Marco Salvatore Alfonso

Jinkai Yuan

Franco Tardani

Wilfrid Neri

A Colin

P Poulin

Absence of giant dielectric permittivity in graphene oxide materials

Keywords: Graphene oxide, Dielectric spectroscopy, Permittivity, Electrode polarization, Dielectric constant, Liquid crystals

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Introduction

Graphene oxide (GO) is an oxidized form of graphene [START_REF] Zhu | Graphene and Graphene Oxide: Synthesis, Properties, and Applications[END_REF][START_REF] Dreyer | The Chemistry of Graphene Oxide[END_REF]. It is generally obtained by acid treatment of graphitic materials. By contrast to graphene, GO is insulating due to heavy functionalization with oxygen groups, and to the subsequent disruption of the conjugated network of sp 2 carbon atoms. Nevertheless, GO displays particularly appealing features as such. It is soluble in polar solvents, including water, and can be easily processed into films and fibers [START_REF] Huang | Graphene-based Composites[END_REF][START_REF] Kuilla | Recent Advances in Graphene Based Polymer Composites[END_REF][START_REF] Naficy | Graphene Oxide Dispersions: Tuning Rheology to Enable Fabrication[END_REF].

It has a high flexibility along with a high Young's modulus [START_REF] Poulin | Superflexibility of Graphene Oxide[END_REF]. In addition, it can be used as a precursor for graphene based materials since electronic conjugation and conductivity can be partially restored after reduction [START_REF] Zhu | Graphene and Graphene Oxide: Synthesis, Properties, and Applications[END_REF][START_REF] Dreyer | The Chemistry of Graphene Oxide[END_REF].

GO is also often considered as promising materials for electronics and energy storage applications in which high-permittivity materials are sought after. Actually, according to various studies, GO materials could even display a giant permittivity with values which vary in a wide range and usually greater of 10 3 .

For example, Liu et al. [START_REF] Liu | Graphene-based Thin Film Supercapacitor with Graphene Oxide as Dielectric Spacer[END_REF] reported that GO can be used as a high-k material for energy storage applications with a dielectric constant greater than 1000 at 100 Hz. Hong et al. drew the same conclusion [START_REF] Hong | Graphite Oxide Paper as a Polarizable Electrical Conductor in the Through-Thickness Direction[END_REF] and Huang et al. [START_REF] Huang | Temperature-Dependent Electrical Property Transition of Graphene Oxide Paper[END_REF] reported values of relative permittivity above 100 at 100 Hz. Kumar et al. have reported even greater values, up to 10 6 at 1 KHz [START_REF] Kumar | A New Single/Few-Layered Graphene Oxide with a High Dielectric Constant of 10 6 : contribution of defects and functional groups[END_REF].

As stated by the authors, these giant dielectric constants would arise from the orientational polarization of oxygen-containing groups at the surface of GO planes and along the edges [START_REF] Kumar | A New Single/Few-Layered Graphene Oxide with a High Dielectric Constant of 10 6 : contribution of defects and functional groups[END_REF][START_REF] Hou | Graphene Oxide Foams: the Simplest Carbon-Air Prototypes for Unique Variable Dielectrics[END_REF]. Very large permittivities have also been reported for GO based foams and sensors in absence of humidity [START_REF] Hou | Graphene Oxide Foams: the Simplest Carbon-Air Prototypes for Unique Variable Dielectrics[END_REF][START_REF] Wan | Graphene Oxide as High-Performance Dielectric Materials for Capacitive Pressure Sensors[END_REF][START_REF] Wang | A Water-Dielectric Capacitor Using Hydrated Graphene Oxide Film[END_REF][START_REF] Gao | Direct Laser Writing of Micro-Supercapacitors on Hydrated Graphite Oxide Films[END_REF]. According to the authors, the capacitive behavior of GO paper for sensing shows a strong dependence on temperature and relative humidity. A strong increase of relative permittivity by around 2 orders of magnitude occurs with decreasing humidity with capacitance values from several hundreds of pF to several hundreds of nF [START_REF] Bayer | Tunable Mixed Ionic/Electronic Conductivity and Permittivity of Graphene Oxide Paper for Electrochemical Energy Conversion[END_REF].

Large values of relative dielectric permittivity are also suggested from characterizations of composite materials in which a small fraction of GO added to a polymer enhances the effective permittivity of the composites [START_REF] Mensah | A Study of Graphene Oxide-Reinforced Rubber Nanocomposite[END_REF][START_REF] Chen | Dielectric and Mechanical Properties and Thermal Stability of Polyimide-Graphene Oxide Composite films[END_REF][START_REF] Tantis | Functionalized Graphene -Poly(vinyl alcohol) Nnocomposites: Physical and Dielectric Properties[END_REF][START_REF] Wang | Transparent PVDF-TrFE/Graphene Oxide Ultrathin Films with Enhanced Energy Harvesting Performance[END_REF][START_REF] Sadasivuni | Dielectric Properties of Modified Graphene Oxide Filled Polyurethane Nanocomposites and its Correlation with Rheology[END_REF][START_REF] Kuwahara | Enhancement of Alternating Current Electroluminescence Properties by the addition of Graphene Oxide Nanosheets as Dielectric Materials[END_REF][START_REF] Liu | Thermal and Dielectric Properties of Graphene Oxide/Polyimide Resin Composite[END_REF]. By contrast, other studies have reported more moderate values of permittivity, of about 3, or less, in frequency ranges typically from 100 Hz to 1000 Hz [START_REF] Salomao | Determination of High-Frequency Dielectric Constant and Surface Potential of Graphene Oxide and Influence of Humidity by Kelvin Probe Force Microscopy[END_REF][START_REF] Kavinkumar | Effect of Functional Groups on Dielectric, Optical Gas Sensing Properties of Graphene Oxide and Reduced Graphene Oxide at Room Temperature[END_REF][START_REF] Yasin | Synthesis and Electrical Characterization of Graphene Oxide Films[END_REF][START_REF] Zhang | Anomalous Capacitive Behaviors of Graphene Oxide Based Solid-State Supercapacitors[END_REF]. Uses of GO as dielectrics in transistors led also to the conclusion that GO has a relative permittivity typically between 3 and 5 [START_REF] Standley | Graphene-Graphite Oxide Field-Effect Transistors[END_REF][START_REF] Lee | All Graphene-Based Thin Film Transistors on Flexible Plastic Substrates[END_REF][START_REF] Eda | Graphene Oxide Gate Dielectric for Graphene-Based Monolithic Field Effect Transistors[END_REF].

Enhancements of permittivity are, as for neat GO based materials, deduced from measurements of large capacitances. Nevertheless, the origin of such large dielectric permittivities remains unclear. Al-Zangana et al. [START_REF] Al-Zangana | Dielectric spectroscopy of isotropic liquids and liquid crystal phases with dispersed graphene oxide[END_REF] have recently investigated dielectric relaxation processes of GO solutions in isotropic fluids or liquid crystals. Relaxations in 2-Propanol are associated with very large amplitudes, well above 1000 in relative permittivity at 1 KHz. Such a giant amplitude would reflect a giant intrinsic dielectric constant of GO particles.

