Paul Carneau
email: paul.carneau@enpc.fr

Romain Mesnil

Nicolas Roussel

Olivier Baverel

An exploration of 3d printing design space inspired by masonry

Keywords: 3d printing, cementitious material, funicular shell, masonry, digital fabrication, robotics, form finding

This paper describes an approach of a process-aware exploration of the design space of building components constructed by extrusion of cementitious material (or concrete 3d printing). In an attempt to broaden the geometries that are being printed today, and to build cantilevers, vaults and domes without using temporary support, we look back at construction techniques used in masonry structures, and analyse the stability of the fabrication process at the scale of the brick, the layer and the final structure. Those strategies are applied to 3d printing with regards to the material properties and the technological aspects of the process. Three examples are actually printed to validate the approach, namely a Nubian vault, a dome and an arch, all without temporary support. Finally a framework for generation of printable geometries and corresponding toolpaths is proposed.

1 Introduction

Context

Additive fabrication of cementitious material -or mortar 3d printing -has known a rapid expansion in the last decade which has led to multiple innovations in term of material rheology, printing technology (robotics and extrusion head) and their applications. Research has focused on finding the best mortar for printing, on the formation of "cold joints" at the interface between layers or on the potential reinforcement of the printed structures. A field that is yet to explore is the design space of printable geometries. 3d printed geometries are limited by the resistance of the fresh material and the printing process. Non standard geometries have been achieved by use of external supports [START_REF] Lim | Modelling curved-layered printing paths for fabricating large-scale construction components[END_REF]. However, it is not satisfying as a sustainable solution due to costs and environmental impact of such temporary structures. The challenge is hence to solve the problem of printing free of temporary support and yet without loosing architectural freedom or structural efficiency. This paper proposes a geometry exploration of printable structure that does not require external support. In a first part we look at 3d printing processes that have been developed so far based on two asymptotic strategies in terms of material and technology. Then in a second part, we present three examples of ancient masonry brickworks and their implementation using 3d printing process. Finally we propose a framework for generation of geometries and corresponding toolpaths.

Previous works

Based on what has been done so far by the different actors in the field of 3d printing -academics or industrials actors -the strategies in place can be bounded by two asymptotic strategies in terms of material properties described by Roussel in [START_REF] Roussel | Rheological requirements for printable concretes[END_REF], robotic complexity and control of the process.

• a pure extrusion process which is the most common strategy in use up to this day.

It consists in printing with a 3-axis robot (namely a printing head mounted on a gantry bridge) moving in Cartesian coordinates, extruding a material that does not go through a phase change and which has an initial yield stress relatively high (500 Pa to 1kPa). The shape of the layer is given entirely by the shape of the nozzle. The extrusion occurs in successive horizontal planes (some talks of 2.5D printing) and the only way for this technology to create global cantilever is by creating corbels with the layers (local cantilevers).

• a sheared material deposition is the opposite strategy using a 6-axis robot to print layers that are not inevitably horizontal. This strategy is introduced in [START_REF] Gosselin | Largescale 3d printing of ultra-high performance concrete a new processing route for architects and builders[END_REF]. The mortar used goes through a phase change in the extrusion head by addition of additives giving to the material a low initial yield stress (100 Pa) but a high structuration rate. The layer is shaped by squeezing it between the nozzle and the previous layer. Different layer's heights can be obtained just by changing the printing parameters.

As Pegna mentioned in [START_REF] Pegna | Exploratory investigation of solid freeform construction[END_REF], 3d printing can be seen as a new approach to masonry. 3d printing and masonry are additive manufacturing processes sharing the disadvantage of using a material with a poor tensile strength. Historically, masonry builders have developed strategies to overcome this issue and build cantilevered structures without the use of temporary supports. From these strategies, many of which are detailed in [START_REF] Choisy | L'art de batir chez les Byzantins[END_REF] by Auguste Choisy, we can cite the creation of corbels, the inclination of layers of a dome, the construction of barrel vault by successive inclined arches or the development of squinches by the Persians.

Method statement

Due to the poor mechanical properties of the fresh mortar, 3d printing main difficulty lies in the stability of the structure during the fabrication. By analogy to masonry, this issue can be tackled at different scales. The scale of the section when the material is extruded, the scale of the layer and the scale of the overall structure in its final configuration. In this paper, we explore similarities between masonry and 3d printing through three examples. Each representing a topology and a construction strategy developed by masonry builders. We analyse the printing parameters at play for the realisation of such structures, in terms of material properties and fabrication technology. The aim is to push the boundaries of the design space of printable geometries and propose a framework for the design of such structures and the generation of corresponding toolpaths.

2 Masonry and 3d printing 2.1 Planar cantilever

The geometry

The first example is an opening in a wall, so a cantilever in two dimensions. The crossing takes the shape of a pointed arch. The wall is made of a double layer for two purposes. First, this is a way to have a closed toolpath and avoid having to stop and start the printing set up. Secondly, it allows for the overall thickness of the wall to evolve with the height. It is wider at the base than at the top, providing more stability to the wall. Fig. 1 illustrates the geometry of this two dimensional crossing. The main difficulty in this example is to ensure the local stability of a layer around the opening, by avoiding the material to flow after the extrusion. The strategy chosen to tackle this problem is to incline the layers in order to avoid any local cantilever in the layer which would create bending. By doing so, the contact surface is maximised between two consecutive layers, and the material only sees pure shear stress. This is achieved by slicing the geometry not horizontally as it is usually the case, but by offsetting the layers on the object surface with a constant distance, until the two half-arches connect. This can be done using Adiels et al. slicing algorithm described in [START_REF] Adiels | Brick patterns on shells using geodesic coordinates[END_REF] and initially intended for masonry structures. The result is a set of layers that bend downwards near the opening (see Fig. 1.b). In the upper part (the cover ), the distance between layers increases locally to compensate gradually for the non-planarity of the layers until reaching a horizontal level.

The printing process

In practice, the printing head connected to the robot rotates so that the extrusion nozzle remains tangent to the surface of the object at all times. This makes full use of the 6-axis robot and its capacity to reach any target in space with different orientations. The strategy described above ensure the stability of the material locally, just after the extrusion. The overall structural system of each half of the wall that ensure global stability until the arch is complete, is inspired by a construction technique used to build Gothic vault without formwork (see Fig. 2.left). The cantilever structure itself remains in compression thanks to the stone-weighted rope device, working in tension. In our case, the tensile stresses are located in the walls. For such a structure, reinforced concrete could help achieve much higher cantilevers. In practice the printing process can be divided in two part, one for the bottom and another for the upper part of the object.

• in the lower part, all parameters -printing speed and extrusion flow rate -are kept constant.

• in the upper part, the flow rate q c is kept constant. In order to compensate for the thickness increase in the layer and keep the same width d, the speed c r is decreased locally with respect to mass conservation relation 1.

Q c = c r • h • d (1)
with Q c the flow of concrete, c r the speed of the robot, h and d the height and width of the layer.

Observations and limitations

In this example, we managed to print a two dimensional arch (see Fig. 3) with a global inclination of 37 • and a maximum local inclination up to 60 • in the upper part of the arch. This demonstrates the local stability of a layer even with high inclination can be achieved given that the material's initial yield stress is sufficient. Consequently, a global cantilever can be obtained by 3d printed.

Translated vault

The second example is a barrel vault, built with a technique developed by the Nubians in Egypt, examples of which are still standing after over 3300 years. This technique allows to built barrel vault without any temporary support. The construction can then be assimilated to a horizontal extrusion of an inclined arch, where each layer is supported by the previous ones (see Fig. 2.right).

Construction sequence

Throughout construction of a Nubian vault, each layer experiences three phases in terms of stresses. By considering one layer in particular, the three phases are as follows:

• The first phase corresponds to the extrusion of the material. At this stage, it has to withstand its own self-weight to avoid local collapse of the layer. Since the layer is inclined, the stress in the material is higher than when extruded horizontally. The material's initial yield stress must be adapted to the layer inclination angle θ.

• once the layer is fully in place, it starts behaving as an arch in compression. In the same time, it has to withstand the loads brought by successive new layers. In 3d printing, that means, the structuration rate of the material A thix has to be sufficient so that the stress accumulated by the addition of layers stays below the yield stress τ c at all times. The higher the angle with the vertical θ is, the higher is the influence of the new layers.

• the third phase appear when the addition of new layers has no influence anymore on the stress state of our considered layer. Since the structure evolves horizontally, the layers are not continuously loaded as in vertical extrusion. This characteristic of Nubian vault fabrication make them not limited in length.

To summarise, if θ = 0 (layers are vertical), the initial stress in the material is maximum but the layer reaches its final state quickly. With a high inclination angle θ, the initial stress is lower but the permanent regime is reached later.

Toolpath generation

The geometry of the vault is generated using form-finding method leading to a funicular shape under self-weight. The vault once finished, is fully in compression. This is permitted by the absolute precision of a robot in space. Indeed, for a masonry vault built manually, the shape has to be approximated by arcs of circle, so that a mason can follow the geometry using simple tools (wire, level, etc.). This shows that architectural freedom brought by digital fabrication can support a design process where shape follows forces. In masonry, the Nubian vault usually takes support on a vertical gable wall (see Fig. 2.right). This lead to discontinuous first layers and it has to be avoided for 3d printing process due technical difficulty to stop and restart the set-up. The 3d printing vault then has to start from a fully horizontal layer before shifting to inclined arches. Fig. 4 shows the dimensions of the printed vault. In this example, the layers are all planar. They come from the intersection of the vault surface with a set of planes. In the transition part, the planes inclination gradually increase from horizontal to 40 • . This inclination is then kept constant all the way through the cylindrical part (see Fig. 4.right). The frames describing the orientation of the robot are generated from those layer curves. For each point on the curve, we define the vector tangent to the curve and the vector normal to the vault surface. The extrusion nozzle orientation is given by the cross product of those two vectors (illustrated in Fig. 5). This way, the contact surface between two layers is maximised and always the same. In practice, by adding the robot accessibility constraints, the maximum inclination of the frame with the horizontal is set to 60 • (this angle is reached at the apex of the vault). As a result, the path of the robot is planar, but its orientation gives torsion to the layers. The final step in the generation of the toolpath is the setting of the printing parameters. For this vault, the speed of the robot in the cylindrical part is set at 200mm/s, keeping a constant inter-layer time of 10s.

TIME transition part cylindrical part

Results

This example validate the possibility to 3d print barrel vaults without temporary support in a similar way it is done in masonry. It shows that the material can be printed with an inclination up to 60 • without collapsing, pointing out in the same time the pertinence of a 6-axis robot to broaden the range of printable geometries.

Dome

The last example is a dome with a square base. Circular squinches are generating at each corner and spandrels link them together to reach a horizontal circular perimeter (see Fig. 6). From there up, the geometry consists in a simple hemispherical dome, described by Cowan as a true dome in [START_REF] Cowan | A history of masonry and concrete domes in building construction[END_REF]. The slope of the squinches in the corners have an inclination of 45 • . The goal of this example is to show the potential and imitations of printing with a 3-axis robot, moving in Cartesian coordinates and hence unable to modify the orientation of the nozzle. The layers are all horizontal, and the height between them is constant. In addition to that, all the parameters remain unchanged during the printing. Consequently, the cantilever is made by corbelling layers on top of each other and creating local cantilevers. In this configuration, the material requires a high enough initial yield stress. Once a layer is complete, the stability is ensured structurally by the apparition of compressive rings. The slicing strategy used for the generation of the geometry and the toolpath is simple and fast compare to the ones described in previous parts. However, it comes with limitations:

• the local cantilevers of the layers induce bending in the material. Since this cantilever increases with the inclination of the surface of the dome (see Fig. 7.left), the stress in the material gets higher as the print goes. This leads to a local collapse of the layers closed to the top of the dome.

