 . The resulting estimator is denoted by ŝM0 and is called the least squares estimator over S M0 . S M0 is called the histogram model associated with M 0 . It is a linear space with finite dimension D M0 = |M 0 |, where |M 0 | is the number of cells of the partition M 0 . Denoting . n the Euclidean norm on R n scaled by a factor n -1/2 , and denoting for any function u ∈ R X u n = (u(x i )) 1≤i≤n n , the quadratic risk of ŝM0 , E s -ŝM0 2 n , is the sum of two terms, respectively called bias and variance:

E s -ŝM0 2 n = inf u∈SM 0 s -u 2 n + τ 2 n |M 0 | where τ 2 = E(ε 2 i ).
We see in this expression of the risk of ŝM0 that ŝM0 behaves poorly when M 0 has a too large number of cells and that we should rather choose a partition M built from M 0 which makes a better trade-off between the bias inf u∈SM s -u 2 n and the variance τ 2 n |M |.

In this paper, we propose a procedure to select a partition M (or equivalently a histogram model S M ) according to the data, and we estimate s by ŝ M the least squares estimator over S M . There are already many works on data-driven histogram selection not only for regression [START_REF] Breiman | Classification And Regression Trees[END_REF][START_REF] Catoni | Universal aggregation rules with sharp oracle inequalities[END_REF][START_REF] Lebarbier | Quelques approches pour la détection de ruptures à horizon fini[END_REF][START_REF] Nobel | Histogram regression estimation using data-dependent partitions[END_REF] but also for classification [START_REF] Breiman | Classification And Regression Trees[END_REF][START_REF] Lugosi | Consistency of data-driven histogram methods for density estimation and classification[END_REF] and density estimation [START_REF] Birgé | How many bins should be put in a regular histogram[END_REF][START_REF] Castellan | Modified Akaike's criterion for histogram density estimation[END_REF][START_REF] Catoni | Universal aggregation rules with sharp oracle inequalities[END_REF][START_REF] Lugosi | Consistency of data-driven histogram methods for density estimation and classification[END_REF]. We focus here on the regression framework and we adopt the non asymptotic approach of model selection via penalization developed by Birgé and Massart. Our estimation procedure is as follows. We consider a collection M n of partitions of X all built from an initial partition M 0 , and the corresponding collection (S M ) M∈Mn of histogram models. Denoting ŝM the least squares estimator over the model S M , the best model S M * is the one which minimizes E s -ŝM 2 n among (S M ) M∈Mn . Unfortunately this model depends on the unknown function s, and ŝM * can not be used as an estimator of s. Our aim is to find a data driven criterion whose minimizer S M is an approximately best model, and to estimate s by ŝ M . We select a model S M by minimizing over M n a penalized least squares criterion crit(M ) = Y -ŝM The estimator ŝ M is called the penalized least squares estimator. The penalty pen has to be chosen such that the model S M is close to the optimal model, more precisely such that

E s -ŝ M 2 n ≤ C inf M∈Mn E s -ŝM 2 n . (2)
The inequality (2) will be referred to as the oracle inequality. It bounds the risk of the penalized least squares estimator by the infimum of the risks on a given model up to a constant C. The main result of this paper determines a penalty pen for which the associated penalized least squares estimator satisfies an oracle type inequality.

One of the first penalized least squares criterion is Mallows' C p criterion [START_REF] Mallows | Some comments on cp[END_REF] which corresponds to pen(M ) = 2 τ 2 n D M It leads to an oracle type inequality when the number of models with a given dimension D is a polynomial function of D (as proved by [START_REF] Birgé | Gaussian model selection[END_REF][START_REF] Birgé | Minimal penalties for gaussian model selection[END_REF] in the Gaussian case, and by [START_REF] Baraud | Model selection for regression on a fixed design[END_REF] in a very general case), but it can lead to very bad results when the collection of models is more complex (see [START_REF] Birgé | Minimal penalties for gaussian model selection[END_REF]). In this paper, we determine a penalty which allows to deal with large collections of histogram models. The proposed penalty pen(M ) is the sum of two terms: the first one is proportional to DM n = |M| n and the second one depends on the complexity of the collection M n . It has the same form as the penalty obtained by Birgé and Massart [START_REF] Birgé | Minimal penalties for gaussian model selection[END_REF] in the Gaussian case and those obtained by Baraud, Comte and Viennet [START_REF] Baraud | Model Selection for (auto-)regression with dependent data[END_REF] in the sub-Gaussian case, for any collections of models (not only for histogram models). But we do not assume the (ε i ) 1≤i≤n to be Gaussian nor sub-Gaussian, we only suppose that they have exponential moments around 0.

From a technical point of view, we follow the same ideas as [START_REF] Baraud | Model selection for regression on a fixed design[END_REF][START_REF] Birgé | Minimal penalties for gaussian model selection[END_REF]. In this paper like in [START_REF] Baraud | Model selection for regression on a fixed design[END_REF][START_REF] Birgé | Minimal penalties for gaussian model selection[END_REF], in order to get an adequate penalty, the main point is to control the random variables

χ 2 M = ε M 2 n where ε M = arg min u∈SM ε -u 2 n
for all M ∈ M n simultaneously. The difference between our work, [START_REF] Baraud | Model selection for regression on a fixed design[END_REF], and [START_REF] Birgé | Minimal penalties for gaussian model selection[END_REF], is located in the construction of sharp deviation inequalities for the (χ 2 M ) M∈Mn , which remain sharp when summing them over all M ∈ M n . In the Gaussian case (handled in [START_REF] Birgé | Minimal penalties for gaussian model selection[END_REF]), the n τ 2 χ 2 M 's are χ 2 distributed. But in the non Gaussian case, it is much more difficult to study the deviations of these variables around their expectations. Under a mild integrability condition on the (ε i ) 1≤i≤n (assuming that E (|ε i | p ) < +∞ for some p ≥ 2), Baraud [START_REF] Baraud | Model selection for regression on a fixed design[END_REF] gives a polynomial deviation inequality for the χ 2 M 's. This inequality allows him to prove that penalties pen(M ) = K τ 2 n D M , with K > 1, lead to oracle type inequalities when the number of models with a given dimension D is a polynomial function of D. In order to deal with bigger collections of models, we need exponential deviation inequalities for the χ 2 M 's. By writing χ M = sup u∈BM < ε, u > n , with B M = {u ∈ S M ; u n ≤ 1}, we can apply Bousquet's exponential concentration inequality for a supremum of an empirical process [START_REF] Bousquet | Concentration Inequalities for Sub-Additive Functions Using the Entropy Method[END_REF]. Unfortunately this general result is not sufficient here. Instead of viewing χ M as a supremum, we can view χ 2 M as a χ 2 like random variable and write it as a sum of squares. For histogram models, we can then build adequate exponential deviation inequalities by hand, using only Bernstein's inequality. This is the reason why we determine a penalty which we prove to lead to an oracle inequality only for histogram models.

Thanks to this penalty, given a collection M n of partitions, we get an estimator ŝ M which is simple, easy to interpret and close to the optimal one among the collection of piecewise constant estimators (ŝ M ) M∈Mn . Unfortunately, since the estimators (ŝ M ) M∈Mn are sharply discontinuous, even the best one may not provide an accurate estimation of s.

Histogram model selection may not lead to an accurate estimation of the regression function s, but it has important applications to clustering and it enables to detect the change-points of s. In the framework [START_REF] Baraud | Model selection for regression on a fixed design[END_REF] with X = [0, 1] and x i = i n , in order to detect the change-points of s, Lebarbier [10, chapter 2] considers the collection M n of all partitions with endpoints belonging to the grid (x i ) 1≤i≤n , and the corresponding collection (S M ) M∈Mn of histogram models. Then, she selects a partition M among M n by minimizing a penalized least squares criterion. For this collection

M n , |{M ∈ M n ; |M | = D}| = n -1 D -1
, therefore Baraud's result [START_REF] Baraud | Model selection for regression on a fixed design[END_REF] does not apply to this case, and penalties proportional to |M| n are not sufficient. Assuming the perturbations ε i to be Gaussian, Lebarbier applies the model selection result of Birgé and Massart [START_REF] Birgé | Gaussian model selection[END_REF] to the collection (S M ) M∈Mn . She gets a penalty defined up to two multiplicative constants. Then she proposes a method to calibrate them according to the data and therefore gives a procedure to automatically detect the change-points of a Gaussian signal according to the data. Thanks to our result, this procedure can be extended to detect the change-points of a regression function without assuming the perturbations ε i to be Gaussian.

One of the most famous statistical issues is variable selection. In the classical linear regression framework:

Y i = p j=1 β j x j i + ε i , 1 ≤ i ≤ n,
selecting a small subset of variables V ⊂ {x 1 , . . . , x p } which explain "at best" the response Y is equivalent to choosing the "best" model S V of functions linear in {x j ∈ V }. Instead of considering linear interaction between x 1 , . . . , x p and Y , we can use histogram models. First, we associate one or more histogram models to each subset of variables. Then, we get a large collection of models (at least 2 p models), among which we select one by minimizing a penalized least squares criterion. And finally, we keep the subset of variables which is associated with the selected model. This idea is used by Sauvé and Tuleau [START_REF] Sauvé | Variable selection through CART[END_REF], who propose a variable selection procedure based on histogram model selection.

The CART algorithm (Classification And Regression Trees), proposed by Breiman et al. [START_REF] Breiman | Classification And Regression Trees[END_REF], involves histogram models. Our result allows to validate the pruning step of CART in a non Gaussian regression framework.

The paper is organized as follows. The section 2 presents the statistical framework and some notations. The section 3 gives the main result. To get this result, we have to control a χ 2 like random variable. The section 4 is more technical, it exposes a deviation inequality for a χ 2 like random variable and explains why the existing deviation inequality, due to Bousquet, is not sufficient. Sections 5 and 6 are devoted to the proofs.

The statistical framework

In this paper, we consider the regression framework defined by (1) and we look for a best or approximately best piecewise constant estimator of s. In this section, we precise the integrability condition that should satisfy the random perturbations (ε i ) 1≤i≤n involved in (1), then we define the piecewise constant estimators of s and their risk. We give here some notations needed in the rest of the paper.

The random perturbations

As noted above in the introduction, we assume that the random perturbations (ε i ) 1≤i≤n have finite exponential moments around 0. This assumption is equivalent to the existence of two constants b ∈ R + and σ ∈ R * + such that

∀ λ ∈ (-1/b, 1/b) log E e λεi ≤ σ 2 λ 2 2(1 -b|λ|) (3)
σ 2 is necessarily greater than E(ε 2 i ) and can be chosen as close to E(ε 2 i ) as we want, but at the price of a larger b.

Remark 2.1. Under assumption (3), we have

∀ λ ∈ (-1/2b, 1/2b) log E e λεi ≤ σ 2 λ 2
but we prefer inequality (3) to this last inequality because with the last one we loose a factor 2 in the variance term.