One can indeed expect, in the most optimistic case, that the GO dispersions form a pseudo-lamellar phase in the investigated concentration ranges [START_REF] Zamora-Ledezma | Liquid Crystallinity and Dimensions of Surfactant-Stabilized Sheets of Reduced Graphene Oxide[END_REF][START_REF] Yuan | Graphene Liquid Crystal Retarded Percolation for New High-k Materials[END_REF]. In this case, the system can be modelled as a heterogeneous medium with stacked insulating GO layers separated from each other by solvent layers as shown in Figure 1.

Each GO layer has thickness 𝛿 and complex conductivity 𝜎 𝐺𝑂 * = 2𝜋𝑗𝜈𝜀 0 𝜀 𝐺𝑂 while the solvent layers have a thickness t and complex conductivity 𝜎 𝑆𝑜𝑙𝑣 * = 𝜎 𝑆𝑜𝑙𝑣 + 2𝜋𝑗𝜈𝜀 0 𝜀 𝑆𝑜𝑙𝑣 , where 𝜀 0 is the permittivity of vacuum, 𝜀 𝐺𝑂 and 𝜀 𝑠𝑜𝑙𝑣 the dielectric constants of GO and of the solvent, 𝜎 𝑠𝑜𝑙𝑣 the conductivity of the solvent and 𝜈 the frequency.

Considering that t >> 𝛿 and that the normal of the layers is parallel to the electric field, in order to predict the greatest possible permittivity, the effective complex permittivity of the solution 𝜀 𝑒𝑓𝑓 is given by

𝜀 𝑒𝑓𝑓 * = Δ𝜀 1+𝑗 𝜈 𝜈 0 + 𝜀 𝑆𝑜𝑙𝑣 with Δ𝜀 = 𝜀 𝐺𝑂 𝑡 𝛿 and𝜈 0 = 𝛿𝜎 𝑆𝑜𝑙𝑣 2𝜋𝜀 0 𝑡𝜀 𝐺𝑂 .
The above assumptions are reasonable considering the actual size of GO particles and the fact that they display thermal undulations of small amplitude [START_REF] Poulin | Superflexibility of Graphene Oxide[END_REF].

The inclusion of GO in a given fluid is therefore expected to generate a dielectric relaxation with an amplitude that is proportional to its intrinsic dielectric constant. This is why very large amplitudes should reflect a giant dielectric constant of GO. The same actually would hold for various GO polymer composites developed for energy storage applications, and which display very large effective dielectric constants.

Nevertheless, we have to keep in mind that the large permittivities reported in the literature are deduced from capacitance measurements in complex electrical circuits.

As already discussed by some authors [START_REF] Mensah | A Study of Graphene Oxide-Reinforced Rubber Nanocomposite[END_REF][START_REF] Chen | Dielectric and Mechanical Properties and Thermal Stability of Polyimide-Graphene Oxide Composite films[END_REF][START_REF] Tantis | Functionalized Graphene -Poly(vinyl alcohol) Nnocomposites: Physical and Dielectric Properties[END_REF], other mechanisms, and in particular polarization of electrodes could contribute to the capacitance measurements.

Separating the contribution of electrode polarization from the volumetric one is often challenging. GO by contrast to neat graphene or other graphitic particles, can be viewed as a polyelectrolyte, which carries a huge number of counter-ions. The latter, be they in liquids or polymers, can form ionic double layers at the surface of electrodes and generate a large effective capacitance. Moreover, GO can also adsorb at the electrode interface and contribute to the electrode polarization [START_REF] Zhang | Anomalous Capacitive Behaviors of Graphene Oxide Based Solid-State Supercapacitors[END_REF].

In order to clarify the situation and determine whether GO displays a giant permittivity or not, we propose in this work a simple approach, which consists in measuring the impedance of dielectric cells filled with GO solutions at different concentrations. This approach, called impedance spectroscopy, is broadly used to determine the permittivity and conductivity of materials. We model the system using an equivalent circuit by considering both the contribution of electrodes, for an interfacial capacitance, and that of the dielectric medium, for the volumetric capacitance [START_REF] Barsoukov | Impedance Spectroscopy: Theory, Experiment, and Applications[END_REF]. We use a dielectric cell designed to enhance the contribution of the bulk compared to that of the electrode surface. The present cell is made of platinum rectangular electrodes and has a very low constant cell.

As previously observed in the literature, experiments both in Milli-Q water and in 2-Propanol confirm the measurements of very large capacitance at low frequencies. However, our simple modeling also demonstrates that the observed capacitances can be simply explained by polarization of electrodes, without considering neither giant permittivity of GO nor specific relaxation modes related to the rotations of the particles [START_REF] Al-Zangana | Dielectric spectroscopy of isotropic liquids and liquid crystal phases with dispersed graphene oxide[END_REF]. We therefore conclude that GO is not a high-permittivity material potentially useful for energy storage applications. However, it remains a unique precursor, easily processable, to make graphene-based materials, after reduction [START_REF] Zhu | Graphene and Graphene Oxide: Synthesis, Properties, and Applications[END_REF][START_REF] Dreyer | The Chemistry of Graphene Oxide[END_REF]. Indeed, as already well established, reduced graphene oxide (rGO) is a conducting material. As such, it can enhance the permittivity of composites in particular or concentrations in near-percolated regimes [START_REF] Mancinelli | Electrical AC and DC Behavior of Epoxy Nanocomposites Containing Graphene Oxide[END_REF][START_REF] Cui | Graphene-Based Composite Materials with High Dielectric Permittivity via an in situ Reduction Method[END_REF][START_REF] Hong | Electric Permittivity of Reduced Graphite Oxide[END_REF][START_REF] Kuang | Chemical Reduction Dependent Dielectric Properties and Dielectric Loss Mechanism of Reduced Graphene Oxide[END_REF][START_REF] Toselli | In Situ Thermal Reduction of Graphene Oxide Forming Epoxy Nanocomposites and Their Dielectric Properties[END_REF][START_REF] Xing | Mechanical, Thermal Conductive, and Dielectric Properties of Fluoroelastomer/Reduced Graphene Oxide Composites In Situ Prepared by Solvent Thermal Reduction[END_REF][START_REF] Yousefi | Highly Aligned Graphene/Polymer Nanocomposites with Excellent Dielectric Properties for High-Performance Electromagnetic Interference Shielding[END_REF].

A c c e p t e d M a n u s c r i p t EXPERIMENTAL SECTION

Materials

Commercial GO solution in water with a concentration of 4 mg/ml was purchased from Graphenea (Spain). This dispersion is prepared by chemically processing of raw graphite material in water via the modified Hummer's method [START_REF] Hummers | Preparation of graphitic oxide[END_REF]. According to the provider, the monolayer content, measured with 0.05 wt% dispersion, reaches values up to 95% and the single layer has a sheet dimension of 5-7 µm (see Figure S1 and S2 in Supplementary Information). Note that because of polydispersity, particles greater than 7 µm or smaller than 5 µm can be found in the samples. The degree of oxidation is around 40 % as measured by XPS analysis (Figure S3). The oxidized functions include approximately C-OH 284 eV, C=O 288 eV, O=C-OH 291 eV [START_REF] Hu | A Reduced Graphene Oxide Quantum Dot-Based Adsorbent for Efficiently Binding with Organic Pollutants[END_REF]. Milli-Q water (Q-Gard 00D2) with a resistivity above 18.2 MΩ•cm at 25 °C and 2-Propanol (IPA) absolute 99.9% were purchased from Sigma Aldrich and used as received.