• in addition to the increasing stress in the material, the geometry of the structure makes the layer's length decreases the more it goes up (Fig. 7.right). Since the robot speed is constant, the inter-layer time decreases proportionnally to the length. Thus the last layers have less time to build thixotropy and increase their yield stress accentuating the risk of failure.

To summarise, this strategy advantages lay in the simplicity to generate the toolpath for the robot and the easy movement of the robot reducing the risks of clashes and simplifying the calibration process. On the other hand, the slicing eventually leads to an increase of the stresses in the material, accentuated by an increase of the loading rate.

Results

The examples presented in this paper help to show the similarities between masonry construction techniques and mortar 3d printing process. Using different structural typologies, namely a two dimensional opening, a barrel vault and a dome, we can draw general conclusions in regards to printable geometries and toolpath generation. The challenges being to ensure stability of the printed object during fabrication and mechanical efficiency of the final structure accounting for the material used. Those examples show that 3d printing process faces challenges of stability at different scale. The local scale of the component, where the material must withstand its on weight as soon as it is extruded. The strategy chosen to perform the global cantilever, is translated at the layer's scale to either a local cantilever inducing bending moment in the material, or an inclined layer, where contact is maximised between layers and the stress is mainly shear. At the scale of the layer, a structural subsystem is set in place -arches for the barrel vault, or compression rings for the dome -that ensure its stability by making the material work mainly in compression. Finally, both for masonry and 3d printing, the targeted final geometry must be as closed as possible to the funicular of the structure under its own weight. The structure is then fully in compression, and the lack of reinforcement can be justified. The aim of the authors is to generalised this process to free-formed shapes in order to extend furthermore the design space of printable geometries.

Conclusion

This paper presents an exploration of printable geometries, aiming at building structures without the use of temporary supports. It shows the possibility to print cantilever structures even with a material with poor tensile strength and in this way, validates the approach of finding inspiration in masonry techniques. The problem of stability of the structure is tackled by dividing it in different scales, the scale of the material just after the extrusion, the scale of the layer and the scale of the final structure. Eventually, expending the boundaries of printable geometries can address the need for architectural freedom while providing answers to environmental issues such as wastes generated by complex temporary structures and a solution to reduce fabrication costs and delays.

Figure 1 :

 1 Figure 1: Toolpath of the 2D crossing. a.3D view, b. front view with distinction between upper and lower parts, c. top view

Figure 2 :

 2 Figure 2: left: The stone-weighted rope device for erecting Gothic vaults (John Fitchen, The Construction of Gothic Cathedrals, 1961, fig. 69, p.182 ), right: Nubian vault construction by Auguste Choisy from "L'Art de btir chez les Byzantins"

Figure 3 :

 3 Figure 3: Wall opening

Figure 4 :

 4 Figure 4: left: 3d printed Nubian vault and its dimensions -height, width, and thickness, right: slicing of the vault

Figure 5 :

 5 Figure 5: Printing path and local orientation of the extrusion nozzle

Figure 6 :

 6 Figure 6: Dome on squinches

Figure 7 :

 7 Figure 7: left: effect of corbelling on a spherical dome with gradual increase of the cantilever, right: decrease of the layer's length (in mm)


Jean-Marc Bardet
email: bardet@univ-paris1.fr

Hatem Bibi
email: hatem.bibi@malix.univ-paris1.fr

Abdellatif Jouini
email: abdellatif.jouini@fst.mu.tn

Adaptive wavelet based estimator of the memory parameter for stationary Gaussian processes

This work is intended as a contribution to a wavelet-based adaptive estimator of the memory parameter in the classical semi-parametric framework for Gaussian stationary processes. In particular we introduce and develop the choice of a data-driven optimal bandwidth. Moreover, we establish a central limit theorem for the estimator of the memory parameter with the minimax rate of convergence (up to a logarithm factor).

The quality of the estimators are attested by simulations.

Introduction

Let X = (X t ) t∈Z be a second-order zero-mean stationary process and its covariogram be defined

r(t) = E(X 0 • X t ), for t ∈ Z.
Assume the spectral density f of X, with

f (λ) = 1 2π • k∈Z r(k) • e -ik ,
exists and represents a continuous function on [-π, 0)[∪]0, π]. Consequently, the spectral density of X should satisfy the asymptotic property,

f (λ) ∼ C • 1 λ D when λ → 0,
with D < 1 called the "memory parameter" and C > 0. If D ∈ (0, 1), the process X is a so-called long-memory process, if not X is called a short memory process (see [START_REF] Doukhan | Theory and applications of long-range dependence[END_REF], for more details).

This paper deals with two semi-parametric frameworks which are:

• Assumption A1: X is a zero mean stationary Gaussian process with spectral density satisfying

f (λ) = |λ| -D • f * (λ) for all λ ∈ [-π, 0)[∪]0, π],
with f * (0) > 0 and f * ∈ H(D ′ , C D ′ ) where 0 < D ′ , 0 < C D ′ and

H(D ′ , C D ′ ) = g : [-π, π] → R + such that |g(λ) -g(0)| ≤ C D ′ • |λ| D ′ for all λ ∈ [-π, π] .
• Assumption A1': X is a zero-mean stationary Gaussian process with spectral density satisfying

f (λ) = |λ| -D • f * (λ) for all λ ∈ [-π, 0)[∪]0, π],
with f * (0) > 0 and f * ∈ H ′ (D ′ , C D ′ ) where 0 < D ′ , C D ′ > 0 and

H ′ (D ′ , C D ′ ) = g : [-π, π] → R + such that g(λ) = g(0) + C D ′ |λ| D ′ + o |λ| D ′ when λ → 0 .
Remark 1 A great number of earlier works concerning the estimation of the long range parameter in a semiparametric framework (see for instance [START_REF] Giraitis | Rate optimal semi-parametric estimation of the memory parameter of the Gaussian time series with long range dependence[END_REF][START_REF] Giraitis | Adaptive semiparametric estimation of the memory parameter[END_REF] are based on Assumption A1 or equivalent assumption on f . Another expression (see [START_REF] Robinson | Gaussian semiparametric estimation long range dependence[END_REF] times differentiable around λ = 0 with f * (s) satisfying a Lipschitzian condition of degree 0 < ℓ < 1 around 0, then D ′ ≤ s + ℓ.So for our purpose, D ′ is a more pertinent parameter than s + ℓ (which is often used in no-parametric literature). Finally, the Assumption A1' is a necessary condition to study the following adaptive estimator of D.

We have

H ′ (D ′ , C D ′ ) ⊂ H(D ′ , C D ′ ).
Fractional Gaussian noises (with D ′ = 2) and FARIMA[p,d,q] processes (with also D ′ = 2) represent the first and well known examples of processes satisfying Assumption A1' (and therefore Assumption A1).

Remark 2 In [START_REF] Andrews | Adaptive local polynomial Whittle estimation of long-range dependence[END_REF], an adaptive procedure covers a more general class of functions than H(D ′ , C D ′ ), i.e. H AS (D ′ , C D ′ ) defined by:

H AS (D ′ , C D ′ ) = g : [-π, π] → R + such that, as λ → 0 g(λ) = g(0) + k i=0 C ′ i λ 2i + C D ′ |λ| D ′ + o |λ| D ′ with 2k < D ′ ≤ 2k + 2
.

Unfortunately, the adaptive wavelet based estimator defined below, as local or global log-periodogram estimators, is unable to be adapted to such a class (and therefore, when D ′ > 2, its convergence rate will be the same than if the spectral density is included in H AS (2, C 2 ), at the contrary to Andrew and Sun estimator).

This work is to provide a wavelet-based semi-parametric estimation of the parameter D. This method has been introduced by Flandrin (1989) and numerically developed by Abry et al. (1998Abry et al. ( , 2001) ) and [START_REF] Veitch | On the Automatic Selection of the Onset of Scaling[END_REF]. Asymptotic results are reported in [START_REF] Bardet | Wavelet estimator of long range-dependant processes[END_REF] and more recently in [START_REF] Moulines | On the spectral density of the wavelet coefficients of long memory time series with application to the log-regression estimation of the memory parameter[END_REF].

Taking into account these papers, two points of our work can be highlighted : first, a central limit theorem based on conditions which are weaker than those in [START_REF] Bardet | Wavelet estimator of long range-dependant processes[END_REF]. Secondly, we define an auto-driven estimator Dn of D (its definition being different than in [START_REF] Veitch | On the Automatic Selection of the Onset of Scaling[END_REF]. This results in a central limit theorem followed by Dn and this estimator is proved rate optimal up to a logarithm factor (see below). Below we shall develop this point.

Define the usual Sobolev space W (β, L) for β > 0 and L > 0,

W (β, L) = g(λ) = ℓ∈Z g ℓ e 2πiℓλ ∈ L 2 ([0, 1]) / ℓ∈Z (1 + |ℓ|) β |g ℓ | < ∞ and ℓ∈Z |g ℓ | 2 ≤ L .
Let ψ be a "mother" wavelet satisfying the following assumption:

Assumption W (∞) : ψ : R → R with [0, 1]-support and such that 1. ψ is included in the Sobolev class W (∞, L) with L > 0;

2. The function ψ is a smooth compactly supported function (the interval [0, 1] is meant for better readability, but the following results can be extended to another interval) with its m first vanishing moments. If D ′ ≤ 2 and 0 < D < 1 in Assumptions A1, Assumption W (∞) can be replaced by a weaker assumption:

Assumption W (5/2) : ψ : R → R with [0, 1]-support and such that 1. ψ is included in the Sobolev class W (5/2, L) with L > 0;

2.

1 0 ψ(t) dt = 0 and ψ(0) = ψ(1) = 0.

Remark 3 The choice of a wavelet satisfying Assumption W (∞) is quite restricted because of the required smoothness of ψ. For instance, the function ψ(t) = (t 2t + a) exp(-1/t(1t)) and a ≃ 0.23087577 satisfies Assumption W (∞). The class of "wavelet" checking Assumption W (5/2) is larger. For instance, ψ can be a dilated Daubechies "mother" wavelet of order d with d ≥ 6 to ensure the smoothness of the function ψ.It is also possible to apply the following theory to "essentially" compactly supported "mother" wavelet like the Lemarié-Meyer wavelet. Note that it is not necessary to choose ψ being a "mother" wavelet associated to a multi-resolution analysis of L 2 (R) as in the recent paper of [START_REF] Moulines | On the spectral density of the wavelet coefficients of long memory time series with application to the log-regression estimation of the memory parameter[END_REF]. The whole theory can be developed without this assumption, in which case the choice of ψ is larger.

If Y = (Y t ) t∈R is a continuous-time process for (a, b) ∈ R * + × R, the "classical" wavelet coefficient d(a, b) of the process Y for the scale a and the shift b is d(a, b) = 1 √ a R ψ( t a -b)Y t dt. (1)
However, this formula (1) of a wavelet coefficient cannot be computed from a time series. The support of ψ being [0, 1], let us take the following approximation of formula (1) and define the wavelet coefficients of

X = (X t ) t∈Z by e(a, b) = 1 √ a a k=1 ψ( k a )X k+ab , (2)
for (a, b) ∈ N * + × Z. Note that this approximation is the same as the wavelet coefficient computed from Mallat algorithm for an orthogonal discrete wavelet basis (for instance with Daubechies mother wavelet).

Remark 4 Here a continuous wavelet transform is considered. The discrete wavelet transform where a = 2 j , in other words numerically very interesting (using Mallat cascade algorithm) is just a particular case. The main point in studing a continuous transform is to offer a larger number of "scales" for computing the data-driven optimal bandwidth (see below).