Remark 2.2. Thanks to assumption (3) and Cramer-Chernoff method (see [13, section 2.1]), we can easily get deviation inequalities for any linear combination of the (ε i ) 1≤i≤n . First, since the (ε i ) 1≤i≤n are independent, we get from inequality (3) similar inequalities for any linear combination

n i=1 α i ε i . Denoting α ∞ = max 1≤i≤n |α i | and v = σ 2 n i=1 α 2 i , ∀ λ ∈ 0, 1 b α ∞ log E e λ n i=1 αiεi ≤ vλ 2 2(1 -b α ∞ λ) . (4)
We denote by ψ(λ) the right term of (4) and by h(u) = 1 + u -√ 1 + 2u for any u ∈ R * + . Then, applying Cramer-Chernoff method, since for any x > 0

sup 0<λ< 1 b α ∞ {λx -ψ(λ)} = v b 2 α 2 ∞ h b α ∞ x v ,
we get for any x > 0:

P n i=1 α i ε i ≥ x ≤ exp - v b 2 α 2 ∞ h b α ∞ x v .
Finally we deduce the two following inequalities:

• Since h is inversible with h -1 (u) = u + √ 2u, P n i=1 α i ε i ≥ √ 2vx + b α ∞ x ≤ e -x . • Since h(u) ≥ u 2 2(1+u) , P n i=1 α i ε i ≥ x ≤ exp -x 2 2 (v + b α ∞ x) .

The piecewise constant estimators

For a given partition M of X whose cells contain at least one point x i , we denote S M the space of piecewise constant functions defined on the partition M and ŝM the least squares estimator over S M . ŝM = arg min

u∈SM γ n (u) with γ n (u) = Y -u 2 n = 1 n n i=1 (Y i -u(x i )) 2
where . n denotes the Euclidean norm on R n scaled by a factor n -1/2 , Y = (Y i ) 1≤i≤n , and for u ∈ S M , the vector (u(x i )) 1≤i≤n ∈ R n is denoted by u too. S M is the histogram model associated with M and ŝM is the piecewise constant estimator belonging to S M which plays the role of benchmark among all the estimators in S M .

Let now calculate the quadratic risk of ŝM : E s -ŝM

2 n = E 1 n n i=1 (s(x i ) -ŝM (x i )) 2 .
To this end, we denote by

s M = arg min u∈SM s -u 2 n , ε M = arg min u∈SM ε -u 2
n where ε = (ε i ) 1≤i≤n , |M | the number of elements of the partition M . ŝM , s M and ε M are respectively the orthogonal projections of Y , s and ε on the space S M according to . n . Thanks to Pythagore's equality, we get that:

E s -ŝM 2 n = s -s M 2 n + E ε M 2 n .
For any element J of the partition M , we denote by |J| = |{1 ≤ i ≤ n; x i ∈ J}| and by 1I J :

x → 1 if x ∈ J and 0 if x / ∈ J. Since n |J| 1I J J∈M
is an orthonormal basis of (S M , . n ), we have

ε M 2 n = J∈M ε, n |J| 1I J 2 n = 1 n J∈M xi∈J ε i 2 |J| . (5)
Since (ε i ) 1≤i≤n are centered, independent and identically distributed random variables with E(ε 2 i ) ≤ σ 2 , we get that

E ε M 2 n = E(ε 2 1 ) |M | n ≤ σ 2 |M | n . Therefore E s -ŝM 2 n = s -s M 2 n + E(ε 2 1 ) |M | n ≤ s -s M 2 n + σ 2 |M | n .
Remark 2.3. In the following, the random variable ε M 2 n is denoted by χ 2 M . Thanks to the decomposition (5), we can see χ 2 M as a χ 2 like random variable. If the (ε i ) 1≤i≤n were Gaussian variables with variance τ 2 , then the variables n τ 2 χ 2 M would be χ 2 (|M |)-distributed.

The main theorem

Let M 0 a partition of X and M n a family of partitions of X built from M 0 , i.e. for any M ∈ M n and any element J of M , J is the union of elements of M 0 . In the following theorem, we assume that the initial partition M 0 is not too fine in the sense that the elements of the partition M 0 contain a minimal number of points x i . We measure the fineness of the partition M 0 by the number N min = inf

J∈M0 |J| where |J| = |{1 ≤ i ≤ n; x i ∈ J}|.
The ideal partition M * minimizes the quadratic risk E s -ŝM 2 n over all the partitions M ∈ M n . Unfortunately M * depends on the unknown regression function s and ŝM * can not be used as an estimator of s. The purpose of model selection is to propose a data driven criterion which selects a partition M whose associated piecewise constant estimator ŝ M performs approximately as well as ŝM * in terms of risks. We select a partition M by minimizing a penalized least squares criterion crit(M ) = Y -ŝM It remains to provide a penalty pen such that the partition M is close to the optimal partition, in the sense that the penalized least squares estiamtor ŝ M satisfies an oracle inequality like [START_REF] Baraud | Model Selection for (auto-)regression with dependent data[END_REF]. The following theorem determines a general form of penalty pen which leads to an oracle type inequality for any family of partitions built from a partition M 0 not too fine. We compare our result to those of Birgé and Massart [START_REF] Birgé | Minimal penalties for gaussian model selection[END_REF] and those of Baraud [START_REF] Baraud | Model selection for regression on a fixed design[END_REF], and we study in more detail two particular families of partitions which are too large to apply Baraud's result.

Example 1: We consider X = [0, 1] and a grid on [0, 1] such that there are at least N min points x i between two consecutive grid points. For example, we can take the grid (v j ) 1≤j≤[n/Nmin]-1 with v j = x jNmin . We define M 0 as the partition associated with the whole grid, and M 1 n as the family of all partitions of [0, 1] with endpoints belonging to the grid. M 1 n corresponds to the collection of partitions used by Lebarbier [START_REF] Lebarbier | Quelques approches pour la détection de ruptures à horizon fini[END_REF] to detect the change-points of a Gaussian signal s : [0, 1] -→ R.

Example 2: We consider some set X . We build a partition M 0 by splitting recursively X and the obtained subsets in two different parts as long as each subset contains at least N min points x i . A useful representation of this construction is a tree of maximal depth, called maximal tree and denoted by T max . The leaves of T max are the elements of the partition M 0 . Every pruned subtree of T max gives a partition of X built from M 0 . We denote by M 2 n this second family of partitions. M 2 n corresponds to the family of partitions obtained via the first step of the CART algorithm.

Theorem 3.1. Let b ∈ R + and σ ∈ R * + such that inequality (3) holds. Let M 0 a partition of X such that N min = inf J∈M0 |J| satisfies: N min ≥ (log n) 2 if b = 0 and N min ≥ 1 if b = 0.
Let M n a family of partitions of X built from M 0 and (w M ) M∈Mn a family of weights such that

M∈Mn e -wM ≤ Σ ∈ R * + .
Assume s ∞ ≤ R, with R a positive constant, unless b = 0. Let θ ∈ (0, 1) and K > 2 -θ two numbers.

Taking a penalty satisfying

pen(M ) ≥ K σ 2 n |M | + 8 √ 2(2 -θ) σ 2 n |M |w M + 4(2 -θ) + 2 θ σ 2 n + 4 √ θ Rb n w M we have E s -ŝ M 2 n ≤ 1 1 - √ θ inf M s -s M 2 n + pen(M ) + 1 1 -θ 8(2 -θ) 1 + 8(2 -θ) K + θ -2 + 4 θ + 2 √ θ σ 2 n Σ + 12 1 -θ Rb n Σ +C(b, σ 2 , R) 1I b =0 n(log n) 3
where C(b, σ 2 , R) is a positive constant which depends only on b, σ 2 and R.

This theorem gives the general form of the penalty function

pen(M ) = K σ 2 n |M | + κ 1 (θ) σ 2 n |M |w M + κ 2 (θ) σ 2 n + 4Rb n w M
The penalty is the sum of two terms: the first one is proportional to |M| n and the second one depends on the complexity of the family M n via the weights (w M ) M∈Mn . For θ ∈ (0, 1) and K > 2 -θ, the penalized least squares estimator ŝ M satisfies an oracle type inequality with an additional term tending to 0 like 1/n when n → +∞:

E s -ŝ M 2 n ≤ C 1 inf M s -s M 2 n + pen(M ) + C 2 n
where the constant C 1 only depends on θ, whereas C 2 depends on the integrability condition of (ε i ) 1≤i≤n (via σ 2 and b), on s (via R) unless b = 0, and on the family of partitions (via Σ).

For the two particular families M n quoted above, we calculate adequate weights (w M ) M∈Mn and we get a simpler form of penalty. Before studying these two examples, we compare the general result with those of Birgé and Massart [START_REF] Birgé | Minimal penalties for gaussian model selection[END_REF], those of Baraud, Comte and Viennet [START_REF] Baraud | Model Selection for (auto-)regression with dependent data[END_REF] and those of Baraud [START_REF] Baraud | Model selection for regression on a fixed design[END_REF].

If b can be taken equal to zero in (3), then the variables (ε i ) 1≤i≤n are said to be sub-Gaussian. In this case, we do not need any assumption on the regression function s and the minimal number N min of observations in each element of the partition M 0 is only supposed to be positive. And taking a penalty satisfying

pen(M ) ≥ K σ 2 n |M | + 8 √ 2(2 -θ) σ 2 n |M |w M + 4(2 -θ) + 2 θ σ 2 n w M we have E s -ŝ M 2 n ≤ 2 1 -θ inf M s -s M 2 n + pen(M ) + 1 1 -θ 8(2 -θ) 1 + 8(2 -θ) K + θ -2 + 4 θ + 2 σ 2 n Σ
Up to some small differences in the constants (which can be improved by looking more precisely at the proof), this is the result obtained by Birgé and Massart in the Gaussian case. Using the inequality 2 |M |w M ≤ |M | + w M , we recover the result obtained by Baraud, Comte and Viennet [START_REF] Baraud | Model Selection for (auto-)regression with dependent data[END_REF] in the sub-Gaussian case.

Baraud [START_REF] Baraud | Model selection for regression on a fixed design[END_REF] studies the non Gaussian regression framework as defined in (1) with a milder integrability condition on the random perturbations than ours. For a collection of histogram models (S M ) M∈Mn whose complexity is polynomial, our theorem and those of Baraud both validate penalties pen(M ) proportional to |M |/n through an oracle type inequality with an additional term tending to 0 like 1/n when n → +∞. Thanks to Baraud's result, if |{M ∈ M n ; |M | = D}| ≤ ΓD r for some constants Γ ∈ R * + and r ∈ N, one only needs to assume that the random perturbations have a finite absolute moment of order p > 2r + 6. The minimal admissible value of p increases with the degree r of the polynomial complexity. And, whatever p, having a finite absolute moment of order p seems to be not enough to deal with collections of exponential complexity. Our assumption on the exponential moments is too strong when the complexity is polynomial, but it allows us to propose a general form of penalty which is still valid when the complexity is exponential.

Let now see which form of penalty is adapted to the two collections of partitions quoted above. The complexity of the two corresponding collections of histogram models is exponential, and therefore Baraud's result does not apply to this case.