Sample preparation

The GO dispersions in Milli-Q water and IPA were obtained through a solvent exchange process. First, aqueous GO solution was centrifuged at 50000g for 1h (Sorval RC 6+, rotor SE-12). The collected slurry was then dissolved in pure IPA. The obtained diluted dispersions in IPA were then centrifuged at 1400g for 20 minutes (Jouan B4i Multifunction, rotor S40) in order to precipitate any large aggregates. The supernatant was collected and treated using the same procedures as above to perform the second and third solvent exchanges. Afterwards, the achieved GO/IPA dispersion was sonicated with a tip Branson Sonifier 450A for 30 minutes at 135 W to further improve the dispersion state of GO flakes. This procedure allows the unbundling of the small remaining aggregates. Note that sonication presumably induces a slight reduction of the size of the flakes [START_REF] Coleman | Manipulation and Quantification of Graphene Oxide Flake Size: Photoluminescence and Cytotoxicity[END_REF].

Nevertheless, as shown further, the GO particles remain sufficiently large in diameter to form liquid crystal phases at relatively low concentration. The obtained dispersions were then subjected to a final ultracentrifuge process at 50000g for 1 hour to achieve GO gel with a high concentration of 65 mg/mL. The concentration was determined by dry extract characterizations of the samples. 1 g of the final gel was dried in a muffle furnace at 90 °C under vacuum for a duration of 2 h. Samples with different weight percentage (wt%) of GO flakes were prepared by dilution in IPA.

A c c e p t e d M a n u s c r i p t

The GO dispersions in Milli-Q water were prepared with exactly the same protocols. Note that at high concentration, the GO solutions display a high viscosity and a yield stress [5, 6 -44]. The systems are kept at room temperature during the present study. Increase of temperature can induce reduction of the particles, making them hydrophobic and electrically conductive.

Related materials have been characterized by X-ray diffraction in previous studies, and have been shown to be relatively stable against reduction [START_REF] Poulin | Superflexibility of Graphene Oxide[END_REF][START_REF] Zamora-Ledezma | Liquid Crystallinity and Dimensions of Surfactant-Stabilized Sheets of Reduced Graphene Oxide[END_REF]. The pH of the solutions, measured on litmus paper, varies typically between 5 and 6 in the investigated concentration range. There was not significant effect of pH observed.

Characterization

Polarizing optical micrographs (Leica DM 2500P) of GO dispersions in Milli-Q water and IPA were taken to investigate their birefringence at room temperature. The dielectric properties of the obtained dispersions were studied by using an impedance analyzer (7260 Impedance Analyzer, Materials Mates Italia) under a voltage of 50 mV in a frequency range of 1 Hz -10 6 Hz. The measurements were performed using a homebuilt cell with two platinum electrodes, as shown in Figure 2.

The size of the electrodes is 24 mm × 4 mm and the separation between them is 500 µm.

The cell thickness was calibrated by measurements of sodium chloride water solution with difference concentrations with known conductivity (see Supplementary Information Figure S9). The large surface area of the electrodes and the thin thickness were designed to minimize the strong effect of the polarization of the electrodes and maximize the contribution of the bulk [START_REF] Ben Ishai | Electrode polarization in dielectric measurements: a review[END_REF]. The present cell has a cell constant of only 5.2x10 -3 mm -1 . In the nematic phase (Figure 3 D,E andI,J), no aggregation is detected. Note that the presence or absence of aggregates is more clearly evaluated using unpolarized light (Figure 3 A'-J') compared to images using crossed polarizers (Figure 3 A-J). The flakes alignment is increasing, as evidenced by the increasing size of birefringent domains.

Due to the large size dispersity, no clear lamellar phase is observed, as expected for plate-like nanoparticles systems.

Nevertheless, the formed nematic phase can be viewed as a pseudo-lamellar phase with a regular spacing between the flakes [START_REF] Poulin | Superflexibility of Graphene Oxide[END_REF][START_REF] Zamora-Ledezma | Liquid Crystallinity and Dimensions of Surfactant-Stabilized Sheets of Reduced Graphene Oxide[END_REF]. The present phase behavior is consistent with that observed in previous reports [START_REF] Zamora-Ledezma | Liquid Crystallinity and Dimensions of Surfactant-Stabilized Sheets of Reduced Graphene Oxide[END_REF][START_REF] Zakri | Liquid Crystals of Carbon Nanotubes and Graphene[END_REF][START_REF] Kumar | Rheological Properties of Graphene Oxide Liquid Crystal[END_REF][START_REF] Kim | Graphene Oxide Liquid Crystals[END_REF] and was not investigated in more depth in the present work.

GO liquid crystals (LC) contain a rather large amount of ionic species which affect the determination of their dielectric properties due to a strong interfacial polarization, also called electrode polarization (EP). GO contains indeed a large fraction of functional groups such as carboxylic groups and dissociated counter ions. The polar solvents contain protons and hydroxyl ions.

They can also contain ions due to the dissolution of CO2. The electrode polarization is due to the adsorption of ions and formation of a diffuse double layer [START_REF] Ben Ishai | Electrode polarization in dielectric measurements: a review[END_REF] at the metal/solution interface. This phenomenon induces a large capacitive behavior at low frequencies, when the electric field inversion is slow enough so that the motion of ionic charges is hindered by the charging of the electric double layer. By contrast, at high frequencies the polarity inversion is fast enough to keep the charge carriers moving.

For a quantitative analysis of the system and in order to differentiate the electric doublelayer capacitance from the dielectric behavior of the bulk, one can use an effective equivalent electrical circuit (EEC) model [START_REF] Barsoukov | Impedance Spectroscopy: Theory, Experiment, and Applications[END_REF][START_REF] Bard | Electrochemical methods: fundamentals and applications[END_REF] (see Supplementary Information Figure S5). Because no particular ionic or electronic relaxation is expected at a molecular scale on the frequency range of investigation, we consider the circuit elements as constant.

Several models for GO-based LCs have been reported in the literature. They vary in complexity but were found in most cases suitable to explain experimental measurements in a certain frequency range [START_REF] Garcìa-Garcìa | The peculiar electrical response of liquid crystal-carbon nanotube systems as seen by impedance spectroscopy[END_REF].

First of all, it is necessary to ensure that the volumetric capacitance of the bulk sample is sufficiently large so that it can be measured by the experimental instrumentation. This is why we presently use a cell with a very low constant cell (5.2x10 -3 mm -1 ), taking into account that the smallest capacitance accurately measurable with our instrument is on the order of one pF, a common value for commercially available impedance analyzers.

We perform measurements between 100 Hz and 1 MHz, also a typical frequency range use in impedance spectroscopy. It is preferable that relaxation modes fall in this frequency range in order to achieve more accurate fitting of experimental data.

Herein, we adopted a simple EEC model that physically represents the mechanisms involved in the charge storage and transport of the material. This model is an extension of the classical Randles cell model [START_REF] Barsoukov | Impedance Spectroscopy: Theory, Experiment, and Applications[END_REF][START_REF] Lee | Microfluidic Electrochemical Impedance Spectroscopy of Carbon Composite Nanofluids[END_REF][START_REF] Hubálek | Iterative Precise Conductivity Measurements with IDEs[END_REF].