Under Assumption A1, for all b ∈ Z, the asymptotic behavior of the variance of e(a, b) is a power law in scale a (when a → ∞). Indeed, for all a ∈ N * , (e(a, b)) b∈Z is a Gaussian stationary process and (see Section more details in 2):

E(e 2 (a, 0)) ∼ K (ψ,D) • a D when a → ∞, (3)
with a constant K (ψ,D) such that,

K (ψ,α) = ∞ -∞ | ψ(u)| 2 • |u| -α du > 0 for all α < 1, ( 4
)
where ψ is the Fourier transform of ψ (the existence of K (ψ,α) is established in Section 5). Note that (3) is also checked without the Gaussian hypothesis in Assumption A1 (the existence of the second moment order of X is sufficient).

The principle of the wavelet-based estimation of D is linked to this power law a D . Indeed, let (X 1 , . . . , X N ) be a sampled path of X and define T N (a) a sample variance of e(a, .) obtained from an appropriate choice of shifts b, i.e.

T N (a) = 1 [N/a] [N/a] k=1 e 2 (a, k -1). (5)
Then, when

a = a N → ∞ satisfies lim N →∞ a N • N -1/(2D ′ +1) = ∞, a central limit theorem for log( T N (a N ))
can be proved. More precisely we get

log( T N (a N )) = D log(a N ) + log(f * (0)K (ψ,D) ) + a N N • ε N , with ε N L -→ N →∞ N (0, σ 2 (ψ,D)
) and σ 2 (ψ,D) > 0. As a consequence, using different scales (r 1 a N , . . . , r ℓ a N )) where (r 1 , . . . , r ℓ ) ∈ (N * ) ℓ with a N a "large enough" scale, a linear regression of (log( T N (r i a N )) i by (log(r i a N )) i provides an estimator D(a N ) which satisfies at the same time a central limit theorem with a convergence rate N aN .

But the main problem is : how to select a large enough scale a N considering that the smaller a N , the faster the convergence rate of D(a N ). An optimal solution would be to chose a N larger but closer to N 1/(2D ′ +1) , but the parameter D ′ is supposed to be unknown. In [START_REF] Veitch | On the Automatic Selection of the Onset of Scaling[END_REF], an automatic selection procedure is proposed using a chi-squared goodness of fit statistic. This procedure is applied successfully on a large number of numerical examples without any theoretical proofs however. Our present method is close to the latter. Roughly speaking, the "optimal" choice of scale (a N ) is based on the "best" linear regression among all the possible linear regressions of ℓ consecutive points (a, log( T N (a))), where ℓ is a fixed integer number.

Formally speaking, a contrast is minimized and the chosen scale ãN satisfies:

log(ã N ) log N P -→ N →∞ 1 2D ′ + 1 .
Thus, the adaptive estimator DN of D for this scale ãN is such that :

N ãN ( DN -D) L -→ N →∞ N (0, σ 2 D ),
with σ 2 D > 0. Consequently, the minimax rate of convergence N D ′ /(1+2D ′ ) , up to a logarithm factor, for the estimation of the long memory parameter D in this semi-parametric setting (see [START_REF] Giraitis | Rate optimal semi-parametric estimation of the memory parameter of the Gaussian time series with long range dependence[END_REF] is given by DN .

Such a rate of convergence can also be obtained by other adaptive estimators (for more details see below).

However, DN has several "theoretic" advantages: firstly, it can be applied to all D < -1 and D ′ > 0 (which are very general conditions covering long and short memory, in fact larger conditions than those usually required for adaptive log-periodogram or local Whittle estimators) with a nearly optimal convergence rate. Secondly, DN satisfies a central limit theorem and sharp confidence intervals for D can be computed (in such a case, the asymptotic σ 2 D is replaced by σ 2

DN

, for more details see below). Finally, under additive assumptions on ψ (ψ is supposed to have its first m vanishing moments), DN can also be applied to a process with a polynomial trend of degree ≤ m -1.

We then give a several simulations in order to appreciate empirical properties of the adaptive estimator DN .

First, using a benchmark composed of 5 different "test" processes satisfying Assumption A1' (see below), the ondly it is a very robust estimator: it is not sensitive to possible polynomial trends and seems to be consistent in non-Gaussian cases. Finally, the graph of the log-log regression of sample variance of wavelet coefficients is meaningful and may lead us to model data with more general processes like locally fractional Gaussian noise (see [START_REF] Bardet | Identification of the multiscale fractional Brownian motion with biomechanical applications[END_REF].

The central limit theorem for sample variance of wavelet coefficient is subject of section 2.Section 3 is concerned with the automatic selection of the scale as well as the asymptotic behavior of DN . Finally simulations are given in section 4 and proofs in section 5.

A central limit theorem for the sample variance of wavelet coefficients

The following asymptotic behavior of the variance of wavelet coefficients is the basis of all further developments.

The first point that explains all that follows is the Property 1 Under Assumption A1 and Assumption W (∞), for a ∈ N * , (e(a, b)) b∈Z is a zero mean Gaussian stationary process and it exists M > 0 not depending on a such that, for all a ∈ N * ,

E(e 2 (a, 0)) -f * (0)K (ψ,D) • a D ≤ M • a D-D ′ . (6)
Please see Section 5 for the proofs. The paper of [START_REF] Moulines | On the spectral density of the wavelet coefficients of long memory time series with application to the log-regression estimation of the memory parameter[END_REF] gives similar results for multi-resolution wavelet analysis. The special case of long memory process can also be studied with weaker Assumption W (5/2), Property 2 Under Assumption W (5/2) and Assumption A1 with 0 < D < 1 and 0 < D ′ ≤ 2, for a ∈ N * , (e(a, b)) b∈Z is a zero mean Gaussian stationary process and (6) holds.

Two corollaries can be added to both those properties. First, under Assumption A1' a more precise result can be established.

Corollary 1 Under:

• Assumption A1' and Assumption W (∞);

• or Assumption A1' with 0 < D < 1, 0 < D ′ ≤ 2 and Assumption W (5/2);

then (e(a, b)) b∈Z is a zero mean Gaussian stationary process and

E(e 2 (a, 0)) = f * (0) K (ψ,D) • a D + C D ′ K (ψ,D-D ′ ) • a D-D ′ + o a D-D ′ when a → ∞. ( 7
)
This corollary is key point for the estimation of an appropriated sequence of scale a = (a N ). Indeed, when

f * ∈ H ′ (D ′ , C D ′ ), then f * ∈ H(D ′′ , C D ′′ ) for all D ′′ satisfying 0 < D ′′ ≤ D ′ . Therefore, Assumption A1'
is required for obtaining the optimal choice of a N , i.e. a N ≃ N 1/(2D ′ +1) (see below for more details). The following corollary generalizes the above Properties 1 and 2.

Corollary 2 Properties 1 and 2 are also checked when the Gaussian hypothesis of X is replaced by EX 2 k < ∞ for all k ∈ Z.

Remark 5 In this paper, the Gaussian hypothesis has been taken into account merely to insure the convergence of the sample variance (5) of wavelet coefficients following a central limit theorem (see below). Such a convergence can also be obtained for more general processes using a different proof of the central limit theorem, for instance for linear processes (see a forthcoming work).

As mentioned in the introduction, this property allows an estimation of D from a log-log regression, as soon as a consistant estimator of E(e 2 (a, 0)) is provided from a sample (X 1 , . . . , X N ) of the time series X. Define then the normalized wavelet coefficient such that

ẽ(a, b) = e(a, b) f * (0)K (ψ,D) • a D 1/2 for a ∈ N * and b ∈ Z. (8)
From property 1, it is obvious that under Assumptions A1 it exists M ′ > 0 satisfying for all a ∈ N * ,

E(ẽ 2 (a, 0)) -1 ≤ M ′ • 1 a D ′ .
To use this formula to estimate D by a log-log regression, an estimator of the variance of e(a, 0) should be considered (let us remember that a sample (X 1 , . . . , X N ) of is supposed to be known, but parameters (D,

D ′ , C D ′ ) are unknown).
Consider the sample variance and the normalized sample variance of the wavelet coefficient, for 1 ≤ a < N ,

T N (a) = 1 [ N a ] [ N a ] k=1 e 2 (a, k -1) and TN (a) = 1 [ N a ] [ N a ] k=1 ẽ2 (a, k -1). (9)
The following proposition specifies a central limit theorem satisfied by log TN (a), which provides the first step for obtaining the asymptotic properties of the estimator by log-log regression. More generally, the following multidimensional central limit theorem for a vector (log TN (a i )) i can be established.

Proposition 1 Define ℓ ∈ N \ {0, 1} and (r 1 , • • • , r ℓ ) ∈ (N * ) ℓ . Let (a n ) n∈N be such that N/a N -→ N →∞ ∞ and a N • N -1/(1+2D ′ ) -→ N →∞ ∞. Under Assumption A1 and Assumption W (∞), N a N log TN (r i a N ) 1≤i≤ℓ D -→ N →∞ N ℓ 0 ; Γ(r 1 , • • • , r ℓ , ψ, D) , (10)
with

Γ(r 1 , • • • , r ℓ , ψ, D) = (γ ij ) 1≤i,j≤ℓ the covariance matrix such that γ ij = 8(r i r j ) 2-D K 2 (ψ,D) d ij ∞ m=-∞ ∞ 0 ψ(ur i ) ψ(ur j ) u D cos(u d ij m) du 2 . ( 11
)
The same result under weaker assumptions on ψ can be also established when X is a long memory process.

Proposition 2 Define ℓ ∈ N \ {0, 1} and (r 1 , • • • , r ℓ ) ∈ (N * ) ℓ . Let (a n ) n∈N be such that N/a N -→ N →∞
∞ and

a N • N -1/(1+2D ′ ) -→ N →∞
∞. Under Assumption W (5/2) and Assumption A1 with D ∈ (0, 1) and D ′ ∈ (0, 2), the CLT (10) holds.

These results can be easily generalized for processes with polynomial trends if ψ is considered having its first m vanishing moments. i.e, Corollary 3 Given the same hypothesis as in Proposition 1 or 2 and if ψ is such that m ∈ N \ {0, 1} is satisfying, t p ψ(t) dt = 0 for all p ∈ {0, 1, . . . , m -1} the CLT (10) also holds for any process

X ′ = (X ′ t ) t∈Z such that for all t ∈ Z, EX ′ t = P m (t) with P m (t) = a 0 + a 1 t + • • • + a m-1 t m-1 is a polynomial function and
(a i ) 0≤i≤m-1 are real numbers.

3 Adaptive estimator of memory parameter using data driven optimal scales

The CLT [START_REF] Bhansali | Estimation of the memory parameter by fitting fractionally-differenced autoregressive models[END_REF] implies the following CLT for the vector (log

T N (r i a N )) i , N a N log T N (r i a N ) -D log(r i a N ) -log(f * (0)K (ψ,D) ) 1≤i≤ℓ D -→ N →∞ N ℓ 0 ; Γ(r 1 , • • • , r ℓ , ψ, D) .
and therefore, log

T N (r i a N ) 1≤i≤ℓ = A N •    D K    + 1 N/a N ε i 1≤i≤ℓ , with A N =        log(r 1 a N ) 1 : : log(r ℓ a N ) 1        , K = -log(f * (0) • K (ψ,D) ) and ε i 1≤i≤ℓ D -→ N →∞ N ℓ 0 ; Γ(r 1 , • • • , r ℓ , ψ, D) .
Therefore, a log-log regression of T N (r i a N ) 1≤i≤ℓ on scales r i a N 1≤i≤ℓ provides an estimator

D(a N ) K(a N ) of D K such that D(a N ) K(a N ) = (A ′ N • A N ) -1 • A ′ N • Y (r1,...,r ℓ ) aN with Y (r1,...,r ℓ ) aN = log T N (r i a N )) 1≤i≤ℓ , (12)
which satisfies the following CLT, Proposition 3 Under the Assumptions of the Proposition 1,

N a N D(a N ) K(a N ) - D K D -→ N →∞ N 2 (0 ; (A ′ • A) -1 • A ′ • Γ(r 1 , • • • , r ℓ , ψ, D) • A • (A ′ • A) -1 ), ( 13
)
with A =        log(r 1 ) 1 : : log(r ℓ ) 1        and Γ(r 1 , • • • , r ℓ , ψ, D)
given by [START_REF] Devore | Constructive approximations[END_REF].