Example 1: Since M ∈ M 1 n ; |M | = D = D 0 -1 D -1 ≤ eD0 D D
, where D 0 -1 is the number of grid points,

taking w M = |M | a + log D0 |M| with a > 1 leads to M∈M 1 n e -wM ≤ e a-1 -1 -1 ∈ R * + .
We deduce from the above theorem that: taking a penalty

pen(M ) = σ 2 + Rb n |M | α log |M 0 | |M | + β
with α and β big enough, we have

E s -ŝ M 2 n ≤ C 1 (α, β)inf M s -s M 2 n + σ 2 + Rb n |M | log |M 0 | |M | + 1 + C 2 (α, β) σ 2 + Rb n +C(b, σ 2 , R) 1I b =0 n(log n) 3/2
Since σ 2 , b and R are unknown, we consider penalties of the form pen(M ) = |M| n α log |M0| |M| + β and we determine the right constants α and β according to the data by using, for example, the same technique as Lebarbier [START_REF] Lebarbier | Quelques approches pour la détection de ruptures à horizon fini[END_REF]. We get a data driven criterion which selects a close to optimal partition M . The endpoints of the partition M provide estimators of the change points of the signal s.

; |M | = D ≤ 1 D 2(D -1) D -1 ≤ 2 2D D .
Thus taking w M = a|M | with a > 2 log 2, we get M∈M 2 n e -wM ≤ -log 1 -e -(a-2 log 2) ∈ R * + . We deduce from the above theorem that: taking a penalty

pen(M ) = α σ 2 + Rb n |M |
with α big enough, we have

E s -ŝ M 2 n ≤ C 1 (α)inf M s -s M 2 n + σ 2 + Rb n |M | + C 2 (α) σ 2 + Rb n + C(b, σ 2 , R) 1I b =0 n(log n) 3/2
For this second example, we recommend a penalty pen(M ) proportional to |M| n . For such a penalty, the selected model satisfies an oracle inequality with an additional term tending to 0 like 1/n when n → +∞. This result validates the CART pruning step which involves a penalized least squares criterion with pen(M ) = α |M| n . The last step of CART consists in choosing the right parameter α via cross-validation or test sample. Remark 3.2. Theorem 3.1 do not determine completely the right penalty to be used to get an approximately best estimator. It only gives a form of penalty. The multiplicative constants depend on absolute constants θ ∈ (0; 1) and K > 2 -θ (whose optimal values are unknown), and on unknown parameters σ 2 , b and R. In practice the multiplicative constants (α and β in example 1, or α in example 2) are determined according to the data using for example one of the methods developped in [START_REF] Lebarbier | Quelques approches pour la détection de ruptures à horizon fini[END_REF] or in [15, sections 4.4 and 4.5] Remark 3.3. In theorem 3.1, the assumption "all partitions M ∈ M n are built from some initial partition M 0 " allows to deal with large collections of partitions M n , but is useless for collections M n satisfying |{M ∈ M n ; |M | = D}| ΓD r with Γ ∈ R * + and r ∈ N. (For more details see remark 6.2 at the end of the proof of theorem 3.1.) Such collections have already been handled by Baraud [START_REF] Baraud | Model selection for regression on a fixed design[END_REF] under a mild integrability condition on the ε i and without assuming s ∞ ≤ R. Our result allows to recover Baraud's result (under stronger assumptions), but is only interesting for larger collections.

Remark 3.4. In this paper, the points (x i ) 1≤i≤n of the design are deterministic. If the points of the design were random points (X i ) 1≤i≤n , then with the same approach, working first conditionnally to (X i ) 1≤i≤n ,we would get a similar result. This work is done in a second paper [START_REF] Sauvé | Variable selection through CART[END_REF].

Two deviation inequalities

As will be seen in the proof of theorem 3.1 (via equality [START_REF] Lebarbier | Quelques approches pour la détection de ruptures à horizon fini[END_REF]), in order that the penalized least squares estimator ŝ M satisfies an oracle type inequality, the penalty pen(M ) has to compensate the deviations of the random variables:

χ 2 M = ε M 2 n and ε, s -s M n
for all partitions M ∈ M n simultaneously, without being too large. Therefore we need sharp deviation inequalities for these random variables. This section is composed of two lemmas, which give appropriate deviation inequalities for respectively ε, s -s M n and χ 2 M . First, in lemma 4.1, we give the deviation inequality for ε, s -s M n , which is the most easy to obtain. Then we recall two known deviation inequalities for χ 2

M . The first one is only valid if the ε i 's are Gaussian variables. The second one is very general and is due to Bousquet [START_REF] Bousquet | Concentration Inequalities for Sub-Additive Functions Using the Entropy Method[END_REF], but unfortunately it is not sharp enough. Finally, in lemma 4.2, we give our own deviation inequality for χ 2 M . The proof of lemma 4.2 is quite long and thus postponed to section 5.

Let begin with the deviation inequality for ε, s -s M n : Lemma 4.1. Let b ∈ R + and σ ∈ R * + such that inequality (3) holds. For any partition M and for any x > 0

P ± < ε, s -s M > n ≥ σ √ n s -s M n √ 2x + b n max 1≤i≤n |s(x i ) -s M (x i )| x ≤ e -x ,
and if s ∞ ≤ R then

P ± < ε, s -s M > n ≥ σ √ n s -s M n √ 2x + 2Rb n x ≤ e -x .
Proof. The first inequality is obtained by applying Cramer-Chernoff method as explained in remark 2.2.

If s ∞ ≤ R then max 1≤i≤n |s(x i ) -s M (x i )| ≤ 2R
, and thus we get the second inequality.

It remains to study the deviations of the random variable χ 2 M around its expectation. If the perturbations (ε i ) 1≤i≤n were Gaussian variables with variance τ 2 , then the variable n τ 2 χ 2 M would be χ 2 (|M |)-distributed (see remark 2.3). Thus, according to [4, lemma 1], χ 2 M would satisfy for any x > 0 the following deviation inequality:

P χ 2 M ≥ τ 2 n |M | + 2 τ 2 n |M |x + 2 τ 2 n x ≤ e -x . (6)
Let remark that the term which is linear in x: 2 τ 2 n x, does not depend on M . (We will see in the proof of theorem 3.1 that this property is necessary to define a penalty which compensates the deviations of the χ 2 M for all M ∈ M n simultaneously. ) The square root of χ 2 M : χ M , would then satisfy

P χ M ≥ τ √ n |M | + τ √ n √ 2x ≤ e -x . (7)
Recall that χ M = ε M n where ε M is the orthogonal projection of ε = (ε i ) 1≤i≤n on S M (more pecisely on {u(x i ) 1≤i≤n ; u ∈ S M }). According to Cauchy-Schwarz inequality, we can write χ M as the supremum of a random process:

χ M = ε M n = sup u∈SM u n=1 < ε, u > n = 1 n sup u∈SM u n=1 n i=1 u i ε i . (8)
Therefore, if the (ε i ) 1≤i≤n were Gaussian variables, we could apply the concentration inequality for the supremum of a Gaussian process due to Cirel'son, Ibragimov and Sudakov (see [13, chapter 3]). Then we would recover inequality [START_REF] Breiman | Classification And Regression Trees[END_REF], which is a little bit less sharp than inequality [START_REF] Bousquet | Concentration Inequalities for Sub-Additive Functions Using the Entropy Method[END_REF].

Here, the (ε i ) 1≤i≤n are not Gaussian variables, they are only supposed to have exponential moments around 0, but the expression (8) of χ M is still valid. Our first idea was to consider expression [START_REF] Castellan | Modified Akaike's criterion for histogram density estimation[END_REF] and use Bousquet's concentration inequality for the supremum of an empirical process instead of the Gaussian result of Cirel'son, Ibragimov and Sudakov. Thanks to Bousquet's result [START_REF] Bousquet | Concentration Inequalities for Sub-Additive Functions Using the Entropy Method[END_REF], we have for any x > 0 and any γ > 0:

P χ M ≥ (1 + γ)E(χ M ) + 1 n √ 2vx + 1 n (2 + γ -1 )bcx ≤ e -x
where c = sup u∈SM Var(u i ε i ).

c = n inf J∈M |J| , v = n|M |τ 2 with τ 2 = E(ε 2 i ) ≤ σ 2 , and E(χ M ) ≤ E(χ 2 M ) = τ √ n
|M |. Thus we get that:

P χ M ≥ (1 + γ) τ √ n |M | + τ √ n 2|M |x + 1 n (2 + γ -1 )bcx ≤ e -x . (9)
If the ε i 's are Gaussian variables, then inequality ( 9) is sub-optimal. Indeed, if we compare inequality (7) (obtained in the case b = 0 and ε i Gaussian) with inequality (9), we see that the variance term

v = n i=1 sup u Var(u i ε i ) is much too large. We should have v = sup u n i=1 Var(u i ε i ) instead of v = n i=1 sup u Var(u i ε i ).
With such a refinement, we would obtain here v = nτ 2 instead of v = n|M |τ 2 and the term τ √ n 2|M |x in [START_REF] Catoni | Universal aggregation rules with sharp oracle inequalities[END_REF] would be replaced by τ √ n √ 2x like in [START_REF] Breiman | Classification And Regression Trees[END_REF]. Moreover we would be able to get rid of the term which is linear in x:

P χ M 1I Ω δ ≥ (1 + γ)E(χ M ) + τ √ n √ 2x + (2 + γ -1 )δb √ 2 τ √ n √ x ≤ e -x .
Bousquet's result leads to a deviation inequality for χ M which is not sharp enough for our problem (the term " τ √ n 2|M |x " in ( 9) is in particular too large and its square becomes a linear term in x which depends on M in the deviation inequality for χ 2 M ). Since the refinement of Bousquet's general result seems difficult to obtain, we build our own deviation inequality for χ 2 M by hand. Instead of considering expression [START_REF] Castellan | Modified Akaike's criterion for histogram density estimation[END_REF] where χ M is written as a supremum, we wiew χ 2 M = ε M 2 n as a χ 2 like random variable and we write it as a sum of squares (see expression [START_REF] Birgé | How many bins should be put in a regular histogram[END_REF]). Then, by truncating χ 2 M and applying Bernstein inequality [13, section 2.2.3], we get the following lemma. Let δ > 0 and Ω δ = ∀J ∈ M 0 ;

xi∈J ε i ≤ δσ 2 |J| For any partition M built from M 0 and for any x > 0

P χ 2 M 1I Ω δ ≥ σ 2 n |M | + 4 σ 2 n (1 + bδ) 2|M |x + 2 σ 2 n (1 + bδ)x ≤ e -x
and

P (Ω c δ ) ≤ 2 n N min exp -δ 2 σ 2 N min 2(1 + bδ) If b = 0, we do not need to truncate χ 2
M with Ω δ and for any x > 0

P χ 2 M ≥ σ 2 n |M | + 4 σ 2 n 2|M |x + 2 σ 2 n x ≤ e -x
In lemma 4.2, the (ε i ) 1≤i≤n are only supposed to have exponential moments around 0. In this case, by truncating χ 2 M , we get a deviation inequality which differs from inequality (6) (corresponding to the Gaussian case) only in the multiplicative constants. The set Ω δ on which we control the deviations of χ 2 M is very large. More precisely, if N min satisfies N min ≥ (log n) 2 then, for all k ∈ N, P (Ω c δ ) = o 1 n k when n → +∞. Remark 4.3. In the context of histogram density estimation, Castellan [START_REF] Castellan | Modified Akaike's criterion for histogram density estimation[END_REF] has to control an other χ 2 like random variable. Like here the main point is to truncate the variable. While she concludes by applying a Talagrand inequality to the truncated variable, we use Bernstein inequality. The detailed proof of lemma 4.2 is given in the next section.