A c c e p t e d M a n u s c r i p t

The found values of capacitances exceed well 10 pF and are associated, for most of them, to relaxations clearly observed between 100 Hz and 1 MHz. In details, this simple model comprises a finite diffusion Warburg element W [START_REF] Song | Effects of Nanoparticle Geometry and Size Distribution on Diffusion Impedance of Battery Electrodes[END_REF] to account for the diffusion of charged species through the liquid crystal. This element is in series with a resistor R p to model the resistance associated to the polarization of the electrodes.

Both of them are in parallel with a capacitor C dl to take into account the possible accumulation of charges at the interface for an adsorbed double-layer capacitance. The capacitor C bulk , represents the volumetric capacitance of the bulk dispersions. This capacitance is directly linked to the permittivity of the sample, whereas the resistor R s is the resistance of bulk dispersion due to the ionic species in solution.

A c c e p t e d M a n u s c r i p t

The theoretical impedances were calculated based on the EEC model. The impedances associated to the circuit elements are given in Figure S5. Such EEC model gives us a satisfying fitting from 100 Hz to 1 MHz. However, this model is not able at describing the properties at frequencies well below 100 Hz, or well above 1MHz, due to the crude assumption of considering the impedance of each circuit element independent on frequency. In particular, the strong and dominant polarization of electrodes at low frequencies is expected to display a more complex behavior. Calculated values are then compared with the experimental data.

Figure 4 shows the frequency dependence of the real and imaginary part of the impedance for GO dispersions in 2-Propanol and Milli-Q water. It is observed that impedances are larger for IPA than for Milli-Q water. This is due to the greater permittivity of water compared to IPA, and to the greater ionic conductivity in water. The experimental impedances of samples at different GO concentrations matched well with the theoretical curves (colored circles represent the experimental values whereas the continuous line represents the calculated ones). This agreement suggests that the proposed electrical circuit model is suitable and accurate enough to well describe the electrical behavior of our systems. We also stress that the fits are accurate enough to determine the volumetric capacitance. As shown in supplementary information (Fig. S6), slight changes of the volumetric capacitance lead to poor fitting. The real part of the impedance for dispersions in 2-Propanol, as can be seen in Figure 4 (a), decreases by an order of magnitude with increasing concentration of GO. This trend reflects how the resistance R s of the samples decreases with increasing the GO flake concentration. The imaginary part of the impedance, shown in Figure 4 (b), also decreases as the concentration of the GO flakes increases, showing a maximum peak starting from 10 kHz, which reflects a capacitive mode behavior. These peaks shift at high frequency with increasing GO concentration.

Concerning the GO dispersions in Milli-Q water, the behaviors of the real and imaginary parts of the impedance are slightly different. This can be attributed to the higher polarity and permittivity of the solvent, which lead to relaxation modes slightly above 1 MHz.

The real part of the impedance, shown in Figure 4 (c), decreases with the GO concentration. This result reflects again a decrease of resistance of the bulk, which confirms that GO acts as a simple electrolyte in solution. In addition, the real part of the impedance in Milli-Q water shows a relaxation at frequency above 1 MHz, in agreement with other studies [START_REF] Hubálek | Iterative Precise Conductivity Measurements with IDEs[END_REF]. Nevertheless, the lack of full relaxation in the investigated range of frequencies makes the fitting with our model slightly more difficult. In parallel, the maximum of the imaginary part of the impedance, showed in Figure 4 (d) is not completely visible in the investigated frequency range. The values of each element of the EEC model that the best fitting given in Table 1.

A c c e p t e d M a n u s c r i p t

Figure 5 shows the trends of the bulk resistance, R s , and the related conductivity 𝜎 for GO dispersions in 2-Propanol (a) and Milli-Q water (c), calculated through the following expression:

𝜎 = 𝑑 𝑅 𝑠 𝐴 (1)
with, 𝑑 equal to 500 μm and 𝐴 equal to 96 mm 2 .

A c c e p t e d M a n u s c r i p t

The linear increase in conductivity with GO concentration shows that the GO particles behave as a classical electrolyte in solution.

Ionic conductivity values of bulk dispersions are shown in Table 2. It is interesting to note that the ionic conductivity of the supernatant is rather large, even greater than that of NaCl solutions (see Table S2). This large ionic conductivity comes from the dissociation of chemical functionalities at the surface of the GO particles.

The fitting of the data allows also the volumetric capacitance of the bulk and the double layer capacitance to be determined. For the dispersions in IPA, the electric double layer capacitance increases from 1 nF, for the supernatant, to 2.5 nF for a concentration of 6.5 wt% of GO.

This increase can be ascribed to the greater concentration of ionic species at the surface of the electrodes. The bulk capacitance evolves from 30 pF for the supernatant, to 58 pF for a concentration of 6.5 wt% of GO.

Regarding a GO dispersions in Milli-Q water, the interface capacitance evolves from 1 nF to 100 nF for a dispersion of 6.5 wt% of GO. The bulk capacitance, increases from 136 pF for the supernatant to 180 pF for the highest GO concentration. The relative permittivity of the samples can be deduced from the capacitance of the bulk [START_REF] Zhang | Anomalous Capacitive Behaviors of Graphene Oxide Based Solid-State Supercapacitors[END_REF] using equation 2:

𝐶 𝐵𝑢𝑙𝑘 = 𝜀 0 𝜀 𝑟 𝐴 𝑑 (2)
In Figure 6, it is possible to observe how the permittivity of the dispersions grows with the concentration of GO. This increase is clear but not as large as the increase which could have been expected if GO would have had a giant permittivity. As mentioned previously, GO liquid crystals can be considered as a heterogeneous system of planar 2D particles A c c e p t e d M a n u s c r i p t in a pseudo-lamellar phase in which the amplitude Δ𝜀 (understood as the difference between the Ɛ r of GO at 6.5 wt% and the one of the supernatant) is related to the dielectric constant of the particles 𝜀 𝑝 as follows:

Δ𝜀 = 𝜀 𝑝 𝑡 𝛿 (3)
where, t is the thickness of the solvent layer and 𝛿 the thickness of the particle. Using this relation, we can estimate the value of the dielectric constant of the GO particles. The ratio of the respective thickness of the GO layer and solvent layer can be estimated through the volume fraction 𝜙 𝑉 , defined by the ratio of the volume of the particle and the total volume.

From this relation, by using the respective density of the solvents equal to 0.79 g/mL for 2-Propanol and 1.83 g/mL for GO we obtain a value of 𝜙 𝑉 ~ 0.0291 for a weight fraction of 6.5 wt%. We deduce from this value that the dielectric constant of the particle equal to ~1.0 for the dispersions in IPA. On the other hand, considering the volume fraction for Milli-Q water, equal to 𝜙 𝑉 ~ 0.0361 at a weight fraction of 6.5 wt%, we find a value of 2.3 for the dielectric constant of the GO particles in Milli-Q water.