Moreover, under Assumption A1' and if D ∈ (-1, 1), D(a N ) is a semi-parametric estimator of D and its asymptotic mean square error can be minimized with an appropriate scales sequence (a N ) reaching the wellknown minimax rate of convergence for memory parameter D in this semi-parametric setting (see for instance [START_REF] Giraitis | Rate optimal semi-parametric estimation of the memory parameter of the Gaussian time series with long range dependence[END_REF][START_REF] Bardet | Testing for the presence of self-similarity of Gaussian time series having stationary increments[END_REF]. Indeed,

Proposition 4 Let X satisfy Assumption A1' with D ∈ (-1, 1) and ψ the assumption W (∞). Let (a N ) be a sequence such that a N = [N 1/(1+2D ′ ) ].
Then, the estimator D(a N ) is rate optimal in the minimax sense, i.e. lim sup

N →∞ sup D∈(-1,1) sup f * ∈H(D ′ ,C D ′ ) N 2D ′ 1+2D ′ • E[ D(a N ) -D) 2 ] < +∞.
Remark 6 As far as we know, there are no theoretic results of optimality in case of D ≤ -1, but according to the usual following non-parametric theory, such minimax results can also be obtained. Moreover, in case of long-memory processes (if D ∈ (0, 1)), under Assumption A1' for X and Assumption W (5/2) for ψ, the estimator D(a N ) is also rate optimal in the minimax sense.

In the previous Propositions 1 and 3, the rate of convergence of scale a N obeys to the following condition,

N a N -→ N →∞ ∞ and a N N 1/(1+2D ′ ) -→ N →∞ ∞ with D ′ ∈ (0, ∞).
Now, for better readability, take a N = N α . Then, the above condition goes as follow:

a N = N α with α * < α < 1 and α * = 1 1 + 2D ′ . ( 14
)
Thus an optimal choice (leading to a faster convergence rate of the estimator) is obtained for

α = α * + ε with ε → 0+. But α * depends
(log T N (r i a N ), log(r i a N ))
. It seems to be an efficient and interesting numerical way to estimate D, but without theoretical proofs (contrary to global or local log-periodogram procedures which are proved to reach the minimax convergence rate, see for instance [START_REF] Moulines | Semiparametric spectral estimation for fractionnal processes[END_REF].

We suggest a new procedure for the data-driven selection of optimal scales, i.e. optimal α. Let us consider an important parameter, the number of considered scales ℓ ∈ N \ {0, 1, 2} and set (r 1 , . . . , r ℓ ) = (1, . . . , ℓ).

For α ∈ (0, 1), define also

• the vector Y N (α) = log T N (i • N α ) 1≤i≤ℓ ; • the matrix A N (α) =        log(N α ) 1 : : log(ℓ • N α ) 1        ; • the contrast, Q N (α, D, K) = Y N (α) -A N (α) • D K ′ • Y N (α) -A N (α) • D K . Q N (α, D, K) corresponds to a squared distance between the ℓ points log(i • N α ) , log T N (i • N α ) i and a line.
The point is to minimize this contrast for these three parameters. It is obvious that for a fixed α ∈ (0, 1) Q is minimized from the previous least square regression and therefore,

Q N ( α N , D(a N ), K(a N )) = min α∈(0,1),D<1,K∈R Q N (α, D, K).
with ( D(a N ), K(a N )) obtained as in relation [START_REF] Doukhan | Theory and applications of long-range dependence[END_REF]. However, since α N has to be obtained from numerical computations, the interval (0, 1) can be discretized as follows, Consequently, take

α N ∈ A N = 2 log N , 3 log N , . . . , log[N/ℓ] log N . Hence, if α ∈ A N , it exists k ∈ {2, 3, . . . , log[N/ℓ]} such that k = α • log N .
Q N (α) = Q N (α, D(a N ), K(a N )); then, minimize Q N for variables (α, D, K) is equivalent to minimize Q N for variable α ∈ A N , that is Q N ( α N ) = min α∈AN Q N (α).
From this central limit theorem derives Proposition 5 Let X satisfy Assumption A1' and ψ Assumption W (∞) (or Assumption W (5/2) if 0 < D < 1 and 0 < D ′ ≤ 2). Then,

α N = log a N log N P -→ N →∞ α * = 1 1 + 2D ′ .
This proves also the consistency of an estimator D ′ N of the parameter D ′ , Corollary 4 Taking the hypothesis of Proposition 5, we have

D ′ N = 1 -α N 2 α N P -→ N →∞ D ′ .
The estimator α N defines the selected scale a N such that a N = N αN . From a straightforward application of the proof of Proposition 5 (see the details in the proof of Theorem 1), the asymptotic behavior of a N can be specified, that is,

Pr N α * (log N ) λ ≤ N αN ≤ N α * • (log N ) µ P -→ N →∞ 1, (15)
for all positive real numbers λ and µ such that λ >

αN = α N + 3 (ℓ -2) D ′ N • log log N log N , ãN = N αN = N αN • log N 3 (ℓ-2) D ′ N and DN = D(ã N ). Then, with σ 2 D = (1 0) • (A ′ • A) -1 • A ′ • Γ(1, • • • , ℓ, ψ, D) • A • (A ′ • A) -1 • (1 0) ′ , N N αN DN -D D -→ N →∞ N (0 ; σ 2 D ) and ∀ρ > 2(1 + 3D ′ ) (ℓ -2)D ′ , N D ′ 1+2D ′ (log N ) ρ • DN -D P -→ N →∞ 0. ( 16
)
Remark 8 Both the adaptive estimators D N and DN converge to D with a rate of convergence rate equal to the minimax rate of convergence N D ′ 1+2D ′ up to a logarithm factor (this result being classical within this semi-parametric framework). Unfortunately, our method cannot prove that the mean square error of both these estimators reaches the optimal rate and therefore to be oracles.

To conclude this theoretic approach, the main properties satisfied by the estimators D N and DN can be summarized as follows:

1. Both the adaptive estimators D N and DN converge at D with a rate of convergence rate equal to the minimax rate of convergence N D ′ 1+2D ′ up to a logarithm factor for all D < -1 and D ′ > 0 (this being very general conditions covering long and short memory, even larger than usual conditions required for adaptive log-periodogram or local Whittle estimators) whith X considered a Gaussian process.

2. The estimator DN satisfies the CLT ( 16) and therefore sharp confidence intervals for D can be computed (in which case, the asymptotic matrix Γ(1, . . . , ℓ, ψ, D) is replaced by Γ(1, . . . , ℓ, ψ, D N )). This is not applicable to an adaptive log-periodogram or local Whittle estimators.

3. The main Property 1 is also satisfied without the Gaussian hypothesis. Therefore, adaptive estimators D N and DN can also be interesting estimators of D for non-Gaussian processes like linear or more general processes (but a CLT similar to Theorem 1 has to be established...).

4. Under additive assumptions on ψ (ψ is supposed to have its first m vanishing moments), both estimators D N and DN can also be used for a process X with a polynomial trend of degree ≤ m -1, which again cannot be yielded with an adaptive log-periodogram or local Whittle estimators.

Simulations

The adaptive wavelet basis estimators D N and DN are new estimators of the memory parameter D in the semi-parametric frame. Different estimators of this kind are also reported in other research works to have proved optimal. In this paper, some theoretic advantages of adaptive wavelet basis estimators have been highlighted. But what about concrete procedure and results of such estimators applied to an observed sample?

The following simulations will help to answer this question.

First, the properties (consistency, robustness, choice of the parameter ℓ and mother wavelet function ψ) of D N and DN are investigated. Secondly, in cases of Gaussian long-memory processes (with D ∈ (0, 1) and

D ′ ≤ 2)
, the simulation results of the estimator D N are compared to those obtained with the best known semi-parametric long-memory estimators.

To begin with, the simulations conditions have to be specified. The results are obtained from 100 generated independent samples of each process belonging to the following "benchmark". The concrete procedures of generation of these processes are obtained from the circulant matrix method, as detailed in Doukhan et al. 3. the Gaussian stationary process X (D,D ′ ) , such that its spectral density is

f 3 (λ) = 1 λ D (1 + λ D ′ ) for λ ∈ [-π, π], (17)
with D ∈ (-∞, 1) and

D ′ ∈ (0, ∞). Therefore f * 3 = 1 + λ D ′ ∈ H(D ′ , 1) with D ′ ∈ (0, ∞).
In the long memory frame, a "benchmark" of processes is considered for D = 0.1, 0.3, 0.5, 0.7, 0.9:

• fGn processes with parameters H = (D + 1)/2 and σ 2 = 1;

• FARIMA[0,d,0] processes with d = D/2 and standard Gaussian innovations;

• • X (D,D ′ ) Gaussian processes with D ′ = 1.

Properties of adaptive wavelet basis estimators from simulations

Below, we give the different properties of the adaptive wavelet based method.

Choice of the mother wavelet ψ: For short memory processes (D ≤ 0), let the wavelet ψ SM be such

that ψ SM (t) = (t 2 -t + a) exp(-1/t(1 -t)) with a ≃ 0.23087577. It satisfies Assumption W (∞). Lemarié-
Meyer wavelets can be also investigated but this will lead to quite different theoretic studies since its support is not bounded (but "essentially" compact).

For long memory processes (0 < D < 1), let the mother wavelet ψ LM be such that ψ LM (t) = 100 • t 2 (t -1) 2 (t 2t + 3/14)I 0≤t≤1 which satisfies Assumption W (5/2). Note that Daubechies mother wavelet or ψ SM lead to "similar" results (but not as good).

Choice of the parameter ℓ: This parameter is very important to estimate the "beginning" of the linear part of the graph drawn by points (log(a i ), log T (a i )) i . On the one hand, if ℓ is a too small a number (for instance ℓ = 3), another small linear part of this graph (even before the "true" beginning N α * ) may be chosen; consequently, the √ M SE (square root of the mean square error) of α N and therefore of D N or DN will be too large. On the other hand, if ℓ is a too large a number (for instance ℓ = 50 for N = 1000), the estimator α N will certainly satisfy α N < α * since it will not be possible to consider ℓ different scales larger than N α * (if D ′ = 1 therefore α ′ = 1/3, then a N has to satisfy: N/(50a N ) = 20/a N is a large number and (a N > N 1/3 = 10; this is not really possible). Moreover, it is possible that a "good" choice of ℓ depends on the "flatness" of the spectral density f , i.e. on D ′ . We have proceeded to simulations for each different values of ℓ (and N and D). Only √ M SE of estimators are presented. The results are specified in Table 1.

In Table 1, two phenomena can be distinguished: the detection of α * and the estimation of D:

• To estimate α * , ℓ has to be small enough, especially because of "D ′ close to 0" and so "α ′ close to 1" is possible. However, our simulations indicate that ℓ must not be too small (for instance ℓ = 5 leads to an important MSE for α N implying an important MSE for D N ) and seems to be independent of N (cases N = 1000 and N = 10000 are quite similar). Hence, our choice is ℓ 1 = 15 to estimate α * for any N .

• To estimate D, once α * is estimated, a second value ℓ 2 of ℓ can be chosen. We use an adaptive procedure which, roughly speaking, consists in determining the "end" of the acceptable linear zone. Firstly, we use again the same procedure than for estimating a N but with scales (a N /i) 1≤i≤ℓ1 and ℓ 1 = 15. It provides an estimator b N corresponding to the maximum of acceptable (for a linear regression) scales.