Proof of lemma 4.2

Let M a partition built from M 0 and denote, for any J ∈ M ,

Z J = i∈J ε i 2 |J| ∧ δ 2 σ 4 |J| (Z J ) J∈M are independent random variables, E (Z J ) ≤ E ε 2 1 ≤ σ 2
, and for any k ≥ 2 we have

E |Z J | k = 1 |J| k E   i∈J ε i ∧ δσ 2 |J| 2k   = 1 |J| k +∞ 0 2kx 2k-1 P i∈J ε i ∧ δσ 2 |J| ≥ x dx = 1 |J| k δσ 2 |J| 0 2kx 2k-1 P i∈J ε i ≥ x dx
We deduce from assumption (3) and Cramer-Chernoff method (see remark 2.2) that for any x > 0

P i∈J ε i ≥ x ≤ 2 exp -x 2 2(σ 2 |J| + bx) Thus E |Z J | k ≤ 1 |J| k δσ 2 |J| 0 2kx 2k-1 2 exp -x 2 2(σ 2 |J| + bx) dx ≤ 4k |J| k +∞ 0 x 2k-1 exp -x 2 2σ 2 |J|(1 + bδ) dx
Integrating part by part, we get

E |Z J | k ≤ k! 2 4σ 2 (1 + bδ) 2 2σ 2 (1 + bδ) k-2
Thanks to Bernstein inequality [13, section 2.2.3], we obtain that for any x > 0

P J∈M Z J ≥ σ 2 |M | + 4σ 2 (1 + bδ) 2|M |x + 2σ 2 (1 + bδ)x ≤ e -x Since 1 n J∈M Z J = χ 2 M on the set Ω δ , P χ 2 M 1I Ω δ ≥ σ 2 n |M | + 4 σ 2 n (1 + bδ) 2|M |x + 2 σ 2 n (1 + bδ)x ≤ e -x
Thanks to assumption (3), for any J ∈ M 0 , we have

P i∈J ε i ≥ δσ 2 |J| ≤ 2 exp -δ 2 σ 2 |J| 2(1 + bδ) ≤ 2 exp -δ 2 σ 2 N min 2(1 + bδ)
Summing these inequalities over J ∈ M 0 , we get

P (Ω c δ ) ≤ 2|M 0 | exp -δ 2 σ 2 N min 2(1 + bδ) ≤ 2 n N min exp -δ 2 σ 2 N min 2(1 + bδ)

Proof of the theorem

Let θ ∈ (0, 1) and K > 2 -θ.

According to the definition of M : M = arg min

M∈Mn Y -ŝM 2 n + pen(M ) , we have

s -ŝ M 2 n = -2 ε, s -ŝ M n -pen( M ) + inf M∈Mn s -ŝM 2 n + 2 ε, s -ŝM n + pen(M ) . Since ŝM = s M + ε M , ε, s -ŝM n = ε, s -s M n -ε M 2 n and s -ŝM 2 n = s -s M 2 n + ε M 2 n . Thus s -ŝ M 2 n = 2 ε M 2 n -2 ε, s -s M n -pen( M ) + inf M∈Mn s -s M 2 n -ε M 2 n + 2 ε, s -s M n + pen(M ) and s -ŝ M 2 n = s -s M 2 n + ε M 2 n .
We deduce from these two last equalities that,

(1 -θ) s -ŝ M 2 n = (2 -θ) ε M 2 n -2 ε, s -s M n -θ s -s M 2 n -pen( M ) + inf M∈Mn s -s M 2 n -ε M 2 n + 2 ε, s -s M n + pen(M ) , or equivalently, (1 -θ) s -ŝ M 2 n = ∆ M + inf M∈Mn R M (10)
where

∆ M = (2 -θ) ε M 2 n -2 < ε, s -s M > n -θ s -s M 2 n -pen(M ) R M = s -s M 2 n -ε M 2 n + 2 < ε, s -s M > n +pen(M ) Let denote Ω = ∀J ∈ M 0 ; i∈J ε i ≤ σ 2 b |J|
Thanks to lemma 4.2,

P (Ω c ) ≤ 2 n N min exp -σ 2 N min 4b 2
and, for any M ∈ M n and any x > 0,

P ε M 2 n 1I Ω ≥ σ 2 n |M | + 8 σ 2 n 2|M |x + 4 σ 2 n x ≤ e -x (11)
Thanks to lemma 4.1, for any M ∈ M n and any x > 0,

P -< ε, s -s M > n ≥ σ √ n s -s M n √ 2x + 2Rb n x ≤ e -x (12)
Setting x = w M + ξ with ξ > 0, and summing all inequalities [START_REF] Lugosi | Consistency of data-driven histogram methods for density estimation and classification[END_REF] and [START_REF] Mallows | Some comments on cp[END_REF] with respect to M ∈ M n , we derive a set E ξ such that: Thanks to lemma 4.1, for any M and any x > 0

• P E c ξ ≤ e -ξ
P ε, s -s M n ≥ σ √ n s -s M n √ 2x + 2Rb n x ≤ e -x
Thus we derive a set F ξ such that

• P F c ξ ≤ e -ξ Σ
• on the set F ξ , for any M , Finally we have the following result: Taking a penalty which satisfies for all M ∈ M n pen(M ) ≥ K

≤ (1 + √ θ) s -s M E s -ŝ M 2 n 1I Ω c = E s -s M 2 n 1I Ω c + E ε M 2 n 1I Ω c ≤ E s 2 n 1I Ω c + E ε M0 2 n 1I Ω c ≤ R 2 P (Ω c ) + E ( ε M0

2 n

 2 + pen(M ) .

2 n

 2 + pen(M ) over M n : M = arg min M∈Mn Y -ŝM 2 n + pen(M ) .

Example 2 :

 2 Thanks to Catalan inequality, M ∈ M 2 n

u

  ∞ and the variance term v = n i=1 sup u∈SM u n=1

Lemma 4 . 2 .

 42 Let b ∈ R + and σ ∈ R * + such that inequality (3) holds. Let M 0 a partition of X and denote N min = inf J∈M0 |J|.

2 n + 2 σ

 22 < ε, s -s M > n ≤ σ √ n s -s M n 2 (w M + ξ) + 2Rb n (w M + ξ)It follows from definition of R M that on the set F ξ , for any M ,R M ≤ s -s M √ n s -s M n 2 (w M + ξ) + 4Rb n (w M + ξ) + pen(M ) + ξ) + 4Rb n (w M + ξ) + pen(M )

4 n ) P (Ω c ) By developing ε M0 4 nE s -ŝ M 2 n 2 n 2 C

 4222 , since E ε 2 i ≤ σ 2 and E ε 4 i ≤ C(b, σ 2 ) 2 , we get 1I Ω c ≤ R 2 P (Ω c ) + C (b, σ 2 ) N min P (Ω c )Let us recall thatP (Ω c ) ≤ 2 n N min exp -σ 2 N min 4b 2 For N min ≥ (log n) 2 , P (Ω c ) ≤ C (b, σ 2 ) 1 n 2 (log n) 2 and E s -ŝ M 1I Ω c ≤ R 2 C (b, σ 2 ) n 2 (log n) 2 + C (b, σ 2 ) (log n) (b, σ 2 ) n log n ≤ C(b, σ 2 , R) 1 n(log n) 3

  1 n (2 + γ -1 )bcx, by truncating χ M . More precisely, since the supremum in (8) is achieved with u = εM εM n , denoting Ω δ = ∀J ∈ M 0 ; xi∈J ε i ≤ δτ 2 |J| and truncating χ M with Ω δ χ M ≥ τ

	√ n	√ 2x , we would get:

  2Σ• on the set E ξ Ω, for any M , > 0, we deduce that on the set E ξ Ω, for any M , Taking a penalty pen wich compensates for all the other terms in M , i.e.In other words, on the set E ξ ,

				∆ M 1I Ω ≤ 4(2 -θ) 1 +	8(2 -θ) K + θ -2	+	2 θ	σ 2 n	ξ +	4Rb n	ξ
	Integrating with respect to ξ,
			E ∆ M 1I Ω ≤ 2 4(2 -θ) 1 +	8(2 -θ) K + θ -2	+	2 θ	σ 2 n	Σ +	8Rb n	Σ	(13)
	We are going now to control E inf
			∆ M ≤ (2 -θ)	σ 2 n	|M | + 8(2 -θ)	σ 2 n	2|M |(w M + ξ) + 4(2 -θ)	σ 2 n	(w M + ξ)
				+2	σ √ n	s -s M n 2(w M + ξ) +	4Rb n	(w M + ξ)
				-θ s -s M	2 n -pen(M )
	Using the two following inequalities
	2 4 √	2	K+θ-2 2-θ
			∆ M ≤ (2 -θ)	σ 2 n	|M | + 8(2 -θ)	σ 2 n	2|M |(w M + ξ)
			+ 4(2 -θ) +	2 θ	σ 2 n	(w M + ξ) +	4Rb n	(w M + ξ)
			-pen(M ) ≤ K σ 2 n |M | + 8	√ 2(2 -θ)	σ 2 n	|M |w M + 4(2 -θ) +	2 θ	σ 2 n	w M +	4Rb n	w M
			+ 4(2 -θ) 1 +	8(2 -θ) K + θ -2	+	2 θ	σ 2 n	ξ +	4Rb n	ξ -pen(M )
		pen(M ) ≥ K	σ 2 n	|M | + 8	√ 2(2 -θ)	σ 2 n	|M |w M +	4(2 -θ) +	2 θ	σ 2 n	+	4 √ θ	Rb n	w M
	we get that, on the set E ξ Ω,
					∆ M ≤ 4(2 -θ) 1 +	8(2 -θ) K + θ -2	+	2 θ	σ 2 n	ξ +	4Rb n	ξ

σ √ n s -s M n 2(w M + ξ) ≤ θ s -s M 2 n + 2 θ σ 2 n (w M + ξ), 8(2 -θ) σ 2 n 2|M |(w M + ξ) ≤ 8 √ 2(2 -θ) σ 2 n |M |w M + 4 √ 2(2 -θ) σ 2 n η|M | + η -1 ξ with η = 1 M R M 1I Ω .

n 1I Ω c , except if b = 0 in which case it is finished.

Remark 6.1. Starting from equality [START_REF] Lebarbier | Quelques approches pour la détection de ruptures à horizon fini[END_REF], the deviation inequalities for the χ 2 M M∈Mn and the ( ε, s -s M n ) M∈Mn (lemmas 4.1 and 4.2) are the key to determine the adequate form of penalty and to prove theorem 3.1. The weights (w M ) M∈Mn satisfying M∈Mn e -wM ≤ Σ ∈ R * + allow to sum all deviation inequalities for χ 2

M M∈Mn

and ( ε, s -s M n ) M∈Mn with respect to M ∈ M n , and to control χ 2 M and ε, s -s M n for all M ∈ M n simultaneously. The right penalty pen is thus the sum of two terms: one proportional to |M| n , which upperbounds E(χ 2 M ) = τ 2 |M| n , and a second depending on w M , which compensates the deviations of

The assumption "all partitions M ∈ M n are built from some initial partition M 0 " allows to define a large set Ω on which we control all variables χ 2 M , M ∈ M n , simultaneously, whatever the complexity of

+ and r ∈ N, then one can remove the assumption "all partitions M ∈ M n are built from some initial partition M 0 " from theorem 3.1, and only suppose that


Fanny Godet
email: fanny.godet@math.univ-nantes.fr

LINEAR PREDICTION OF LONG-RANGE DEPENDENT TIME SERIES

Keywords: Mathematics Subject Classification. 62M20; 62M10 Long-memory, linear model, autoregressive process, forecast error

We present two approaches for linear prediction of long-memory time series. The first approach consists in truncating the Wiener-Kolmogorov predictor by restricting the observations to the last k terms, which are the only available data in practice. We derive the asymptotic behaviour of the mean-squared error as k tends to +∞. The second predictor is the finite linear least-squares predictor i.e. the projection of the forecast value on the last k observations. It is shown that these two predictors converges to the Wiener Kolmogorov predictor at the same rate k -1 .