Even if the two values found in Milli-Q water and IPA are just estimates of the dielectric constant of the particle, they are still significantly lower than the giant permittivity found in many other papers [START_REF] Liu | Graphene-based Thin Film Supercapacitor with Graphene Oxide as Dielectric Spacer[END_REF][START_REF] Hong | Graphite Oxide Paper as a Polarizable Electrical Conductor in the Through-Thickness Direction[END_REF][START_REF] Huang | Temperature-Dependent Electrical Property Transition of Graphene Oxide Paper[END_REF][START_REF] Kumar | A New Single/Few-Layered Graphene Oxide with a High Dielectric Constant of 10 6 : contribution of defects and functional groups[END_REF][START_REF] Hou | Graphene Oxide Foams: the Simplest Carbon-Air Prototypes for Unique Variable Dielectrics[END_REF][START_REF] Wan | Graphene Oxide as High-Performance Dielectric Materials for Capacitive Pressure Sensors[END_REF][START_REF] Wang | A Water-Dielectric Capacitor Using Hydrated Graphene Oxide Film[END_REF][START_REF] Gao | Direct Laser Writing of Micro-Supercapacitors on Hydrated Graphite Oxide Films[END_REF][START_REF] Bayer | Tunable Mixed Ionic/Electronic Conductivity and Permittivity of Graphene Oxide Paper for Electrochemical Energy Conversion[END_REF][START_REF] Al-Zangana | Dielectric spectroscopy of isotropic liquids and liquid crystal phases with dispersed graphene oxide[END_REF].

It is also interesting to note a steep rise of the capacitance of the double layer with adding a small amount of GO. This increase is presumably due to the adsorption of ions at the surface of the electrodes.

Last, we also note that the dielectric properties of the sample does not show any dependence on the size of the flakes. This is expected as long as t >> 𝛿. To confirm this expectation, we show in Figure S7 the dielectric properties of a solution of small particles made by strong sonication. The results are in agreement with those found for larger

Conclusion

We have characterized the impedance of GO solutions at high concentrations in two different solvents. In both cases, we observed a large polarization of the electrodes. In order, to determine the actual volumetric capacitance of the bulk and therefore the actual permittivity of the solutions, we use a very low-constant cell and a simple equivalent circuit to model the electrical behavior of the system.

The good agreement between the model and the experiments allowed us to determine the permittivity of GO particles. We claim that the permittivity of GO is not particularly high. Typically, it has a relative permittivity varying from 1 to a few unites.

Therefore, we do not consider GOLCs as good candidates for high-k materials.

Nevertheless, GO can be used as a suitable component for devices that exploit an interphase capacitance, like Field Effect Transistors (FETs), as an electrolyte or as precursor for graphene based materials, but not as an additive to raise the permittivity of solvents and composites for electronics (capacitive sensing, energy storage and energy harvesting in variable capacitors).

Figure 1 -

 1 Figure 1 -Schematic representation of GO dispersion in a pseudo-lamellar phase with the normal of the particles parallel to the direction of the electric field and with 𝛿 the thickness of the stacked dielectric insulating GO layers and t the thickness of the separating solvent layers.

Figure 2 -

 2 Figure 2 -The schematic image of the home-built dielectric cell used in this work. The inlet and outlet present on top of the dielectric cell are necessary to insertion of the sample.

  A c c e p t e d M a n u s c r i p tRESULTS AND DISCUSSIONOptical micrographs of GO dispersions in Milli-Q water and IPA are shown in Figure3.GO flakes are homogeneously dispersed at low concentration, with a small number of microscopic aggregates below 0.6 wt% in Milli-Q water and 0.4 wt% in 2-Propanol. Crosspolarizing pictures show increasing birefringence (bright spots) as approaching the biphasic-nematic phase boundary, around 0.8 wt% GO, for both solvents with increasing GO concentration.

Figure 3 -

 3 Figure 3 -Optical polarizing micrographs of GO dispersion in Milli-Q water(A-E) and 2-Propanol (F-L): (A,F) 0.4 wt%, (B,G) 0.6 wt%, (C,H) 1.0 wt%, (D,I) 1.5 wt%, (E,J) 3.0 wt%. (Scale bar 200 µm). The observed textures reflect the formation of small monodomains of nematic LC phase.

Figure 4 -

 4 Figure 4 -Frequency dependences of the real and imaginary part of measured impedance for GO dispersions in IPA (a) and (b) on top, and Milli-Q water on bottom (c) and (d). The colored circles represent the experimental values whereas the continuous lines show the simulated data for the concentrations: 0.4 wt% (red), 0.6wt% (green), 1.0 wt% (blue), 1.5 wt% (pink), 3.0 wt% (purple), 6.5 wt% (orange) and the respective supernatant (black).

Figure 5 -

 5 Figure 5 -Resistance of the bulk dispersions, Rs and Capacitance values of CdL and Cbulk as function of weight fraction for GO dispersions in IPA (a,b) and Milli-Q water (c,d).

  δ/𝑡 ~ 𝜙 𝑉

Figure 6 -

 6 Figure 6 -Deduced permittivity of the dispersions as function of concentration for GO dispersions in 2-Propanol (A) and Milli-Q water (B).

A c c e

  p t e d M a n u s c r i p t particles. In particular Cbulk remains low, which reflects again the absence of giant permittivity of GO.

A c c e p

  t e d M a n u s c r i p t A c c e p t e d M a n u s c r i p t A c c e p t e d M a n u s c r i p t

Table 1 -The values of each element of the Equivalent Electrical Circuit model (EEC) obtained from data fitting.

 1

	Solvent	GO [wt%]	R p [Ω]	W [Ω] -0.5	C dl [F]	R s [Ω]	C bulk [F]
	2-Propanol	Supernatant	800	14000	1.00x10 -9	61000	3.05x10 -11
		0.4 wt%	700	18000	1.20x10 -9	30500	3.80x10 -11
		0.6 wt%	800	14000	1.50x10 -9	20300	4.00x10 -11
		1.0 wt%	700	12000	1.52x10 -9	13700	4.20x10 -11
		1.5 wt%	500	11000	1.80x10 -9	10600	4.60x10 -11
		3.0 wt%	600	7000	2.00x10 -9	6600	5.00x10 -11
		6.5 wt%	800	7000	2.50x10 -9	3070	5.80x10 -11
	Mili-Q water	Supernatant	300	7000	1.00x10 -9	760	1.36x10 -10
		0.4 wt%	20	6000	1.50x10 -8	500	1.40x10 -10
		0.6 wt%	15	4400	5.00x10 -8	320	1.43x10 -10
		1.0 wt%	9	3200	8.20x10 -8	207	1.45x10 -10
		1.5 wt%	7	2400	8.50x10 -8	123	1.47x10 -10
		3.0 wt%	4	1300	9.00x10 -8	62	1.55x10 -10
		6.5 wt%	1	1250	1.00x10 -7	36	1.65x10 -10

Table 2 -Ionic conductivity of the bulk dispersions.

 2

			Supernatant	0.4 wt%	0.6 wt%	1.0 wt%	1.5 wt%	3.0 wt%	6.5 wt%
	2-Propanol	σ [S/m]	8.8x10-5	1.7x10-4	2.6x10-4	3.8x10-4	4.9x10-4	7.8x10-4	1.6x10-4
	Milli-Q water	σ [S/m]	6.8x10-3	1.0x10-2	1.6x10-2	2.5x10-2	4.2x10-2	8.4x10-2	1.4x10-1

Acknowledgments

The authors wish to thank the Cluster of Excellence AMADEus of Université de Bordeaux for the facilities and financial support.