Secondly, the adaptive number of scales ℓ 2 is computed from the formula

ℓ 2 = ℓ = [ b N / a N ].
The simulations carried out with such values of ℓ 1 and ℓ 2 are detailed in Table 1.

As it may be seen in Table 1, the choice of parameters (ℓ 1 = 15, ℓ 2 = ℓ) provides the best results for estimating D, almost uniformly for all processes.

Consistency of the estimators α N and αN : the previous numerical results (here we consider ℓ 1 = 15)

show that α N and αN converge (very slowly) to the optimal rate α * , that is 0.2 for the first four processes and 1/3 for the fifth. Figure 1 illustrates the evolution with N of the log-log plotting and the choice of the onset of scaling.

Figure 1 shows that log T N (i • N α ) is not a linear function of the logarithm of the scales log(i • N α ) when N increases and α < α * (a consequence of Property 1: it means there is a bias). Moreover, if α > α * and α increases, a linear model appears with an increasing error variance.

Consistency and distribution of the estimators D N and DN : The results of Table 1 show the consistency with N of D N and DN only by using ℓ 1 = 15. Figure 2 Consistency in case of short memory: The following Table 2 provides the behavior of D N and DN if D ≤ 0 and D ′ > 0. Two processes are considered in such a frame: a FARIMA(0, d, 0) process with -0.5 < d < 0 and therefore -1 < D ≤ 0 (always with D ′ = 2) and a process X (D,D ′ ) and D < 0 and D ′ > 0.

The results are displayed in Table 4.1 (here N = 1000, N = 10000 and N = 100000, ℓ 1 = 15 and ℓ 2 = [5 N 0.1 ]) for different choices of D and D ′ . Thus it appears that D N and DN can be successively applied to short memory processes as well. Moreover, the larger D ′ , the faster their convergence rates.

Robustness of D N , DN : To conclude with the numerical properties of the estimators, four different processes not satisfying Assumption A1 ′ are considered:

• a FARIMA(0, d, 0) process (denoted P 1) with innovations satisfying a uniform law (and EX 2 i < ∞);

• a FARIMA(0, d, 0) process (denoted P 2) with innovations satisfying a distribution with density w.r.t.

Lebesgue measure f (x) = 3/4 * (1 + |x|) -5/2 for x ∈ R (and therefore

E|X i | 2 = ∞ but E|X i | < ∞);
• a FARIMA(0, d, 0) process (denoted P 3) with innovations satisfying a Cauchy distribution (and

E|X i | = ∞);
• a Gaussian stationary process (denoted P 4) with a spectral density f (λ) = (|λ|π/2) -1/2 for all λ ∈ [-π, π] \ {-π/2, π/2}. The local behavior of f in 0 is f (|λ|) ∼ π/2 |λ| D with D = 0, but the smoothness condition for f in Assumption A1 is not satisfied.

For the first 3 processes, D is varies in {0.1, 0.3, 0.5, 0.7, 0.9} and 100 independent replications are taken into account. The results of these simulations are given in Table 3.

As outlined in the theoretical part of this paper, the estimators D N and DN seem also to be accurate for L 2 -linear processes. For L α -linear processes with 1 ≤ α < 2, they are also convergent with a slower rate of convergence. Despite the spectral density of process P 4 does not satisfies the smoothness hypothesis requires in Assumptions A1 or A1', the convergence rates of D N and DN are still convincing. These results confirm the robustness of wavelet based estimators.

Comparisons with other semi-parametric long-memory parameter estimators from simulations

Here we consider only long-memory Gaussian processes (D ∈ (0, 1)) based on the usual hypothesis 0 < D ′ ≤ 2.

More precisely, the "benchmark" is: 100 generated independent samples of each process with length N = 10 3

and N = 10 4.

Comments on the results of Table 4: These simulations allow to distinguish four "clusters" of estimators.

• D BGK is obtained from a BIC-criterium hierarchical model selection (from 2 to 11 parameters, corresponding to the length of the approximation of the Fourier expansion of the spectral density) using Whittle estimation. For these simulations, the BIC criterion is generally minimal for 5 to 7 parameters to be estimated. Simulation results are not very satisfactory except for D = 0.1 (close to the short memory). Moreover, this procedure is rather time-consuming.

• D GRS offers good results for fGn and FARIMA(0, d, 0). However, this estimator does not converge fast enough for the other processes.

• Estimators D MS and D R have similar properties. They (especially D R ) are very interesting because they offer the same fairly good rates of convergence for all processes of the benchmark.

• Being built on similar principles, estimators D AT V and D N have similar behavior as well. Their convergence rates are the fastest for fGn and FARIMA(0, d, 0) and are almost close to fast ones for the other processes. Their times of computing, especially for D AT V for which the computations of wavelet coefficients with that the Mallat algorithm, are the shortest.

Conclusion:

Which estimator among those studied above has to be chosen in a practical frame, i.e. an observed time series? We propose the following procedure for estimating an eventual long memory parameter:

1. Firstly, since this procedure is very low time consuming and applicable to processes with smooth trends, draw the log-log regression of wavelet coefficients' variances onto scales. If a linear zone appears in this graph, consider the estimator D N (or D AT V ) of D.

2. If a linear zone appears in the previous graph and if the observed time series seems to be without a trend, compute D R .

3. Compare both the estimated value of D from confidence intervals (available for D N or D AT V and D R ).

Proofs

Proof (

). First, for a ∈ N * , E(e 2 (a, 0)) = 1 a a k=1 a k ′ =1 ψ(k/a)ψ(k ′ /a)E(X k X k ′ ) = 1 a a k=1 a k ′ =1 ψ(k/a)ψ(k ′ /a)r(k -k ′ ) = 1 a a k=1 a k ′ =1 ψ(k/a)ψ(k ′ /a) π -π f (λ)e iλ(k-k ′ ) dλ = aπ -aπ f u a × 1 a 2 a k=1 a k ′ =1 ψ k a ψ k ′ a e iu k a -k ′ a du = aπ -aπ f u a × 1 a a k=1 ψ k a cos k a u 2 + 1 a a k=1 ψ k a sin k a u 2 du (18) 2007
Now, it is well known that if ψ ∈ W (β, L) the Sobolev space with parameters β > 1/2 and L > 0, then

sup |u|≤aπ ∆ a (u) ≤ C β,L 1 a β-1/2 with ∆ a (u) := 1 a a k=1 ψ k a e -iu k a - 1 0 ψ(t)e -iut dt , (19)
with C β,L > 0 only depending on β and L (see for instance [START_REF] Devore | Constructive approximations[END_REF]. Therefore if ψ satisfies Assumption W (∞) and X Assumption A1, for all

β > 1/2, since sup u∈R | ψ(u)| < ∞, E(e 2 (a, 0)) - aπ -aπ f u a × | ψ(u)| 2 du ≤ 2C β,L 2 a β-3/2 aπ 0 f u a | ψ(u)| du + C 2 β,L 2 a 2β-2 aπ 0 f u a du ≤ 2 • C 2 β,L 2 a 2β-3 π 0 f (v) dv, (20)
since sup u∈R (1 + u n )| ψ(u)| < ∞ for all n ∈ N. Consequently, if ψ satisfies Assumption W (∞), for all n > 0,
for all a ∈ N * , there exists C(n) > 0 not depending on a such that

E(e 2 (a, 0)) - aπ -aπ f u a × | ψ(u)| 2 du ≤ C(n) 1 a n . ( 21
)
But from Assumption W (∞), for all c < 1,

K (ψ,c) = ∞ -∞ | ψ(u)| 2 |u| c du < ∞, because Assumption W (∞) implies that | ψ(u)| = O(|u|) when u → 0 and there exists p > 1 -c such that sup u∈R | ψ(u)| 2 (1 + |u|) p < ∞. Moreover, for all p > 1 -c, aπ -aπ | ψ(u)| 2 |u| c du -K (ψ,c) = 2 ∞ aπ | ψ(u)| 2 u c du ≤ C • ∞ aπ 1 u p+c du ≤ C ′ • 1 a p+c-1 ,
with C > 0 and C ′ > 0 not depending on a. As a consequence, under Assumption A1, for all p > 1 -D, all n ∈ N and all a ∈ N * ,

E(e 2 (a, 0)) -f * (0) • ∞ -∞ | ψ(u)| 2 |u/a| D du ≤ 2f * (0)a D ∞ aπ | ψ(u)| 2 u D du + C D ′ a D-D ′ aπ -aπ | ψ(u)| 2 |u| D-D ′ du + C(n) 1 a n =⇒ E(e 2 (a, 0)) -f * (0)K (ψ,D) • a D ≤ C ′ f * (0) • a 1-p + C D ′ K (ψ,D-D ′ ) • a D-D ′ .
Now, by choosing p such that 1p < D -D ′ , the inequality (6) is obtained. 2

Proof [Property 2] Using the proof of previous Property 1, with Assumption W (5/2), ψ is included in a Sobolev space W (5/2, L), inequality ( 19) is checked with β = 5/2 and (20) is replaced by

E(e 2 (a, 0)) -a aπ -aπ f u a × | ψ(u)| 2 du ≤ 2 • C 2 5/2,L 2 a 2 π 0 f (v) dv, (22)
since sup u∈R (1 + u 3/2 )| ψ(u)| < ∞. Therefore, inequality (21) is replaced by E(e 2 (a, 0)) -a aπ -aπ f u a × | ψ(u)| 2 du ≤ C(2) 1 a 2 .
The end of the proof is similar to the end of the previous proof, but now K (ψ,c) exists for -2 < c < 1 and

aπ -aπ | ψ(u)| 2 |u| c du -K (ψ,c) ≤ C ′ • 1 a 2+c . Finally, under Assumption A1', for all a ∈ N * , since -2 < D -D ′ < 1, E(e 2 (a, 0)) -f * (0)K (ψ,D) • a D ≤ C D ′ K (ψ,D-D ′ ) • a D-D ′ + C ′ 1 a 2 ,
which achieves the proof. 2

Proof [Corollary 1] Both these proofs provide main arguments to establish [START_REF] Bardet | Statistical Study of the Wavelet Analysis of Fractional Brownian Motion[END_REF]. For better readability , we will consider only Assumption A1' and Assumption W (∞) (the long memory process being similar). The main difference consists in specifying the asymptotic behavior of

aπ -aπ f u a × | ψ(u)| 2 du. But, aπ -aπ f u a × | ψ(u)| 2 du = √ a - √ a f u a × | ψ(u)| 2 du + 2 aπ √ a f u a × | ψ(u)| 2 du. (23)
The asymptotic behavior of ψ(u) when u → ∞ (ψ is considered to satisfy Assumption W (∞)), this behavior

induces that aπ √ a f u a × | ψ(u)| 2 du ≤ Ca D ∞ √ a u -D × | ψ(u)| 2 du ≤ C(n) a n , (24)
for all n ∈ N. Moreover,

√ a - √ a f u a | ψ(u)| 2 du = f * (0) √ a - √ a u a -D + C D ′ u a D ′ -D | ψ(u)| 2 du + √ a - √ a f u a -f * (0) u a -D + C D ′ u a D ′ -D | ψ(u)| 2 du. ( 25
)
From computations of previous proofs,

√ a - √ a u a -D + C D ′ u a D ′ -D | ψ(u)| 2 du = K (ψ,D) • a D + C D ′ K (ψ,D-D ′ ) • a D-D ′ + Λ(a), (26)
and

|Λ(a)| ≤ C(n) a n . Finally, using f (λ) = f * (0) |λ| -D + C D ′ |λ| D ′ -D + o |λ| D ′ -D when λ → 0, we obtain √ a - √ a f u a -f * (0) u a -D + C D ′ u a D ′ -D | ψ(u)| 2 du = √ a - √ a u a D-D ′ f u a -f * (0) u a -D + C D ′ u a D ′ -D | ψ(u)| 2 u a D ′ -D du = a D-D ′ √ a - √ a g(u, a)| ψ(u)| 2 |u| D ′ -D du,
with for all u ∈ [-√ a, √ a], g(u, a) → 0 when a → ∞. Therefore, from Lebesgue Theorem (checked from the asymptotic behavior of ψ),

lim a→∞ a D-D ′ √ a - √ a f u a -f * (0) u a -D + C D ′ u a D ′ -D | ψ(u)| 2 du = 0. ( 27
)
As a consequence, from (23), ( 24), ( 25), ( 26) and ( 27), the corollary is proven. 2

Proof [Proposition 1] This proof can be decomposed into three steps :Step 1, Step 2 and Step 3.