Introduction

ARMA (autoregressive moving-average) processes are often called short-memory processes because their covariances decay geometrically. On the other hand, a long-memory process is characterised by the following feature: the autocovariance function σ decays more slowly in so far as it is not absolutely summable. These processes are so-named because of the strong association between observations widely separated in time. The long-memory time series models have attracted much attention lately and there is now a growing realisation that time series possessing long-memory characteristics arise in areas as diverse as Economics, Geophysics, Hydrology or telecom traffic (see, e.g., [START_REF] Mandelbrot | Some long-run properties of geophysical records[END_REF] and [START_REF] Granger | An introduction to long-memory time series models and fractional differencing[END_REF]). Although there exists substantial literature on the prediction of shortmemory processes (see [START_REF] Brockwell | Simple consistent estimation of the coefficients of a linear filter[END_REF] and [START_REF] Bhansali | Linear prediction by autoregressive model fitting in the time domain[END_REF] for the univariate case or [START_REF] Lewis | Prediction of multivariate time series by autoregressive model fitting[END_REF] for the multivariate case), there are fewer results for long-memory time series. Most prediction methods are successfully applied on long memory data either simulated like in [START_REF] Crato | Model selection and forecasting for long-range dependent processes[END_REF] or on real data like electricity demand (see [START_REF] Soares | Forecasting electricity demand using generalized long memory[END_REF]). In this paper, we consider the question of prediction in long memory models from a theoretical point of view.

More precisely, we compare two prediction methods for long-memory processes. We search linear predictors of X k+h knowing (X 1 , . . . , X k ) which are optimal in the sense that they minimise the mean-squared error. We study the behaviour of the mean-squared errors as k tends to infinity as [START_REF] Pourahmadi | On the convergence of finite linear predictors of stationary processes[END_REF] does for short memory processes. The paper is organised as follows. First we introduce our model and our main assumptions. Then, in Section 2, we study the best linear predictor knowing all the past i.e. the Wiener-Kolmogorov predictor proposed by [START_REF] Whittle | Prediction and regulation by linear least-square methods[END_REF] and by [START_REF] Bhansali | Prediction of long-memory time series: An overview[END_REF] for long-memory time series. In practice, only the last k values of the process are available. Therefore we need to truncate the infinite series in the definition of the predictor and to derive the asymptotic behaviour of the mean-squared error as k tends to +∞. This method is for example used by [START_REF] Barkoulas | Long-memory forecasting of US monetary indices[END_REF] to forecast US monetary indices.

In Section 3 we discuss the asymptotic properties of the forecast error if we project onto the closed span of the last k observations. This approach has been proposed by [START_REF] Ray | Modeling long-memory processes for optimal long-range prediction[END_REF] for fractional noise series and by [START_REF] Crato | Model selection and forecasting for long-range dependent processes[END_REF] for other simulated data. These authors view this approach as fitting a misspecified autoregressive model of order k. The simulations in [START_REF] Ray | Modeling long-memory processes for optimal long-range prediction[END_REF] show that high-order AR-models predict fractional integrated noise very well.

Finally in Section 4 we compare the two previous methods for h-step prediction. We give some asymptotic properties of the mean-squared error of the linear least-squares predictor as h tends to +∞ in the particular case of long-memory processes. Then we study our kth order predictors as k tends to +∞.

Let (X n ) n∈Z be a discrete-time (weakly) stationary process in L 2 with mean 0 and σ its autocovariance function. We assume that the process (X n ) n∈Z admits an infinite moving average representation as follows:

X n = ∞ j=0 b j ε n-j (1)
where (ε n ) n∈Z is a white-noise series consisting of uncorrelated random variables, each with mean zero and variance σ 2 ε , and where (b j ) j∈N is square-summable and b 0 = 1. We shall further assume that (X n ) n∈Z admits an infinite autoregressive representation:

ε n = ∞ j=0 a j X n-j , ( 2
)
where the sequence (a j ) j∈N is absolutely summable and a 0 = 1. We assume also that (a j ) j∈N and (b j ) j∈N , occurring respectively in ( 2) and ( 1), satisfy the following conditions for all δ > 0:

|a j | ≤ C 1 j -d-1+δ
(3)

|b j | ≤ C 2 j d-1+δ , (4)
where C 1 and C 2 are constants and d is a parameter verifying d ∈ ]0, 1/2[. The class of processes satisfying [START_REF] Bhansali | Prediction of long-memory time series: An overview[END_REF] and (4) includes long memory processes i.e. processes with covariances not absolutely summable:

∞ k=-∞ |σ(k)| = ∞.
For example, the FARIMA process (X n ) n∈Z , the most studied long memory process, is the stationary solution to the difference equations:

φ(B)(1 -B) d X n = θ(B)ε n (5)
where (ε n ) n∈Z is a white noise series in L 2 with mean zero, B is the backward shift operator and φ and θ are polynomials with no zero in the closed unit disk. The coefficients verify as j goes to infinity

|a j | ∼ +∞ C 1 j -d-1 |b j | ∼ +∞ C 2 j d-1
and thus (3) and (4) hold.

More generally, conditions (3) and (4) hold when the coefficients are of the form:

|a j | ∼ +∞ L(j)j -d-1 (6) |b j | ∼ +∞ L (j)j d-1 (7)
where L and L are slowly varying functions. A positive function L is a slowly varying function in the sense of [START_REF] Zygmund | Trigonometric series[END_REF] if, for any δ > 0, x → x -δ L(x) is ultimately decreasing and x → x δ L(x) is ultimately increasing.

Under conditions ( 6) and ( 7), (X n ) n∈Z is a long memory process. Indeed [START_REF] Inoue | Regularly varying correlation functions and KMO-Langevin equations[END_REF] proved that [START_REF] Crato | Model selection and forecasting for long-range dependent processes[END_REF] implies that [START_REF] Inoue | Regularly varying correlation functions and KMO-Langevin equations[END_REF] proved that if:

σ(j) ∼ j 2d-1 [L (j) β(d, 1 -2d)] 2
where β is the beta function. Note that the converse is not true. More assumptions about the series (b j ) j∈N are needed in order to get an asymptotic equivalent for (b j ) j∈N using an asymptotic equivalent of (σ(j)) j∈N (see [START_REF] Inoue | Asymptotics for the partial autocorrelation function of a stationary process[END_REF]).

2. Wiener-Kolmogorov next step prediction theory

The Wiener-Kolmogorov predictor

The aim of this part is to compute the best linear one-step predictor (with minimum mean-squared distance from the true random variable) knowing all the past {X k+1-j , j 1}. Our predictor is therefore an infinite linear combination of the infinite past:

X k (1) = ∞ j=0 λ(j)X k-j
where (λ(j)) j∈N are chosen to minimise the mean squared prediction error:

E X k (1) -X k+1 2 . Since ε k+1 is orthogonal to the past [ε] k -∞ = [X] k -∞ , the Wiener-Kolmogorov predictor is equal to X k (1) = - ∞ j=1 a j X k+1-j . (8)

Mean-squared prediction error when the predictor is truncated

In practice, we only know a finite subset of the past, the one which we have observed. So the predictor should only depend on the observations. Assume that we only know the set {X 1 , . . . , X k } and that we replace the unknown values by 0, then we have the following new predictor:

X k (1) = - k j=1 a j X k+1-j . ( 9
)
It is equivalent to say that we have truncated the infinite series (8) to k terms. The following proposition provides the asymptotic properties of the mean-squared prediction error as a function of k.

Theorem 2.1. Let (X n ) n∈Z be a linear stationary process defined by (1), ( 2) and verifying conditions (3) and (4). We can approximate the mean-squared prediction error of X k (1) by:

∀δ > 0, E X k+1 -X k (1) 2 = σ 2 ε + O(k -1+δ ) as k → +∞.
Note that the prediction error is the sum of σ 2 ε , the error in the Wiener-Kolmogorov model, and the error due to the truncation to k terms which is bounded by O(k -1+δ ) for all δ > 0. We denote

σ 2 k := E X k+1 -X k (1)
2 hereafter the sum of these two errors.

The proof of Theorem (2.1) hinges on the following lemma:

Lemma 2.2. Under assumption (4), the autocovariance function σ of the process (X n ) n∈Z verifies:

∀δ > 0, ∃C 3 ∈ R, |σ(j)| ≤ C 3 j 2d-1+δ . ( 10
)
Proof. Notice that it suffices to prove [START_REF] Hosking | Fractional differencing[END_REF] for δ near 0 in order to prove [START_REF] Hosking | Fractional differencing[END_REF] for all δ > 0. Let δ < 1 -2d:

σ(k) = +∞ j=0 b j b j+k |σ(k)| ≤ +∞ j=1 |b j b j+k | + |b 0 b k | ≤ C 2 2 +∞ j=1 j d-1+δ/2 (k + j) d-1+δ/2 + |b 0 b k | ≤ C 2 2 k 2d-1+δ +∞ 0 j d-1+δ/2 (1 + j) d-1+δ/2 dj + C 2 k d-1+δ/2 ≤ C 3 k 2d-1+δ .
We now prove Theorem 2.1.

Proof.

X k+1 -X k (1) = ε k+1 - +∞ j=k+1 a j X k+1-j . ( 11
)
The two parts of the sum [START_REF] Inoue | Regularly varying correlation functions and KMO-Langevin equations[END_REF] are orthogonal with respect to the inner product associated with the mean-squared norm. Consequently:

E X k+1 -X k (1) 2 = σ 2 ε + ∞ j=k+1 ∞ l=k+1 a j a l σ(l -j).
The error due to the truncation, which we have to study, is then equal to:

σ 2 k -σ 2 ε = ∞ j=k+1 ∞ l=k+1 a j a l σ(l -j) ( 12
)
we have:

+∞ j=k+1 +∞ l=k+1 a j a l σ(l -j) = 2 +∞ j=k+1 a j +∞ l=j+1 a l σ(l -j) + +∞ j=k+1 a 2 j σ(0) ≤ 2 +∞ j=k+1 |a j | |a j+1 | |σ(1)| + +∞ j=k+1 a 2 j σ(0) +2 +∞ j=k+1 |a j | +∞ l=j+2 |a l ||σ(l -j)|.
From (3) and ( 10) and the triangular inequality, it follows that:

+∞ j=k+1 +∞ l=k+1 a j a l σ(l -j) ≤ C 2 1 C 3 ⎛ ⎝ 2 +∞ j=k+1 j -d-1+δ (j + 1) -d-1+δ + +∞ j=k+1 j -d-1+δ 2 ⎞ ⎠ +2C 2 1 C 3 +∞ j=k+1 j -d-1+δ +∞ l=j+2 l -d-1+δ |l -j| 2d-1+δ (13)
for all δ > 0. Assume now that δ < 1/2d. For the terms (13), since j → j -d-1+δ (j + 1) -d-1+δ is a positive and decreasing function on R + , we have the following approximations:

2C 2 1 C 3 +∞ j=k+1 j -d-1+δ (j + 1) -d-1+δ ∼ 2C 2 1 C 3 +∞ k j -d-1+δ (j + 1) -d-1+δ dj ∼ 2C 2 1 C 3 1 + 2d -2δ k -2d-1+2δ .
Since the function j → j -d-1+δ 2 is also positive and decreasing, we can establish in a similar way that:

C 2 1 C 3 +∞ j=k+1 j -d-1+δ 2 ∼ C 2 1 C 3 +∞ k j -d-1+δ 2 dj ∼ C 2 1 C 3 1 + 2d -2δ k -2d-1+2δ .
For the infinite double series ( 13), we will similarly compare the series with an integral. In the next lemma, we establish a necessary result for this comparison:

Lemma 2.3. Let g the function (l, j) → j -d-1+δ l -d-1+δ |l -j| 2d-1+δ
. Let m and n be two positive integers.