Karine Bertin
email: karine.bertin@uv.cl

Nicolas Klutchnikoff
email: nicolas.klutchnikoff@univ-rennes2.fr

Adaptive regression with Brownian path covariate

Keywords: Functional regression, Wiener-Itô chaos expansion, Oracle inequalities, Adaptive minimax rates of convergence. AMS Subject Classification: 62G08, 62H12

come

Adaptive regression with Brownian path covariate

Introduction

The problem of regression estimation is one of the most studied in statistics and different models have been considered depending on the nature of the data. In an increasing number of applications, it seems natural to assume that the covariate takes values in a functional space. The book of [START_REF] Ramsay | Functional data analysis[END_REF] provides an overview on the subject of functional data analysis. In this context, several authors studied linear functional regression models (see for example [START_REF] Müller | Generalized functional linear models[END_REF][START_REF] Cai | Prediction in functional linear regression[END_REF]1 Crambes, Kneip, and Sarda, 2009). Non-parametric functional regression models have also been investigated (see Ferraty and Vieu, 2006, and references therein). In this paper, we are interested in such a model where the covariate is a Wiener Process. More precisely, let ε be a real-valued random variable and W = (W (t) : 0 ≤ t ≤ 1) be a standard Brownian motion independent of ε. We define Y = m(W ) + ε where m : C → R is a mapping defined on the set C of all continuous functions w : [0, 1] → R and we assume that both m(W ) and ε are square integrable random variables. Our goal is to estimate the function m using a dataset (Y 1 , W 1 ), . . . , (Y n , W n ) of independent realizations of (Y, W ). Since this framework is a specific case of the more general functional regression framework, usual approaches (which mainly consist in extending classical local methods such as k-nearest neighbors, kernel smoothing or local polynomial smoothing) could be used. However, in our context, these methods are known to lead to slow rates of convergence over classical models (see below for detailed references). Taking advantage of the probabilistic properties of the Wiener coprocess, we aim at defining a new family of models as well as dedicated estimation procedures with faster rates of convergence (in both minimax and adaptive minimax senses).

In usual functional approaches the set C is endowed with a metric d (see for example [START_REF] Ferraty | Nonparametric functional data analysis[END_REF][START_REF] Ferraty | Nonparametric regression on functional data: inference and practical aspects[END_REF][START_REF] Biau | Rates of convergence of the functional k-nearest neighbor estimate[END_REF] which allows to extend several nonparametric estimators. For example a simple version of the Nadaraya-Watson estimator is given, for any function w ∈ C and any bandwidth h > 0, by:

mh (w) = 1 n n i=1 Y i I {d(W i ,w)≤h} n j=1 I {d(W j ,w)≤h}
, where I stands for the indicator function. The properties of these estimators are related to the behavior of a quantity known as the small ball probability defined for w ∈ C and h > 0 by ϕ w (h) = P(d(W, w) ≤ h). Pointwise risks of such methods can be generally bounded, up to a positive factor by

h β + 1 nϕ w (h) 1/2
where β denotes the smoothness of the mapping m measured in a Hölder sense. For example, if 0 < β ≤ 1 it is assumed that there exists L > 0 such that |m(w) -m(w )| ≤ Ld(w, w ) β for any w, w ∈ C. Under additional assumptions similar results can be obtained for integrated risks.

The classical assumption ϕ w (h) h k corresponds roughly to the situation where the covariate W lies in some space of finite dimension k (see [START_REF] Azaïs | Remark on the finite-dimensional character of certain results of functional statistics[END_REF]. This framework corresponds to the usual nonparametric case. The minimax rates of convergence are then given by n -β/(2β+k) (see [START_REF] Alexandre | Introduction to nonparametric estimation[END_REF]. However if W lies in a functional space, the behavior of ϕ w (h) is quite different. In our context, where W is a standard Wiener process, it is well-known (see [START_REF] Li | Gaussian processes: inequalities, small ball probabilities and applications[END_REF]

that log P sup t∈[0,1] |W (t)| ≤ h = log ϕ 0 (h) h→0 -h -2
which leads to slower rates of convergence of the form (log n) -β/2 . We refer the reader to [START_REF] Chagny | Adaptive estimation in the functional nonparametric regression model[END_REF] for recent results with different behavior of ϕ w (h).

In practical situations, since β is unknown, finding adaptive procedures to select the smoothing parameter h is of prime interest. To our best knowledge few papers deal with this problem. Adaptive procedures based on cross validation have been used in [START_REF] Rachdi | Nonparametric regression for functional data: automatic smoothing parameter selection[END_REF]. [START_REF] Chagny | Adaptive estimation in the functional nonparametric regression model[END_REF] also propose an adaptation of the method developed by Goldenshluger and Lepski (see [START_REF] Goldenshluger | Bandwidth selection in kernel density estimation: oracle inequalities and adaptive minimax optimality[END_REF] using an empirical version of the quantity ϕ w (h). Lower bounds have been investigated by [START_REF] Mas | Lower bound in regression for functional data by representation of small ball probabilities[END_REF]. In all these papers, the pointwise risk is studied in terms of ϕ w (h) and theoretical properties are obtained assuming a β-Hölder condition on m with respect to the metric d with smoothness β ∈ (0, 1].

In this paper we follow a different strategy. Taking advantage of probabilistic properties of the Wiener process, similarly to the methodology developed by [START_REF] Cadre | Nonparametric regression estimation onto a Poisson point process covariate[END_REF] and [START_REF] Cadre | Minimax regression estimation for Poisson coprocess[END_REF], we consider the Wiener-Itô chaotic decomposition of m(W ). Indeed, every random variable that belongs to L 2 W = {m(W ) | m : C → R and E(m(W )) 2 < +∞} can be decomposed as a sum of multiple stochastic integrals (see Di Nunno, Øksendal, and Proske, 2009, for more details). There exists a unique sequence of functions (f ) ≥1 such that m(W )

L 2 = E(Y ) + ∞ =1 1 ! I (f )(W ), (1)
where f belongs to L 2 sym (∆ ), the set of symmetric and square integrable real-valued functions defined on ∆ = [0, 1] and

I (f )(W ) = ∆ f dW ⊗ = ∆ f (u 1 , . . . , u )W (du 1 ) • • • W (du ).
The iterated integral I (f )(W ) is called a chaos of order . We define a well-adapted family of models assuming that the summation (1) stops at a finite index L and imposing regularity assumptions on each function f for = 1, . . . , L (see Section 2.1 for a precise definition). Kernel-type estimators f of f can be defined using the Itô's isometry. Using plug-in estimator for each chaos, this lead to a simple estimator of m given by:

m(W ) = 1 n n i=1 Y i + L =1 1 ! I ( f )(W ).
We prove that such dedicated estimators achieve optimal rates of convergence on our models (indeed, these rates are proved to be the minimax ones). Contrary to the classical functional framework (where logarithmic rates are derived) the minimax rates of convergence over our models are polynomial in n with an exponent that depends on the smoothness of the functions f . A data-driven procedure, based on the method developed by [START_REF] Goldenshluger | Bandwidth selection in kernel density estimation: oracle inequalities and adaptive minimax optimality[END_REF], is then defined to tune the bandwidths used in the estimation of the functions f . The resulting estimator of m satisfies an oracle-type inequality that allows us to derive adaptive results. The paper is organized as follows. Section 2 presents the model and the studied problem. Section 3 describes the construction of the estimators. Section 4 gives the main results and Section 5 is dedicated to the proofs.