Step 1. In this part, N a N • Cov( TN (r i a N ), TN (r j a N )) 1≤i,j≤ℓ is proven to converge at an asymptotic covariance matrix Γ. First, for all (i, j) ∈ {1, . . . , ℓ}

i a N ), TN (r j a N )) = 2 1 [N/r i a N ] 1 [N/r j a N ] [N/riaN ] p=1 [N/rjaN ] q=1 Cov(ẽ(r i a N , p), ẽ(r j a N , q) 2 , 2 , Cov( TN (r
because X is a Gaussian process. Therefore, by considering only i = j and p = q, for N and a N large enough,

Cov( TN (r i a N ), TN (r i a N )) ≥ 1 r i N a N . (29)
Now, for (p, q) ∈ {1, . . . ,

[N/r i a N ]} × {1, . . . , [N/r i a N ]}, Cov ẽ(r i a N , p), ẽ(r j a N , q) = a 1-D N (r i r j ) (1-D)/2 f * (0)K (ψ,D) 1 r i a N 1 r j a N riaN k=1 rj aN k ′ =1 ψ k r i a N ψ k ′ r j a N r k -k ′ + a N (r i p -r j q) = a 1-D N (r i r j ) (1-D)/2 f * (0)K (ψ,D) 1 r i a N 1 r j a N riaN k=1 rj aN k ′ =1 ψ k r i a N ψ k ′ r j a N π -π dλ f (λ)e -iλ(k-k ′ +aN (rip-rj q)) = (r i r j ) (1-D)/2 a D N f * (0)K (ψ,D) 1 r i a N 1 r j a N riaN k=1 rj aN k ′ =1 ψ k r i a N ψ k ′ r j a N πaN -πaN du f u a N e -iu( k a N -k ′ a N +rip-rj q) .
Using the same expansion as in (21), under Assumption W (∞) the previous equality becomes, for all n ∈ N * , Cov ẽ(r i a N , p), ẽ(r j a N , q) -

(r i r j ) (1-D)/2 a D N f * (0)K (ψ,D) πaN -πaN du ψ(ur i ) ψ(ur j )f u a N e -iu(rip-rj q) ≤ C(n) a n+D N πaN -πaN du ψ(ur i ) ψ(ur j )f u a N ≤ C ′ (n) a n N ∞ -∞ du |u| -D ψ(ur i ) ψ(ur j ) ≤ C ′′ (n) a n N , (30)
with C(n), C ′ (n), C ′′ (n) > 0 not depending on a N and due the asymptotic behaviors of ψ(u) when u → 0 and

u → ∞. Now, under Assumption A1, πaN -πaN du ψ(ur i ) ψ(ur j )f u a N e -iu(rip-rj q) -a N f * (0) π -π du ψ(ur i a N ) ψ(ur j a N ) |u| D e -iuaN (rip-rj q) ≤ a D-D ′ N f * (0)C D ′ πaN -πaN du ψ(ur i ) ψ(ur j ) |u| D-D ′ ≤ a D-D ′ N f * (0)C D ′ ∞ -∞ du ψ(ur i ) ψ(ur j ) |u| D-D ′ , (31) since ∞
-∞ du ψ(ur i ) ψ(ur j ) |u| D-D ′ < ∞ from Assumption W (∞).
Finally, from (30) and (31), we have C > 0 not

depending on N such that for all a N ∈ N * , Cov ẽ(r i a N , p), ẽ(r j a N , q) - a 1-D N (r i r j ) (1-D)/2 K (ψ,D) π -π du ψ(ur i a N ) ψ(ur j a N ) |u| D e -iuaN (rip-rj q) ≤ C a -D ′ N . (32)
It remains to evaluate

a 1-D N π -π du ψ(ur i a N ) ψ(ur j a N ) |u| D e -iuaN (rip-rj q) = πaN -πaN du ψ(ur i ) ψ(ur j )
|u| D e -iu(rip-rj q) . Thus, if |r i pr j q| ≥ 1, using an integration by parts,

πaN -πaN du ψ(ur i ) ψ(ur j ) |u| D e -iu(rip-rj q) = 1 -i(r i p -r j q) ψ(ur i ) ψ(ur j ) u D e -iu(rip-rj q) πaN -πaN + 1 i(r i p -r j q) πaN -πaN du ∂ ∂u ψ(ur i ) ψ(ur j ) u D e -iu(rip-rj q) ≤ 1 |r i p -r j q| ∞ -∞ D |u| D+1 ψ(ur i ) ψ(ur j ) + 1 |u| D ∂ ∂u ψ(ur i ) ψ(ur j ) du ≤ C 1 |r i p -r j q| (33)
with C < ∞ not depending on N , since:

• ψ(πr i a N ) ψ(πr j a N ) = ψ(-πr i a N ) ψ(-πr j a N ) and sin(πa N (r i pr j q)) = 0;

• from Assumption W (∞), lim sup u→0 u -1 | ψ(u)| < ∞, lim sup u→0 ∂ ∂u ψ(u) < ∞ =⇒ lim sup u→0 u -1 ∂ ∂u ψ(ur i ) ψ(ur j ) < ∞; • from Assumption W (∞), for all n ∈ N, sup u∈R (1 + |u|) n | ψ(u)| < ∞ and sup u∈R (1 + |u|) n ∂ ∂u ψ(u) < ∞.
Moreover, if |r i pr j q| = 0, from Cauchy-Schwartz Inequality and Property 1, for a N large enough Cov ẽ(r i a N , p), ẽ(r j a N , q) ≤ E(ẽ 2 (r i a N , p)) • E( d2 (r j a N , q))

1/2 ≤ 2. ( 34
)
Therefore, using (32), (33) and (34) and the inequality (x + y) 2 ≤ 2(x 2 + y 2 ) for all (x, y) ∈ R 2 , we have C > 0 such that for a N large enough,

Cov 2 ẽ(r i a N , p), ẽ(r j a N , q) ≤ C 1 (1 + |r i p -r j q|) 2 + 1 a 2D ′ N (35)
Hence, with (28),

Cov( TN (r i a N ), TN (r j a N )) ≤ C 1 [N/r i a N ] 1 [N/r j a N ] [N/riaN ] p=1 [N/rj aN ] q=1 1 (1 + |r i p -r j q|) 2 + 1 a 2D ′ N
But, from the theorem of comparison between sums and integrals,

[N/riaN ] p=1 [N/rjaN ] q=1 (1 + |r i p -r j q|) -2 ≤ 1 r i r j N/aN 0 N/aN 0 du dv (1 + |u -v|) 2 ≤ 2 r i r j N/aN 0 N/a N dw (1 + w) 2 ≤ 2 r i r j • N a N .
As a consequence, if a N is such that lim sup

N →∞ N a N 1 a 2D ′ N < ∞ then lim sup N →∞ N a N Cov( TN (r i a N ), TN (r j a N )) < ∞.
More precisely, since this covariance is a sum of positive terms, if lim sup

N →∞ N a N 1 a 2D ′ N = 0, lim N →∞ N a N Cov( SN (r i a N ), SN (r j a N )) 1≤i,j≤ℓ = Γ(r 1 , • • • , r ℓ , ψ, D), (36)
a non null (from (29)) symmetric matrix with Γ(r 1 , • • • , r ℓ , ψ, D) = (γ ij ) 1≤i,j≤ℓ that can be specified. Indeed, from the previous computations, if lim sup

N →∞ N a N 1 a 2D ′ N = 0, γ ij = lim N →∞ 8r i r j a N N [N/riaN ] p=1 [N/rjaN ] q=1 (r i r j ) (1-D)/2 K (ψ,D) ∞ 0 du ψ(ur i ) ψ(ur j ) u D cos(u(r i p -r j q)) 2 = lim N →∞ 8(r i r j ) 2-D a N K 2 (ψ,D) N [N/dijaN ]-1 m=-[N/dijaN ]+1 ( N d ij a N -|m| ∞ 0 du ψ(ur i ) ψ(ur j ) u D cos(u d ij m) 2 = 8(r i r j ) 2-D K 2 (ψ,D) d ij ∞ m=-∞ ∞ 0 ψ(ur i ) ψ(ur j ) u D cos(u d ij m) du 2 ,
with d ij = GCD(r i ; r j ). Therefore, the matrix Γ depends only on on r 1 , • • • , r ℓ , ψ, D.

Step 2.Generaly speaking, the above result is not sufficient to obtain the central limit theorem,

N a N TN (r i a N ) -E(ẽ 2 (r i a N , 0) 1≤i≤ℓ L -→ N →∞ N ℓ (0, Γ(r 1 , • • • , r ℓ , ψ, D)). (37)
However, each TN (r i a N ) is a quadratic form of a Gaussian process. Mutatis mutandis, it is exactly the same framework (i.e. a Lindeberg central limit theorem) as that of Proposition 2.1 in [START_REF] Bardet | Testing for the presence of self-similarity of Gaussian time series having stationary increments[END_REF], and (37) is checked. Moreover, if (a n ) n is such that lim sup

N →∞ N a 1+2D ′ N = 0 then using the asymptotic behavior of E(ẽ 2 (r i a N , 0) provided in Property 1, N a N E(ẽ 2 (r i a N , 0) -→ N →∞ 0.
As a consequence, under those assumptions,

N a N TN (r i a N ) -1 1≤i≤ℓ L -→ N →∞ N ℓ (0, Γ(r 1 , • • • , r ℓ , ψ, D)). (38)
Step 3. The logarithm function (x 1 , .., x ℓ ) ∈ (0, +∞) ℓ → (log x 1 , .., log x m ) is C 2 on (0, +∞) ℓ . As a consequence, using the Delta-method, the central limit theorem [START_REF] Bhansali | Estimation of the memory parameter by fitting fractionally-differenced autoregressive models[END_REF] for the vector log TN (r i a N ) 1≤i≤ℓ follows with the same asymptotical covariance matrix Γ(r 1 , • • • , r ℓ , ψ, D) (because the Jacobian matrix of the function in (1, .., 1) is the identity matrix). 2

Proof [Proposition 2]

There is a perfect identity between this proof and that of Proposition 1, both of which are based on the approximations of Fourier transforms provided in the proof of Property 2. 2

Proof [Corollary 3] It is clear that X ′ t = X t + P m (t) for all t ∈ Z, with X = (X t ) t satisfying Proposition 1 and 2. But, any wavelet coefficient of (P m (t)) t is obviously null from the assumption on ψ. Therefore the statistic T N is the same for X and X ′ . 2

Proof [Proposition 5] Let ε > 0 be a fixed positive real number, such that α * + ε < 1.