We assume that δ < 1 -2d and m

≥ δ-d-1 δ+2d-1 for all δ ∈ 0, δ-d-1 δ+2d-1 . If n ≥ m + 1 then An,m g(l, j) dj dl ≥ g(n + 1, m) where A n,m is the square domain [n, n + 1] × [m, m + 1].
Assume now that δ < 1 -2d without loss of generality. Thanks to the previous Lemma and the asymptotic equivalents of (13), there exists

K ∈ N such that if k > K: +∞ j=k+1 +∞ l=k+1 a j a l σ(l -j) ≤ C +∞ k+1 j -d-1+δ +∞ j l -d-1+δ (l -j) 2d-1+δ dl dj + O k -2d-1+2δ .
By using the substitution jl = l in the integral over l we obtain:

+∞ j=k+1 +∞ l=k+1 a j a l σ(l -j) ≤ C +∞ k+1 j -2+3δ +∞ 1 l -d-1+δ (l -1) 2d-1+δ dldj + O k -2d-1 . Since if δ < (1 -d)/2 +∞ 1 l -d-1+δ (l -1) 2d-1+δ dl < +∞, it follows: +∞ j=k+1 +∞ l=k+1 a j a l σ(l -j) ≤ O k -1+3δ + O k -2d-1 ≤ O k -1+3δ . ( 14
)
If δ > 0, δ < 1 -2d and δ < (1d)/2, we have:

σ 2 k -σ 2 ε = +∞ j=k+1 +∞ l=k+1 a j a l σ(l -j) = O k -1+3δ .
Notice that if the equality is true under the assumptions δ > 0, δ < 1 -2d and δ < (1d)/2, it is also true for any δ > 0. This concludes the proof.

The result of Theorem 2.1 is not interesting for short memory processes. It is proved in [START_REF] Pourahmadi | On the convergence of finite linear predictors of stationary processes[END_REF] that the error due to the truncation ( σ 2 kσ 2 ε ) converges to 0 at an exponential rate for short memory processes. We now prove that there exist long-memory processes whose prediction error attains the rate of convergence k -1 .

When we specify the form of the autoregressive coefficients and the autocovariance function, we obtain an equivalent of the mean-squared prediction error.

Proposition 2.4. Let (X n ) n∈Z be a linear stationary process defined by (1), ( 2) and verifying [START_REF] Brockwell | Time Series: Theory and Methods[END_REF]. Assume that the coefficients (a j ) j∈N * are ultimately of constant sign. We assume also that the autocovariance function σ of the process (X n ) n∈Z verifies:

|σ(j)| ∼ +∞ L (j)j 2d-1
with L a slowly varying function and that the function σ is of a constant sign. Then the rate of convergence

O(k -1
) is optimal i.e. we have the following asymptotic equivalent:

E X k+1 -X k (1) 2 = σ 2 ε + L (k)k -1 + o k -1 as k → +∞ ( 15
)
where L is a slowly varying function.

Proof. In this particular case, we can estimate the prediction error more precisely:

+∞ k+1 +∞ k+1 a j a l σ(l -j) = +∞ k+1 |a j | +∞ j+1 |a l ||σ(l -j)| + +∞ k+1 a 2 j σ(0) = +∞ k+1 j -d-1 L(j) +∞ j+1 l -d-1 L(l)(l -j) 2d-1 L (l -j)dldj + k -2d-1 L 2 (k) 1 + 2d + O k -2d-1 ∼ +∞ k+1 j -2 L 2 (j)L (j) +∞ 1/j+1 l -d-1 L(lj) L(j) (l -1) 2d-1 L (j(l -1)) L (j) dldj ∼ k -1 L 2 (k)L (k)Γ(1 -d)Γ(2d) Γ(1 + d) • ( 16
)
Example 2.5. Assume now that (X n ) n∈Z is fractionally integrated noise, which is the stationary solution of the difference equation:

X n = (1 -B) -d ε n (17)
with B the usual backward shift operator, (ε n ) n∈Z is a white-noise series and d ∈ ]0, 1/2[ (see for example [START_REF] Brockwell | Time Series: Theory and Methods[END_REF]). This is a particular case of FARIMA models defined in [START_REF] Brockwell | Simple consistent estimation of the coefficients of a linear filter[END_REF] with φ = θ = 1. We can compute the coefficients and obtain that:

∀j > 0, a j = Γ(j -d) Γ(j + 1)Γ(-d) < 0 and a j ∼ j -d-1 Γ(-d) as j → ∞.
We can also compute the autocovariance function and we obtain:

∀j ≥ 0, σ(j) = (-1) j Γ(1 -2d) Γ(j -d + 1)Γ(1 -j -d) σ 2 ε > 0 and σ(j) ∼ j 2d-1 Γ(1 -2d) Γ(d)Γ(1 -d) σ 2 ε as j → ∞.
In this particular case, we can estimate the prediction error more precisely and give the explicit expression of L defined in [START_REF] Pourahmadi | On the convergence of finite linear predictors of stationary processes[END_REF]:

+∞ j=k+1 +∞ l=k+1 a j a l σ(l -j) ∼ Γ(1 -2d)Γ(2d)σ 2 ε Γ(-d) 2 Γ(d)Γ(1 + d) k -1 (18)
since in this case the functions L and L are constant and equal respectively to 1 Γ(-d) and to

Γ(1 -2d)σ 2 ε Γ(d)Γ(1 -d) .
In the specific case of fractionally integrated noise, we may write the prediction error as:

E X k+1 -X k (1) 2 = σ 2 ε + C(d)k -1 + o k -1
and we can express C(d) as a function of d:

C(d) = Γ(1 -2d)Γ(2d)σ 2 ε Γ(-d) 2 Γ(d)Γ(1 + d) • ( 19
)
It is easy to prove that C(d) → +∞ as d → 1/2 and we may write the following asymptotic equivalent as d → 1/2:

C(d) ∼ σ 2 ε (1 -2d)Γ(-1/2) 2 Γ(1/2)Γ(3/2) • ( 20
)
As d → 0, C(d) → 0 and we have the following equivalent as d → 0:

C(d) ∼ σ 2 ε d 2 .
As Figure 1 suggests and the asymptotic equivalent given in [START_REF] Zygmund | Trigonometric series[END_REF] proves, the mean-squared error tends to +∞ as d → 1/2. By contrast, the constant C(d) takes small values for d in a large interval of [0, 1/2[. Although the rate of convergence has a constant order k -1 , the forecast error is bigger when d → 1/2. This result is not surprising since the correlation between the predicted variable and the unobserved variables increases when

d → 1/2.
Example 2.6. Long-memory processes with seasonal effects (see for example [START_REF] Viano | Long-range dependence and mixing for discrete time fractional processes[END_REF]) provide another class of long memory processes verifying assumptions of Theorem 2.1 but not those of Proposition 2.4. We can also prove that the rate of convergence k -1 is attained for some seasonal processes. We consider the particular case of GARMA processes introduced by [START_REF] Hosking | Fractional differencing[END_REF] and studied for example by [START_REF] Gray | On generalized fractional processes[END_REF]. We first recall their definition. (X n ) n∈Z is a GARMA process if it is the stationary solution of the equation: [START_REF] Whittle | Prediction and regulation by linear least-square methods[END_REF]. with cos ν = 0, d ∈]0, 1/2[, (ε n ) n∈Z a white noise and B the backward shift operator. (X n ) n∈Z admits an infinite autoregressive representation like in [START_REF] Bhansali | Linear prediction by autoregressive model fitting in the time domain[END_REF]. The coefficients verify:

∀n ∈ Z, X n = (1 -2 cos νB + B 2 ) -d ε n Figure 1. Behaviour of constant C(d), d ∈ [0, 1/2[, defined in
a j = cos[(j -d)ν + (dπ/2)] Γ(-d) sin -d (ν) 2 j 1+d + O j -5/2 . ( 21
)
Its autocovariance function σ is such that:

σ(j) = K cos(jν)j 2d-1 1 + O(j -1 ) ( 22
)
where K is a constant independent of j.

We now give the asymptotic behaviour of the mean-squared error of prediction for these processes.

Proposition 2.7. If (X n ) n∈Z is a GARMA process, then E X k+1 -X k (1) 2 = σ 2 ε + Ck -1 + o k -1 as k → +∞.
Proof. The proof is given in Appendix.

After these results on the truncating Wiener-Kolmogorov predictor, we will propose another predictor. Instead of truncating to k terms the series which defines the Wiener-Kolmogorov predictor it is more efficient to project directly onto the observations in order to minimise the forecast error. We study this method in the following section.

Prediction method

In this section, we study the asymptotic properties of the forecast mean-squared error when we project the forecast random variable X k+1 onto the closed span of the subset {X k , . . . , X 1 }. This is a generalisation of the "autoregressive model fitting" approach developed by [START_REF] Ray | Modeling long-memory processes for optimal long-range prediction[END_REF] in the case of fractionally integrated noise (defined in [START_REF] Soares | Forecasting electricity demand using generalized long memory[END_REF]). Fitting a k-th order autoregressive process is equivalent to projecting onto the span of the subset {X k , . . . , X 1 }.

Definition of the predictor

Here (X n ) n∈Z is a long-memory process which verifies the assumptions (1)-( 4) of Section 1. We choose to define the predictor as the projection mapping onto the closed span of the subset {X k , . . . , X 1 } of the Hilbert space L 2 (Ω, F , P) with inner product X, Y = E(XY ). Then the predictor can be written as:

X k (1) = -a 1,k X k -. . . -a k,k X 1 .
Consequently the coefficients (-a i,k ) 1≤i≤k of the projection verify the kth order Yule-Walker equations:

∀j ∈ 1, k , k i=1 a i,k σ(i -j) = -σ(j). ( 23
)
The mean-squared prediction error is:

E X k (1) -X k+1 2 = σ 2 ε + E ⎡ ⎢ ⎣ ⎛ ⎝ - ∞ j=1 a j X k+1-j + k j=1 a j,k X k+1-j ⎞ ⎠ 2 ⎤ ⎥ ⎦. (24)

Rate of convergence of the error by projection

In the next theorem we derive an asymptotic expression for the prediction error when projecting onto the finite past:

Theorem 3.1. Let (X n ) n∈Z be a linear stationary process defined by (1), ( 2) and verifying conditions (3) and (4) of Section 1. We can approximate the mean-squared prediction error of X k (1) by:

E X k (1) -X k+1 2 -σ 2 ε = O(k -1 ) as k → +∞. ( 25
)
There exist long-memory processes such that:

E X k (1) -X k+1 2 -σ 2 ε ∼ Ck -1 as k → +∞
where C is a constant.