Statistical framework

Model

Let W = (W (t) : 0 ≤ t ≤ 1) be a standard Brownian motion and let ε be a centered real-valued random variable independent of W . We define:

Y = m(W ) + ε
where m : C → R is a given mapping. We assume that m(W ) as well as ε belong to L 2 , the set of square integrable random variables, and that there exists L ∈ N such that

m(W ) = E(Y ) + L =1 1 ! I (f )(W ),
where for ∈ {1, . . . , L}, f belongs to L 2 sym (∆ ). As mentioned in the introduction we also assume that f is a regular function. Below we define precisely the functional classes used to measure the smoothness of each function f . Definition 1 Set ∈ N, s > 0, Λ > 0 and M > 0. The Hölder ball H (s , Λ , M ) is the set of all functions f : ∆ → R that satisfy the following properties:

1. For any α = (α 1 , . . . , α ) ∈ N such that |α| = i α i ≤ s = max{k ∈ N | k < s }, the partial derivative D α f exists where

D α f = ∂ |α| f ∂x α 1 . . . ∂x α .
2. For any x and y in ∆ we have:

|α|= s |D α f (x) -D α f (y)| ≤ Λ |x -y| s-s ,
where | • | stands for the Euclidean norm of R .

3. We have f 2 ∆ ≤ M 2 ! where:

f ∆ = ∆ f 2 (u) du 1/2 .
Equipped with these notations we can define a scale of models for the mapping m. Each model is caraterized by L functional classes H (s , Λ , M ) for = 1, . . . , L. Definition 2 Set s = (s 1 , . . . , s L ) ∈ (0, +∞) L , Λ = (Λ 1 , . . . , Λ L ) ∈ (0, +∞) L and M > 0. We denote by M s,Λ,M the subset of L 2 W defined by:

M s,Λ,M = R ⊕ L =1 1 ! I H (s , Λ , M ) (W )
where

I H (s , Λ , M ) (W ) consists of all random variables I (f )(W ) with f ∈ H (s , Λ , M ). We also denote by M s,Λ,M the corresponding class of mappings m : C → R such that m(W ) ∈ M s,Λ,M .

Adaptive framework

The observations consist in a n-sample (Y 1 , W 1 ), . . . , (Y n , W n ) distributed as and independent of (Y, W ). Our main goal is to investigate the adaptive estimation of m, based on these observations, over the scale of mapping classes

M(s * ) = {M s,Λ,M | s ∈ (0, s * ) L , Λ ∈ (0, +∞) L , M > 0} where
s * > 0 is fixed. To measure the accuracy of an arbitrary estimator mn = m(

• ; (Y 1 , W 1 ), . . . , (Y n , W n ))
of m, we consider the following risk:

R p ( mn , m) = E| mn (W ) -m(W )| p 1/p
where p ≥ 2.

The maximal risk of an arbitrary estimator mn over a given class of mappings M is defined by:

R p ( mn , M) = sup m∈M R p ( mn , m),
whereas the minimax risk is defined, taking the infimum over all possible estimators, by:

Φ

n (M, p) = inf mn R p ( mn , M).
An estimator mn whose maximal risk is asymptotically bounded, up to a multiplicative factor, by Φ n (M, p) is called minimax over M. Such an estimator is well-adapted to the estimation over M but it can performs poorly over another class of mappings. The problem of adaptive estimation consists in finding a single estimation procedure that is simultaneously minimax over a scale of mapping classes. More precisely our goal is to construct a single estimation procedure m * n such that, for any M ∈ M(s * ), the risk R p (m * n , M) is asymptotically bounded, up to a multiplicative constant, by Φ n (M, p). One of the main tools to prove such a result is to find an oracle-type inequality that guarantees that this procedure performs almost as well as the best estimator in a rich family of estimators. Ideally, we would like to have, for any m ∈ M s,Λ,M , an inequality of the following form:

R p (m * n , m) ≤ inf η∈H R p ( mn,η , m), (2)
where { mn,η | η ∈ H} is a family of estimators well-adapted to our problem in the following sense: for any M ∈ M(s * ), there exists η ∈ H such that mn,η is minimax over M. However, in many situations, ( 2) is relaxed and we prove a weaker inequality of the type:

R p (m * n , m) ≤ Υ 1,p inf η∈H R * p (m, η) + Υ 2,p log n n 1/2 , ( 3
)
where Υ 1,p and Υ 2,p are two positive constants and R * p (m, η) is an appropriate quantity to be determined that can be viewed as a tight upper bound on R p ( mn,η , m). Inequalities of the form (3) are called oracle-type inequalities.

Theorems 2 and 3 below correspond respectively to an oracle-type inequality and an adaptive result of these types.

Estimator construction

In this section we present our estimation procedure. To do so, we first recall classical properties satisfied by Wiener chaos which allow us to construct a family of "simple" estimators that depends on a multivariate tuning parameter. Next we construct a procedure which selects, in a data-driven way, this tuning parameter using the methodology developed by [START_REF] Goldenshluger | Bandwidth selection in kernel density estimation: oracle inequalities and adaptive minimax optimality[END_REF].

Classical properties of the chaos

Throughout this paper and in the construction of our statistical procedure, we use the following two fundamental properties satisfied by the iterated integrals.

For , ∈ N, Itô's isometry [START_REF] Di Nunno | Malliavin calculus for Lévy processes with applications to finance[END_REF] ensures that, if g ∈ L 2 sym (∆ ) and g ∈ L 2 sym (∆ ), then

E I (g)(W )I (g )(W ) = δ , ! ∆ g(u)g (u) du (4)
where δ , denotes the Kronecker delta.

The hypercontractivity property [START_REF] Nourdin | Normal approximations with Malliavin calculus[END_REF] will be used to control the concentration of our estimators. Set q ≥ 2 and ∈ N. There exists a positive constant c (q) such that for any g ∈ L 2 sym (∆ ) we have:

(E|I (g)(W )| q ) 1/q ≤ c (q) EI 2 (g)(W ) 1/2 .
(5)

A simple family of estimators

Let k : R → R be a function that satisfies the following properties:

k is continuous inside [0, 1], k(x) = 0 for any x / ∈ [0, 1], 1 0 k(x) dx = 1 and 1 0 x s k(x) dx = 0, s = 1, . . . , s * .
Let ∈ {1, . . . , L}. A natural estimator of the function f is given, for h ∈ (0, 1), by:

f ( ) h (t) = 1 n n i=1 Y i I K ( ) h (t, •) (W i ), t ∈ ∆ (6) where K ( )
h is a multivariate kernel defined by:

K ( ) h (t, u) = 1 h k=1 k ς(t k ) t k -u k h with ς(•) = 2I (1/2,1) (•) -1.
This specific construction allows one to obtain an estimator free of boundary bias (see [START_REF] Bertin | Adaptive density estimation on bounded domains[END_REF], for more details). Indeed, note that for any t ∈ ∆ and under regularity assumptions on f we have:

f (t) (h→0) ≈ ∆ f (u)K ( ) h (t -u) du = E 1 ! I (f )(W )I K ( ) h (t, •) (W ) = E m(W )I K ( ) h (t, •) (W )
where the last two lines are obtained using ( 1) and ( 4). Since ε is centered and independent of W we have:

f (t) (h→0) ≈ E Y I K ( ) h (t, •) (W ) (n→+∞) ≈ f ( ) h (t).
Equipped with these notations we define a family of plugin estimators of the mapping m. For all h = (h 1 , . . . , h L ) ∈ (0, 1) L we set:

mh (W ) = Y n + L =1 1 ! I f ( ) h (W )
where

Y n = n i=1 Y i /n.