I. First, a bound of Pr( α N ≤ α * + ε) is provided. Indeed, Pr α N ≤ α * + ε ≥ Pr Q N (α * + ε/2) ≤ min α≥α * +ε and α∈AN Q N (α) ≥ 1 -Pr α≥α * +ε and α∈AN Q N (α * + ε/2) > Q N (α) ≥ 1 - log[N/ℓ] k=[(α * +ε) log N ] Pr Q N (α * + ε/2) > Q N k log N . (39)
But, for α ≥ α * + 1,

Pr Q N (α * + ε/2) > Q N (α) = Pr P N (α * + ε/2) • Y N (α * + ε/2) 2 > P N (α) • Y N (α) 2 with P N (α) = I ℓ -A N (α) • A ′ N (α) • A N (α) -1 • A N (α)
for all α ∈ (0, 1), i.e. P N (α) is the matrix of an orthogonal projection on the orthogonal subspace (in R ℓ ) generated by A N (α) (and I ℓ is the identity matrix in R ℓ ). From the expression of A N (α), it is obvious that for all α ∈ (0, 1),

P N (α) = P = I ℓ -A • A ′ • A -1 • A, with the matrix A =        log(r 1 ) 1 : : log(r ℓ ) 1       
as in Proposition 3. Thereby,

Pr Q N (α * + ε/2) > Q N (α) = Pr P • Y N (α * + ε/2) 2 > P • Y N (α) 2 = Pr P • N N α * +ε/2 Y N (α * + ε/2) 2 > N α-(α * +ε/2) P • N N α Y N (α) 2 ≤ Pr V N (α * + ε/2) > N (α-(α * +ε/2))/2 + Pr V N (α) ≤ N -(α-(α * +ε/2))/2 with V N (α) = P • N N α Y N (α) 2
for all α ∈ (0, 1). From Proposition 1, for all α > α * , the asymptotic law of P • N N α Y N (α) is a Gaussian law with covariance matrix P • Γ • P ′ . Moreover, the rank of the matrix is

P • Γ • P ′ is ℓ -2 (
this is the rank of P ) and we have 0 < λ -, not depending on N ) such that P • Γ • P ′λ -P • P ′ is a non-negative matrix (0 < λ -< min{λ ∈ Sp(Γ)}). As a consequence, for a large enough N ,

Pr V N (α) ≤ N -(α-(α * +ε/2))/2 ≤ 2 • Pr V -≤ N -(α-(α * +ε/2))/2 ≤ 1 2 ℓ/2-2 Γ(ℓ/2) • N λ - -( ℓ 2 -1) (α-(α * +ε/2)) 2 , with V -∼ λ -• χ 2 (ℓ -2). Moreover, from Markov inequality, Pr V N (α * + ε/2) > N (α-(α * +ε/2))/2 ≤ 2 • Pr exp( V + ) > exp N (α-(α * +ε/2))/4 ≤ 2 • E(exp( V + )) • exp -N (α-(α * +ε/2))/4 with V + ∼ λ + • χ 2 (ℓ -2) and λ + > max{λ ∈ Sp(Γ)} > 0. Like E(exp( V + )) < ∞ does not depend on N , we obtain that M 1 > 0 not depending on N , such that for large enough N , Pr Q N (α * + ε/2) > Q N (α) ≤ M 1 • N -( ℓ 2 -1) (α-(α * +ε/2)) 2 ,
and therefore, the inequality (39) becomes, for N large enough,

Pr α N ≤ α * + ε ≥ 1 -M 1 • log[N/ℓ] k=[(α * +ε) log N ] N -(ℓ-2) 4 k log N -(α * +ε/2) ≥ 1 -M 1 • log N • N -(ℓ-2) 12 ε . (40)
II. Secondly, a bound of Pr( α N ≥ α *ε) is provided. Following the above arguments and notations ,

Pr α N ≥ α * -ε ≥ Pr Q N (α * + 1 -α * 2α * ε) ≤ min α≤α * -ε and α∈AN Q N (α) ≥ 1 - [(α * -ε) log N ]+1 k=2 Pr Q N (α * + 1 -α * 2α * ε) > Q N k log N , (41)
and as above,

Pr Q N (α * + 1 -α * 2α * ε) > Q N (α) = Pr P • N N α * + 1-α * 2α * ε Y N (α * + 1 -α * 2α * ε) 2 > N α-(α * + 1-α * 2α * ε) P • N N α Y N (α) 2 . (42)
Now, in the case a N = N α with α ≤ α * , the sample variance of wavelet coefficients is biased. In this case, from the relation of Corollary 1 under Assumption A1',

Y N (α) 1≤i≤ℓ = C D ′ K (ψ,D-D ′ )) f * (0)K (ψ,D) (iN α ) -D ′ (1 + o i (1)) 1≤i≤ℓ + N α N • ε N (α) 1≤i≤ℓ ,
with o i (1) → 0 when N → ∞ for all i and E(Z N (α)) = 0. As a consequence, for large enough N ,

P • N N α Y N (α) 2 = P • ε N (α) 2 + N α * -α α * P • C D ′ K (ψ,D-D ′ )) f * (0)K (ψ,D) i -D ′ (1 + o i (1)) 1≤i≤ℓ 2 ≥ D • N α * -α α * ,
with D > 0, because the vector (i -D ′ ) 1≤i≤ℓ not in the orthogonal subspace of the subspace generated by the matrix A. Then, the relation (42) becomes,

Pr Q N (α * + 1 -α * 2α * ε) > Q N (α) ≤ Pr P • N N α * + 1-α * 2α * ε Y N (α * + 1 -α * 2α * ε) 2 ≥ D • N α-(α * + 1-α * 2α * ε) • N α * -α α * ≤ Pr V + ≥ D • N 1-α * 2α * (2(α * -α)-ε) ≤ M 2 • N -( ℓ 2 -1) 1-α * 2α * ε , with M 2 > 0, because V + ∼ λ + • χ 2 (ℓ -2) and 1 -α * 2α * (2(α * -α) -ε) ≥ 1 -α * 2α * ε for all α ≤ α * -ε. Hence,
from the inequality (41), for large enough N ,

Pr α N ≥ α * -ε ≥ 1 -M 2 • log N • N -( ℓ 2 -1) 1-α * 2α * ε . (43)
The inequalities (40) and (43) imply that Pr

| α N -α| ≥ ε -→ N →∞ 0. 2
Proof [Theorem 1] The central limit theorem of ( 16) can be established from the following arguments. First,

Pr(α N > α * ) -→ N →∞
1. Following the previous proof, there is for all ε > 0, FARIMA(0, -0.25, 0) X (-1,1) X (-1,3) X (-3,1) X (-3,3) N = 10

Pr α N ≥ α * -ε ≥ 1 -M 2 • log N • N -( ℓ 2 -1) 1-α * 2α * ε . Consequently, if ε N = λ • log log N log N with λ > 2 (ℓ -2)D ′ then, Pr α N ≥ α * -ε N ≥ 1 -M 2 • log N • N -λ (ℓ-2)D ′ 2 • log log N log N ≥ 1 -M 2 • log N 1-λ (ℓ-2)D ′ 2 =⇒ Pr α N + ε N ≥ α * -→ N →∞ 1. N = 10 3 √ MSE ℓ = 5 ℓ = 10 ℓ = 15 ℓ = 20 ℓ = 25      ℓ1 = 15 ℓ2 = ℓ fGn = D+1 2 ) DN ,

,

  [START_REF] Moulines | Semiparametric spectral estimation for fractionnal processes[END_REF] or[START_REF] Moulines | On the spectral density of the wavelet coefficients of long memory time series with application to the log-regression estimation of the memory parameter[END_REF] is f (λ) = |1e iλ | -2d • f * (λ) with f * a function such that |f * (λ)f * (0)| ≤ f * (0) • λ β and 0 < β).It is obvious that for β ≤ 2 such an assumption corresponds to Assumption A1 with D ′ = β. Moreover, following arguments developed in[START_REF] Giraitis | Rate optimal semi-parametric estimation of the memory parameter of the Gaussian time series with long range dependence[END_REF][START_REF] Giraitis | Adaptive semiparametric estimation of the memory parameter[END_REF], if f * ∈ H(D ′ , C D ′ ) with D ′ > 2 is such that f * is s ∈ N *

1 0ψ 1 0

 11 (t) dt = 0 and ψ(0) = ψ(1) = 0. A consequence of the first point of this Assumption is: for all p > 0, sup λ∈R | ψ(λ)|(1 + |λ|) p < ∞, where ψ(u) = ψ(t) e -iut dt is the Fourier transform of ψ. A useful consequence of the second point is ψ(u) ∼ C u for u → 0 with |C| < ∞ a real number not depending on u.

  central limit theorem satisfied by DN is empirically checked. The empirical choice of the parameter ℓ is also studied. Moreover, the robustness of DN is successfully tested. Finally, the adaptive wavelet-based estimator is compared with several existing adaptive estimators of the memory parameter from generated paths of the 5 different "test" processes (Giraitis-Robinson-Samarov adaptive local log-periodogram, Moulines-Soulier adaptive global log-periodogram, Robinson local Whittle, Abry-Taqqu-Veitch data-driven wavelet based, Bhansali-Giraitis-Kokoszka FAR estimators). The simulations results of DN are convincing. The convergence rate of DN is often ranges among the best of the 5 test processes (however the Robinson local Whittle estimator D R provides more uniformly accurate estimations of D). Three other numerical advantages are offered by the adaptive wavelet-based estimator (and not by D R ). Firstly, it is a very low consuming time estimator. Sec-

  on D ′ which is unknown. To solve this problem, Veitch et al. (2003) suggest a chi-square-based test (constructed from a distance between the regression line and the different points

Remark 7

 7 This choice of discretization is implied by the following proof of the consistency of α N . If the interval (0, 1) is stepped in N β points, with β > 0, the used proof cannot attest this consistency. Finally, it is the same framework as the usual discrete wavelet transform (see for instance[START_REF] Veitch | On the Automatic Selection of the Onset of Scaling[END_REF] but less restricted since log N may be replaced in the previous expression of A N by any negligible function of N compared to functions N β with β > 0 (for instance, (log N ) d or d log N can be used).

2 (

 2 ℓ-2)D ′ and µ > 12 ℓ-2 . Consequently, the selected scale is asymptotically equal to N α * up to a logarithm factor. Finally, Proposition 5 can be used to define an adaptive estimator of D. First, define the straightforward estimator D N = D( a N ), which should minimize the mean square error using a N . However, the estimator D N does not attest a CLT since Pr( α N ≤ α * ) > 0 and therefore it can not be asserted that E( N/ a N ( D N -D)) = 0. To establish a CLT satisfied by an adaptive estimator DN of D, an adaptive scale sequence (ã N ) = (N αN ) has to be defined to ensure Pr(α N ≤ α * ) -→ N →∞ 0. The following theorem provides the asymptotic behavior of such an estimator, Theorem 1 Let X satisfy Assumption A1' and ψ Assumption W (∞) (or Assumption W (5/2) if 0 < D < 1 and 0 < D ′ ≤ 2). Define,

(

  2003). The simulations are realized for different values of D, N and processes which satisfy Assumption A1' and therefore Assumption A1 (the article of Moulines et al., 2007, gives a lot of details on this point):1. the fractional Gaussian noise (fGn) of parameter H = (D + 1)/2 (for -1 < D < 1) and σ 2 = 1. The spectral density f f Gn of a fGn is such that f* f Gn is included in H(2, C 2 ) (thus D ′ = 2);2. the FARIMA[p,d,q] process with parameter d such that d = D/2 ∈ (-0.5, 0.5) (therefore -1 < D < 1), the innovation variance σ 2 satisfying σ 2 = 1 and p, q ∈ N. The spectral density f F ARIMA of such a process is such that f * F ARIMA is included in the set H(2, C 2 ) (thus D ′ = 2);

FARIMA[ 1

 1 ,d,0] processes with d = D/2, standard Gaussian innovations and AR coefficient φ = 0.95; • FARIMA[1,d,1] processes with d = D/2, standard Gaussian innovations and AR coefficient φ = -0.3 and MA coefficient φ = 0.7;

Figure 1 :

 1 Figure 1: Log-log graphs for different samples of X (D,D ′ ) with D = 0.5 and D ′ = 1 when N = 10 3 (up and left, D N ≃ 1.04), N = 10 4 (up and right, D N ≃ 0.66), N = 10 5 (down and left, D N ≃ 0.62) and N = 10 6 (down and right, D N ≃ 0.54).

  provides the histograms of D N and DN for 100 independent samples of FARIMA(1, d, 1) processes with D = 0.5 and N = 10 5 . Both the histograms of Figure 2 are similar to Gaussian distribution histograms. It is not surprising for DN since Theorem 1 shows that the asymptotic distribution of DN is a Gaussian distribution with mean equal to D. The asymptotic distribution of D N and the Gaussian distribution seem also to be similar. A Cramer-von Mises test of normality indicates that both distributions of D N and DN can be considered a Gaussian distribution (respectively W ≃ 0.07, pvalue ≃ 0.24 and W ≃ 0.05, pvalue ≃ 0.54).