Proof. Hereafter we will denote

σ 2 k := E X k (1) -X k+1 2 .
Since the projection minimises the forecast error using k observations, the error by using truncation is bigger and since the truncation method involves an error bounded by O k -1 , we obtain (25).

Consequently we only need prove that this rate of convergence is attained. This is the case for the fractionally integrated processes defined in [START_REF] Soares | Forecasting electricity demand using generalized long memory[END_REF]. We want to express the error made by projection in terms of the Wiener-Kolmogorov truncation error. Therefore in the case of a fractionally integrated process we need only show that:

σ 2 k -σ 2 ε ∼ Ck -1 .
Now, setting a j,k = 0 if j > k and developping (24)

σ 2 k -σ 2 ε = +∞ j=0 +∞ l=0 (a j -a j,k )(a l -a l,k )σ(l -j) = +∞ j=0 (a j -a j,k ) +∞ l=0 a l σ(l -j) - k l=0 a l,k +∞ j=0 (a j -a j,k )σ(l -j). ( 26
)
We first study the first term of the sum (26). For any j > 0 , we have +∞ l=0 a l σ(lj) = 0:

ε n = ∞ j=0 a l X n-l E (X n-j ε n ) = ∞ l=0 a l σ(l -j) = 0
as ε n is orthogonal to X n-j for j > 0. We can thus rewrite the first term of (26):

+∞ j=0 (a j -a j,k ) +∞ l=0 a l σ(l -j) = (a 0 -a 0,k ) +∞ l=0 a l σ(l) = 0
since a 0 = a 0,k = 1 according to definition. Next we study the second term of the sum in (26). And we obtain:

k l=0 a l,k +∞ j=0 (a j -a j,k )σ(l -j) = k l=1 (a l,k -a l ) k j=1 (a j -a j,k )σ(l -j) + k l=1 (a l,k -a l ) +∞ j=k+1 a j σ(l -j) + k l=0 a l k j=1 (a j -a j,k )σ(l -j) + k l=0 a l +∞ j=k+1 a j σ(l -j). ( 27
)
Similarly we rewrite the term (27) using the Yule-Walker equations:

k l=1 (a l,k -a l ) +∞ j=k+1 a j σ(l -j) = - k l=1 (a l,k -a l ) k j=0 a j σ(l -j).
We then remark that ( 27) is equal to the opposite of (27). Hence it follows that:

k l=0 a l,k +∞ j=0 (a j -a j,k )σ(l -j) = k l=1 (a l,k -a l ) k j=1 (a j -a j,k )σ(l -j) +2 k l=1 (a l,k -a l ) +∞ j=k+1 a j σ(l -j) + k l=0 a l +∞ j=k+1 a j σ(l -j). ( 28
)
In a similar way we can rewrite the third term of the sum in (28) using Fubini's Theorem and then the Yule-Walker equations:

k l=0 a l +∞ j=k+1 a j σ(l -j) = - +∞ l=k+1 +∞ j=k+1 a j a l σ(l -j).
This third term is therefore equal to the forecast error due to the truncation [START_REF] Inoue | Asymptotics for the partial autocorrelation function of a stationary process[END_REF].

In order to compare the prediction error when truncating the Wiener-Kolmogorov predictor and when projecting onto the finite past for a fractionally integrated process, we need the sign of all the components of the sum in (28). For a fractionally integrated noise, we know the explicit formula for a j and σ(j):

∀j > 0, a j = Γ(j -d) Γ(j + 1)Γ(-d) < 0 and ∀j ≥ 0, σ(j) = (-1) j Γ(1 -2d) Γ(j -d + 1)Γ(1 -j -d) σ 2 ε > 0.
In order to get the sign of a l,ka l we use the explicit formula given in [START_REF] Brockwell | Simple consistent estimation of the coefficients of a linear filter[END_REF] and we easily obtain that a l,ka l is negative for all l ∈ 1, k .

a l -a l,k = Γ(l -d) Γ(l + 1)Γ(-d) - Γ(k + 1)Γ(l -d)Γ(k -d -l + 1) Γ(k -l + 1)Γ(l + 1)Γ(-d)Γ(k -d + 1) = -a l -1 + Γ(k + 1)Γ(k -d -l + 1) Γ(k -l + 1)Γ(k -d + 1) = -a l k...(k -j + 1) (k -d)...(k -d -j + 1) -1 > 0 (29)
since ∀j ∈ N * , a j < 0. To give an asymptotic equivalent for the prediction error, we use the sum given in (28).

We have the sign of the three terms: the first is negative, the second is positive and the last is negative. Moreover the third is equal to the forecast error by truncation and we have proved that this asymptotic equivalent has order O(k -1 ). The prediction error when projecting converges faster to 0 than the error by truncation only if the second term is equivalent to Ck -1 , with C constant. Consequently, we search for a bound for a la l,k using (29):

a l -a l,k = -a l l-1 m=0 1 -m k 1 -m+d k -1 = -a l l-1 m=0 1 + d k 1 -d+m k -1 .
Then we use the following inequality:

∀x ∈ R, 1 + x ≤ exp(x)
which gives:

a l -a l,k ≤ -a l exp l-1 m=0 d k 1 -d+m k -1 ≤ -a l exp d l-1 m=0 1 k -d -m -1 ≤ -a l exp d l-1 m=0 1 k -d -m •
According to the previous inequality, we have:

k l=1 (a l -a l,k ) +∞ j=k+1 -a j σ(l -j) = k-1 l=1 (a l -a l,k ) +∞ j=k+1 -a j σ(l -j) +(a k -a k,k ) +∞ j=k+1 -a j σ(k -j) ≤ k-1 l=1 -a l exp d j-1 m=0 1 k -d -m +∞ j=k+1 -a j σ(l -j) +(-a k ) exp d k-1 m=0 1 k -d -m +∞ j=k+1 -a j σ(k -j).
As the function x → 1 k-d-x is increasing, we can bound the series by the integrals.

k l=1 (a l -a l,k ) +∞ j=k+1 -a j σ(l -j) ≤ k-1 l=1 -a l exp d l 0 1 k -d -m dm +∞ j=k+1 -a j σ(l -j) +(-a k ) exp d k-1 m=0 1 k -d -m dm +∞ j=k+1 -a j σ(k -j).
We now search a bound for exp d k-1 m=0

1 k-d-m dm . We have:

k-1 m=0 1 k -d -m dm ∼ ln(k) ≤ 3 2 ln(k)
for k large enough. Therefore there exists K such that for all k ≥ K:

k l=1 (a l -a l,k ) +∞ j=k+1 -a j σ(l -j) ≤ k-1 l=1 -a l exp d ln k -d k -d -l +∞ j=k+1 -a j σ(l -j) -a k k 3 2 d +∞ j=k+1 -a j σ(0) ≤ C(k -d) d k-1 l=1 l -d-1 (k -d -l) -d +∞ j=k+1 j -d-1 (j -l) 2d-1 +Ck -d-1 k 3 2 d k -d ≤ C (k -d) 2 1 1/(k-d) l -d-1 (1 -l) -d +∞ 1 j -d-1 (j -1) 2d-1 dl dj +Ck -1 2 d-1 ≤ C (k -d) -2+d + Ck -1 2 d-1 Figure 2. Ratio r(k), d ∈ ]0, 1/2[ defined in (30).
and so the positive term has a smaller asymptotic order than the forecast error made by truncating. Therefore we have proved that in the particular case of fractionally integrated processes, the two prediction errors are equivalent to Ck -1 with C constant.

The two approaches to next-step prediction, by truncation to k terms or by projection on the last k observations have consequently a prediction error with the same rate of convergence k -1 . So it is interesting to study how the second approach improves the prediction and if the improvement is negligible with respect to the prediction error due to the truncation. In this aim, we define the following quotient:

r(k) := σ 2 k -σ 2 ε -σ 2 k -σ 2 ε σ 2 k -σ 2 ε ( 30
)
which is the ratio of the difference between the two prediction errors (prediction error due to the truncation [START_REF] Inoue | Asymptotics for the partial autocorrelation function of a stationary process[END_REF] and prediction error due to the projection (26)) and the prediction error due to the truncation. By (28), we obtain:

r(k) = k j=1 (a j,k -a j ) k l=1 (a l -a l,k )σ(j -l) + 2 k j=1 (a j,k -a j ) +∞ l=k+1 a l σ(j -l) +∞ j=k+1 a j +∞ l=k+1 a l σ(j -l) •
We evaluate this ratio in the particular case of a fractionally integrated noise defined in [START_REF] Soares | Forecasting electricity demand using generalized long memory[END_REF]. Figure 2 shows that the prediction by truncation incurs a larger performance loss when d → 1/2. The improvement reaches 50 percent when d > 0.3 and k > 20.

The multistep predictors

After obtaining an asymptotic equivalent for the next step predictor, we will generalise the two methods to h-step prediction and aim to obtain their asymptotic behaviour as k → +∞ but also as h → +∞.

The Wiener-Kolmogorov predictor

Since we assume that the process (X n ) n∈Z has an autoregressive representation (2) and moving average representation (1), the linear least-squares predictor, X k (h), of X k+h based on the infinite past (X j , j ≤ k) is given by:

X k (h) = +∞ j=h b j ε k+h-j
(see for example Th. 5.5.1 of [START_REF] Brockwell | Time Series: Theory and Methods[END_REF]). The corresponding mean squared error of prediction is:

E X k (h) -X k+h 2 = σ 2 ε h-1 j=0 b 2 j .
As the prediction step h tends to infinity, the mean-squared prediction error converges to

σ 2 ε +∞ j=0 b 2 j = σ(0),
which is the the variance of the process (X n ) n∈Z . But if the mean-squared prediction error is equal to σ(0), we have no more interest in the prediction method since its error is equal to the error of predicting the future by 0.

The convergence of the mean-squared error to σ(0) is slower in the long-memory case than in the short-memory case since the sequence b j decays more slowly to 0. More precisely in the case of a long-memory process under the assumption [START_REF] Crato | Model selection and forecasting for long-range dependent processes[END_REF] we can explicit the asymptotic behaviour of the mean-squared error. We have as h tends to infinity:

σ(0) -E X k (h) -X k+h 2 = σ 2 e +∞ j=h b 2 j ∼ +∞ j=h j 2d-2 L 2 (j) ∼ +∞ h j 2d-2 L 2 (j)dj ∼ h→+∞ 1 1 -2d h 2d-1 L 2 (h) (31)
according to Proposition 1.5.10 of [START_REF] Bingham | Regular variation[END_REF].