Figure 2 :

 2 Figure 2: Histograms of D N and DN for 100 samples of FARIMA(1, d, 1) with D = 0.5 for N = 10 5 .

4

 4 and different values of D, D = 0.1, 0.3, 0.5, 0.7, 0.9. Several different semi-parametric estimators of D are considered: • D BGK is an "optimal" parametric Whittle estimator obtained from a BIC criterium model selection of fractionally differenced autoregressive models (introduced by Bhansali it et al., 2006). The required confidence interval of the estimationD BGK is [ D R -2/N 1/4 , D R -2/N 1/4 ]; • D GRS isan adaptive local periodogram estimator introduced by Giraitis et al (2000). It requires two parameters: a bandwidth parameter m, with a procedure of determination provided in this article, and a number of low trimmed frequencies l (satisfying different conditions but without being fixed in this paper; after a number of simulations, l = max(m 1/3 , 10) is chosen); • D MS is an adaptive global periodogram estimator introduced by Moulines and Soulier (1998, 2003), also called FEXP estimator, with bias-variance balance parameter κ = 2; • D R is a local Whittle estimator introduced by Robinson (1995). The trimming parameter is m = N/30; • D AT V is an adaptive wavelet based estimator introduced by Veitch et al. (2003) using a Db4 wavelet (and described above); • D N defined previously with ℓ 1 = 15 and ℓ 2 = N 1-αN /10 and a mother wavelet ψ(t) = 100 •t

[Property 1 ]

 1 The arguments of this proof are similar to those of Abry et al. (1998) or Moulines et al.

4 and 10 5 )

 5 , of D (0.1, 0.3, 0.5, 0.7 and 0.9) and ℓ (5, 10, 15, 20, 25 and (15, ℓ)), 100 independent samples of each process are generated. The √ M SE of each estimator is obtained from a mean of √ M SE obtained for the different values of D.

Table 1 :

 1 Consistency of estimators D N , DN , α N , αN following ℓ from simulations of the different long-memory processes of the benchmark. For each value of N (10 3 , 10

			DN	0.16, 0.75	0.14, 0.19	0.13, 0.17	0.14, 0.15	0.14, 0.15	0.15, 0.18
		αN , αN	0.12, 0.32	0.07, 0.13	0.05, 0.08	0.04, 0.05	0.04, 0.04	0.05, 0.08
	FARIMA(0, D 2 , 0)	DN , DN	0.21, 0.81	0.15, 0.20	0.14, 0.17	0.15, 0.15	0.15, 0.15	0.15, 0.19
		αN , αN	0.14, 0.34	0.07, 0.13	0.05, 0.09	0.05, 0.06	0.04, 0.04	0.05, 0.09
	FARIMA(1, D 2 , 0)	DN , DN	0.30, 0.96	0.28, 0.35	0.27, 0.29	0.29, 0.27	0.30, 0.30	0.31, 0.35
		αN , αN	0.19, 0.44	0.15, 0.24	0.12, 0.17	0.11, 0.15	0.11, 0.12	0.12, 0.17
	FARIMA(1, D 2 , 1)	DN , DN	0.60, 0.92	0.43, 0.41	0.39, 0.35	0.36, 0.35	0.32, 0.33	0.21, 0.20
		αN , αN	0.17, 0.38	0.11, 0.18	0.09, 0.12	0.07, 0.09	0.06, 0.07	0.09, 0.12
	X (D,D ′ ) , D ′ = 1	DN , DN	0.33, 0.68	0.29, 0.28	0.27, 0.26	0.26, 0.27	0.25, 0.25	0.29, 0.30
		αN , αN	0.10, 0.22	0.10, 0.07	0.11, 0.07	0.12, 0.12	0.13, 0.13	0.11, 0.07
		√ MSE	ℓ = 5	ℓ = 10	ℓ = 15	ℓ = 20	ℓ = 25	    	ℓ1 = 15 ℓ2 = ℓ
	fGn (H = D+1 2 )	DN , DN	0.08, 0.26	0.05, 0.05	0.05, 0.05	0.04, 0.04	0.04, 0.04	0.04, 0.04
		αN , αN	0.08, 0.22	0.05, 0.06	0.04, 0.05	0.04, 0.05	0.05, 0.05	0.04, 0.05
	FARIMA(0, D 2 , 0)	DN , DN	0.08, 0.31	0.06, 0.06	0.05, 0.05	0.05, 0.05	0.05, 0.05	0.05, 0.05
	N = 10 4	αN , αN	0.09, 0.24	0.05, 0.07	0.04, 0.05	0.04, 0.05	0.05, 0.05	0.04, 0.05
	FARIMA(1, D 2 , 0)	DN , DN	0.13, 0.57	0.10, 0.10	0.09, 0.08	0.09, 0.08	0.09, 0.09	0.09, 0.08
		αN , αN	0.15, 0.36	0.09, 0.16	0.08, 0.11	0.07, 0.09	0.06, 0.08	0.08, 0.11
	FARIMA(1, D 2 , 1)	DN , DN	0.22, 0.63	0.17, 0.15	0.16, 0.13	0.15, 0.14	0.15, 0.14	0.09 , 0.09
		αN , αN	0.16, 0.38	0.11, 0.17	0.08, 0.11	0.07, 0.09	0.06, 0.07	0.08, 0.11
	X (D,D ′ ) , D ′ = 1	DN , DN	0.23, 0.36	0.19, 0.15	0.18, 0.17	0.17, 0.17	0.15, 0.14	0.15, 0.14
		αN , αN	0.10, 0.18	0.12, 0.08	0.13, 0.12	0.14, 0.14	0.15, 0.15	0.13, 0.12
		√	MSE	ℓ = 5	ℓ = 10	ℓ = 15	ℓ = 20	ℓ = 25	    	ℓ1 = 15 ℓ2 = ℓ
	fGn (H = D+1 2 )	DN , DN	0.04, 0.09	0.03, 0.03	0.02, 0.03	0.02, 0.02	0.02, 0.02	0.02, 0.02
		αN , αN	0.07, 0.16	0.06, 0.04	0.06, 0.06	0.07, 0.07	0.07, 0.07	0.06, 0.06
	FARIMA(0, D 2 , 0)	DN , DN	0.03, 0.13	0.02, 0.02	0.02, 0.02	0.02, 0.02	0.02, 0.02	0.02, 0.02
	N = 10 5	αN , αN	0.07, 0.18	0.04, 0.05	0.04, 0.03	0.04, 0.04	0.05, 0.05	0.04, 0.03
	FARIMA(1, D 2 , 0)	DN , DN	0.05, 0.25	0.05, 0.04	0.04, 0.03	0.04, 0.03	0.04, 0.04	0.03, 0.02
		αN , αN	0.12, 0.30	0.07, 0.12	0.05, 0.07	0.04, 0.06	0.04, 0.05	0.05, 0.07
	FARIMA(1, D 2 , 1)	DN , DN	0.08, 0.30	0.06, 0.04	0.05, 0.04	0.05, 0.04	0.05, 0.05	0.04, 0.03
		αN , αN	0.13, 0.33	0.09, 0.15	0.08, 0.11	0.07, 0.09	0.06, 0.08	0.08, 0.11
	X (D,D ′ ) , D ′ = 1	DN , DN	0.13, 0.19	0.11, 0.08	0.10, 0.08	0.09, 0.09	0.09, 0.09	0.08, 0.07
		αN , αN	0.09, 0.15	0.10, 0.07	0.11, 0.09	0.12, 0.11	0.13, 0.13	0.11, 0.09

Table 2 :

 2 Estimation of the memory parameter from 100 independent samples in case of short memory (D ≤ 0).

	3	√ MSE DN , DN	0.15, 0.20	0.30, 0.30	0.38, 0.37	0.36, 0.37	0.39, 0.38
	N = 10 4	√ MSE DN DN	0.04, 0.04	0.15, 0.14	0.08, 0.08	0.13, 0.14	0.13, 0.13
	N = 10 5	√ MSE DN , DN	0.03, 0.03	0.06, 0.05	0.04, 0.03	0.04, 0.04	0.03, 0.03

Table 3 :

 3 Estimation of the long-memory parameter from 100 independent samples in case of processes P 1 -4 defined above.

			D = 0.1	D = 0.3	D = 0.5	D = 0.7	D = 0.9
	fGn (H = (D + 1)/2)	DBGK	0.089	0.171	0.259	0.341	0.369
		DGRS	0.114	0.132	0.147	0.155	0.175
		DMS	0.163	0.169	0.181	0.195	0.191
		DR	0.211	0.220	0.215	0.218	0.128
		DAT V	0.176	0.153	0.156	0.164	0.162
		DN	0.139	0.147	0.133	0.140	0.150
	FARIMA(0, D 2 , 0)	DBGK	0.094	0.138	0.239	0.326	0.413
		DGRS	0.131	0.139	0.150	0.150	0.162
		DMS	0.172	0.167	0.174	0.197	0.188
		DR	0.246	0.189	0.223	0.234	0.181
		DAT V	0.128	0.107	0.081	0.074	0.065
		DN	0.161	0.146	0.149	0.149	0.161
	FARIMA(1, D 2 , 0)	DBGK	0.146	0.203	0.239	0.236	0.212
		DGRS	0.519	0.545	0.588	0.585	0.830
	N = 10 3 -→	DMS	0.235	0.258	0.256	0.252	0.249
		DR	0.242	0.241	0.234	0.202	0.144
		DAT V	0.248	0.267	0.280	0.268	0.375
		DN	0.340	0.319	0.314	0.315	0.334
	FARIMA(1, D 2 , 1)	DBGK	0.204	0.253	0.342	0.363	0.384
		DGRS	0.901	0.894	0.866	0.870	0.893
		DMS	0.181	0.175	0.180	0.185	0.181
		DR	0.204	0.200	0.200	0.191	0.130
		DAT V	0.392	0.380	0.371	0.343	0.355
		DN	0.170	0.218	0.225	0.226	0.213
	X (D,D ′ ) , D ′ = 1	DBGK	0.090	0.139	0.261	0.328	0.388
		DGRS	0.342	0.339	0.331	0.300	0.315
		DMS	0.176	0.178	0.182	0.166	0.177
		DR	0.219	0.232	0.231	0.173	0.167
		DAT V	0.153	0.161	0.168	0.176	0.176
		DN	0.284	0.294	0.293	0.292	0.288

Table 4 :

 4 Comparison of the different log-memory parameter estimators for processes of the benchmark. For each process and value of D and N , √ M SE are computed from 100 independent generated samples.

Acknowledgments. The authors are very grateful to anonymous referees for many relevant suggestions and corrections that strongly improve the content of the paper.

Now, from Corollary 4,

Secondly, for x ∈ R , lim

To prove the second part of ( 16), we infer deduces from above that

This inequality and the previous central limit theorem result in : for all ρ > ν/2, and ε > 0,