On the contrary, for short memory processes we can obtain an exponential rate of convergence of

E X k (h) -X k+h 2
to the variance of the process σ(0). Consider the classical ARMA processes (see for example [START_REF] Brockwell | Time Series: Theory and Methods[END_REF]). The moving average coefficients b j are bounded by:

|b j | ≤ Cj m-1 ρ -j
where ρ verifies |ρ| < 1 and m is a non negative integer (see for example [START_REF] Brockwell | Time Series: Theory and Methods[END_REF] p. 92). Thus the mean-squared prediction error is equivalent to:

σ(0) -E X k (h) -X k+h 2 ∼ h→+∞ σ 2 ε C +∞ j=h j 2m-2 ρ -2j ∼ h→+∞ σ 2 ε C (2 log(ρ)) 1-2m (2 log(ρ)h) 2m-2 exp (2 log(ρ)h).
We obtain that the rate of convergence is exponential. The mean-squared prediction error goes faster to σ(0)

when the predicting process is ARMA than when the process is a long-memory process.

The h-step prediction is then more interesting for the long-memory process than for the short-memory process, having observed the infinite past. We now consider the truncating effect and the projection effect.

Truncated Wiener-Kolmogorov predictor and the linear least-squares predictor

In practice, we only observe a finite number k of data (X 1 , . . . , X k ) so we cannot evaluate the Wiener-Kolmogorov predictor. We first study what happens when we truncate the series which defines this predictor.

We define the h-step truncated Wiener-Kolmogorov of order k by induction as:

X k (h) = - h-1 j=1 a j X k (h -j) - k j=1 a h-1+j X k+1-j . ( 32
)
The following proposition describes the asymptotic behaviour of the mean-squared error of the predictor (32).

Proposition 4.1. Let (X n ) n∈Z be a linear stationary process defined by (1), ( 2) and verifying (3) and (4). We can approximate the mean-squared prediction error of X k (h) by:

∀δ > 0, E X k (h) -X k+h 2 = σ 2 ε h-1 l=0 b 2 l + R(h, k) ( 33
)
where R(h, k) ≥ 0 and where for all h and k,

R(h, k) ≤ Ch 2d+δ k -1+δ
with C a constant independent of (h, k).

Proof. First we write the difference between the predicted random variable and its predictor:

X k (h) -X k+h = - h-1 j=1 a j X k (h -j) - k l=1 a h-1+j X k+1-j -ε k+h + +∞ j=1 a j X k+h-j = -ε k+h + h-1 j=1 a j X k+h-j -X k (h -j) + k j=1 a h-1+j (X k+1-j -X k+1-j ) + +∞ j=k+1 a h-1+j X k+1-j = -ε k+h + h-1 j=1 a j X k+h-j -X k (h -j) + +∞ j=k+1 a h-1+j X k+1-j .
We proceed by induction on h to show that

X k (h) -X k+h = - h-1 l=0 ⎛ ⎝ j1+j2+...+j h =l (-1) card({i,ji =0}) a j1 a j2 . . . a j h ⎞ ⎠ ε k+h-l + +∞ j=k+1 i1+i2+...+i h =h-1 (-1) card({l>1,i l =0}) a j+i1 a i2 . . . a i h X k+1-j .
For h = 2, we have for example

X k (2) -X k+2 = -(a 0 ε k+2 -a 1 ε k+1 ) + +∞ j=k+1
(-a 1 a j + a j+1 )X k+1-j .

Denote A(z) = 1+ +∞ j=1 a j z j and B(z) = 1+ +∞ j=1 b j z j . Since we have A(z) = B(z) -1 , we obtain the following conditions on the coefficients:

b 1 = -a 1 b 2 = -a 2 + a 2 1 b 3 = -a 3 + 2a 1 a 2 -a 3 1 . . .
So we obtain:

X k (h) -X k+h = - h-1 l=0 b l ε k+h-l + +∞ j=k+1 h-1 m=0 a j+m b h-1-m X k+1-j . ( 34
)
Since the process (ε n ) n∈Z is uncorrelated, the two terms of the sum (34) are orthogonal. We can rewrite the mean-squared error:

E X k (h) -X k+h 2 = h-1 l=0 b 2 l σ 2 ε + E ⎡ ⎣ +∞ j=k+1 h-1 m=0 a j+h-1-m b m X k+1-j ⎤ ⎦ 2 . ( 35
)
The first part of the error (35) is due to the prediction method and the second (35) due to truncating the predictor. We now approximate the error term (35) by using ( 3) and ( 4). We obtain the following upper bound:

∀δ > 0, h-1 m=0 a j+h-1-m b m ≤ h-1 m=1 |a j+h-1-m b m | + |b 0 a j+h-1 | ≤ C 1 C 2 h 0 (j + h -1 -l) -d-1+δ l d-1+δ dl + C 1 (j + h) -d-1 ≤ C 1 C 2 h -1+2δ 1 0 j h + 1 -l -d-1+δ l d-1+δ dl + C 1 (j + h) -d-1 ≤ C 1 C 2 h -1+2δ j -d-1+δ 1 0 1 h + 1 -l j -d-1+δ l d-1+δ dl + C 1 (j + h) -d-1 ≤ C 1 C 2 h d+2δ j -d-1+δ 1 0 l d-1+δ dl + C 1 (j + h) -d-1 h m=0 a j+h-m b m ≤ C 1 C 2 h d+2δ d j -d-1+δ (36)
where the constants C 1 and C 2 are independent of (h, j). We then obtain by (36):

E ⎡ ⎣ +∞ j=k+1 h-1 m=0 a j+h-1-m b m X k+1-j ⎤ ⎦ 2 ≤ +∞ j=k+1 +∞ l=k+1 h-1 m=0 a j+h-1-m b m h-1 m=0 a l+h-1-m b m σ(j -l) ≤ C 1 C 2 h d+2δ d 2 +∞ j=k+1 +∞ l=k+1 j -d-1+δ l -d-1+δ σ(j -l). ( 37
)
The series in (37) is independent of h and using Theorem 2.1 we have that it is bounded by Ck -1+δ with C independent of k. This concludes the proof.

Remark 4.2. Gathering all the above results about the truncated h-step predictor X k (h), the variance of prediction error is

E(X k+h -X k (h)) 2 = σ(0) -σ 2 ε +∞ l=h b 2 l + R(h, k) where σ(0) -σ 2 ε +∞ l=h b 2
l is the optimal mean-squared error of linear prediction while R(h, k) quantifies the truncating effect. Truncation always increases the variance of forecasting error but it is possible to keep it's effect as small as possible. Indeed, taking (31) into account, the condition

h 2d k -1 = o(h 2d-1 ), (38)
which is satisfied if h = o(k), insures that the truncating effect is negligible with respect to the quality of the optimal linear predictor X k (h).

In other words, long-step prediction in a long-memory context is interesting when the prediction is based on a sufficiently large sample (X 1 , . . . , X k ).

We can compare the prediction error of the truncated Wiener-Kolmogorov predictor with the least-squares predictor defined as the projection on the finite past (X 1 , . . . , X k ). Let X k (h) the predictor of X k+h . It then can be written as:

X k (h) = - k j=1 c j,k X k+1-j ,
where the vector (c j,k ) 1≤j≤k minimises the mean-squared error:

E X k (h) -X k+h 2 .
Thus the coefficients (c j,k ) 1≤j≤k satisfy the following relation:

(c j,k ) 1≤j≤k = -Σ -1 k (σ h-1+j ) 1≤j≤k (39)
where Σ k is the autocovariance matrix of the vector (X 1 , . . . , X k ) i.e.

Σ k = ⎛ ⎜ ⎝ σ(0) σ(1) . . . σ(k -1) . . . . . . . . . . . . σ(k -1) σ(k -2) . . . σ(0) ⎞ ⎟ ⎠.
By construction of X k (h), the mean-squared error of prediction is lower than σ(0) for any k and h. Howewer it can be interesting to search a condition on k and h to have the prediction error due to the projection negligible with respect to the information (31) given by the Wiener-Kolmogorov predictor.

As X k (h) is the projection of X k+h onto (X 1 , . . . , X k ), the mean-squared prediction error is also lower than the prediction error of the truncated Wiener-Kolmogorov predictor (see Fig. 3). The mean-squared error of prediction due to the projection onto the span of (X 1 , . . . , X k ) tends at least as fast to zero as the mean-squared error due to truncation of the least-squares predictor. We obtain that under the assumptions of Proposition 4.1 and condition (38), the prediction error due to the projection is negligible with respect to the information (31) given by the Wiener-Kolmogorov predictor. In Figure 3, we show that the comportments of the truncated predictor and of the least-squares predictor are similar. In the both cases, the prediction error due to the prediction method is sufficiently small with respect to the available information (31) to obtain a good predictor in the case the fractionally integrated noise defined in [START_REF] Soares | Forecasting electricity demand using generalized long memory[END_REF] with k = 80 and h ∈ [START_REF] Barkoulas | Long-memory forecasting of US monetary indices[END_REF]150].

Appendix: Proof of Proposition 2.7

We have to find an asymptotic equivalent of We have proved in Proposition 2.4 that (42) is equivalent to Ck -1 . And it is easy to show (by Abel transformations), that (42) and (42) are negligible with respect to (42). This concludes the proof.

Figure 3 .

 3 Figure 3. Mean-squared error of X k (h) (MMSE), X k (h) (TPMSE) and X k (h) (LLSPE) for d = 0.4 and k = 80 for fractionally integrated noise defined in (17).

aaj - 3 / 2 l 2 .

 322 j a l σ(lj). First we replace the coefficients and the autocovariances by their asymptotic expressions given in (21) and (22). We obtain:+∞ j=k+1 +∞ l=k+1 a j a l σ(lj) = j a l σ(lj) = +∞ j=k+1 cos[(jd)ν + (dπ/2)] Γ(-d) sin -d (ν)where B j,l is:B j,l = cos (jd)ν + dπ 2 cos (ld)ν + dπ 2 cos((jl)ν)(jl) -1-d Γ(jl + 2d) Γ(jl + 1) × 1 + O(j d-1/2 ) 1 + O(l d-1/2 ) 1 + O((jl) -1 ) .The first term (40) is bounded by O(k -2d-1 ) and thus negligible with respect to k -1 . For the second term (40): jl)ν)(jl) -1-d Γ(jl + 2d) Γ(jl + 1) jl)ν)(jl) -1-d Γ(jl + 2d) Γ(jl + 1jl)ν)(jl) -1-d Γ(jl + 2d) Γ(jl + 1d)ν + dπ 2 cos (ld)ν + dπ 2 cos((jl)ν)(jl) -1-d Γ(jl + 2d) Γ(jl + 1) O((jl) -1 ).(41)We first prove that the terms (41), (41) and (41) are bounded by O(k d-3/2 ) and thus negligible with respect to k -1 . For example, we study (41): jl)ν)(jl) -1-d Γ(jl + 2d) Γ(jl + 1) O(j d-1/2 ) -1-d (jl) 2d-1 = O k d-3/The calculations for (41) and (41) have the same form. The dominant term of (40) is then (41). (41) can be broken into three parts:d)ν + dπ 2 cos (ld)ν + dπ 2 cos((jl)ν)(jl) -1-d Γ(jl + 2d) Γ(jl + 1-1-d Γ(jl + 2d) Γ(jl + 1) jl)ν)(jl) -1-d Γ(jl + 2d)Γ(jl + 1) (2j -2d)ν + dπ] + cos [(2l -2d)ν + dπ]] (jl) -1-d Γ(jl + 2d) Γ(jl + 1) •(42)

Acknowledgements. I would like to thank Anne Philippe, my PhD advisor, who monitored my work. I am also grateful to the referees for their helpful comments.