Selection procedure

Set τ > 0 and M > 0. Assume that µ 4 = (E|ε| 4 ) 1/4 exists. Let ∈ {1, . . . , L} be fixed and define

H = h ∈ (0, 1) : n -1/(2τ + ) ≤ h ≤ (log n) -1 ∩ {e -k : k ∈ N}. Now, define M ( , h) = ν( ) 1 + 4 log(1/h ) √ nh (7)
where

ν( ) = µ 4 + L k=1 c 4 (k)M b ,2 2 and b ,2 = c 2 (4)2 ! k 2 [0,1] ,
where the constants c (k) are defined in (5). Define for h ∈ H

B( , h) = max h ∈H f ( ) h - f ( ) h∨h -M ( , h ) -M ( , h ∨ h ) + and set ĥ = arg min h∈H {B( , h) + M ( , h)} . ( 8
)
The estimation procedure is the defined by m = mĥ where ĥ = ĥ1 , . . . , ĥL .

Remark 1 This selection rule follows the principles and the ideas developed by Goldenshluger and Lepski in a series of papers (see Goldenshluger andLepski, 2011, 2014, among others). The quantity M ( , h), which is called a majorant in the papers cited above, is a penalized version of the standard deviation of the estimator f ( ) h while the quantity B( , h) is, in some sense, closed to its bias term, see (24). Finding tight majorants is the key point of the method since ĥ is chosen in (8) in order to realize an empirical trade-off between these two quantities.

It is worth noting that the procedure depends on a hyperparameter τ > 0 which can be chosen arbitrary small. The introduction of this parameter is due to technical reasons, see (28) in the proof of Lemma 2. This additional assumption (we would like to take τ = 0) implies some restrictions on Theorem 3 below.

Main results

Our first result proves that the minimax rate of convergence over the class M s,Λ,M is of the same order as:

φ n (s, Λ, M ) = max    Λ /(2s + ) M 2 n s /(2s + ) | = 1, . . . , L    . Theorem 1 Set p ≥ 2, L ∈ N, s ∈ (0, s * ) L , Λ ∈ (0, +∞) L , M > 0 and assume that E|ε| p < +∞. Define h n (s, Λ, M ) = h ( ) n (s, Λ, M ) =1,...,L ∈ (0, 1) L where: h ( ) n (s, Λ, M ) = M 2 Λ 2 n 1 2s + , = 1, . .

. , L.

There exist two positive constants κ * and κ * that depend only on L and s * such that

lim sup n→+∞ φ -1 n (s, Λ, M )R p mhn(s,Λ,M) , M s,Λ,M ≤ κ * (9) and lim inf n→+∞ φ -1 n (s, Λ, M )Φ n (M s,Λ,M , p) ≥ κ * . ( 10
)
Note that this result also ensures that the family of estimators constructed in Section 3.2 is well-adapted to our problem. The next result states an oracle-type inequality satisfied by our data-driven estimator m.

Theorem 2 Assume that for any = 1, . . . , L, f 2 ∆ ≤ M 2 ! and that for any q ≥ 1 the moment µ q = (E|ε| q ) 1/q exists. Then:

R 2 ( m, m) ≤ Υ 1 L =1 inf h∈H    max h ∈H h ≤h E f h -f 2 + M ( , h)    + Υ 2 log n n 1/2
, where Υ 1 and Υ 2 are two positive constants.

Using Theorems 1 and 2 we can derive our last result: the data-driven estimation procedure is adaptive, up to a logarithmic factor, over the scale

{M s,Λ,M : s ∈ [τ, s * ) L , Λ ∈ (0, +∞) L , M > 0}.
Theorem 3 Assume that for any q ≥ 1 the moment µ q = (E|ε| q ) 1/q exists. For any s ∈ [τ, s * ) L , any Λ ∈ (0, +∞) L , any M > 0, we have

lim sup n→+∞ φ-1 n (s)R 2 ( m, M s,Λ,M ) ≤ κ * *
where κ * * is a positive constant and

φn (s) = max    log n n s /(2s + ) | = 1, . . . , L    .
Remark 2 While the minimax rate of convergence over M(s, Λ, M ) is given in Theorem 1 for any risk R p (•, •) with p ≥ 2, Theorems 2 and 3 are only stated for the risk R 2 (•, •). We made this choice to avoid too technical details in the proofs but similar results could be obtained for p > 2 using the hypercontractivity property. Note that in Theorem 2, the quantity

max h ∈H h ≤h E f h -f 2
is a tight upper bound of the bias term of the estimator f h . This result ensures that our data-driven procedure is adaptive, up to a logarithmic factor, over a large scale of mapping classes since τ can be chosen as close to 0 as one wants.

The presence of the extra logarithmic factor in the adaptive rate of convergence is not usual for p = 2. This term is introduced in the definition of M ( , h) to control the deviation of the estimator (6) based on the variables I K ( ) h (t, •) (W i ). See (29) for more details.

Proofs

We first consider some notation and lemmas. Define for i ∈ {1, . . . , n}, ∈ {1, . . . , L} and h ∈ (0, 1)

ξ i, (t, h) = I K h (t -•) (W i ), and ξ (t, h) = I K h (t -•) (W ) and Θ i, = I (f )(W i ) and Θ i, = I (f )(W ).
Lemma 1 We have, for any ∈ {1, . . . , L}, h ∈ H and r ≥ 1

Eξ 2r (t, h) 1/r ≤ b ,r h -with b ,r = c 2 (2r)2 ! k 2 [0,1] . Moreover for ϕ > 0 and q ≥ 1 E |ξ (t, h)| r I |ξ (t,h)|>ϕ ≤ (b ,r ) r/2 (b ,q ) q/2 ϕ -q h -(r+q)/2 .
Lemma 2 Let ∈ {1, . . . , L} and h ∈ H . Let χ, χ 1 , . . . , χ n be i.i.d random variables such that, for any r ≥ 1 E|χ| 2r 1/(2r) ≤ a r < +∞.

Define

U(t) = 1 n n i=1 χ i ξ i, (t, h) -E (χ i ξ i, (t, h))
and

T = (1 + δ) a 2 b ,2 2 √ nh with δ = 4 log(1/h ).
Then there exists a positive constant C > 0 such that

E { U 2 -T } 2 + ≤ Cn -1 . ( 11
)
The following Lemma recalls the Bousquet's version of Talagrand's concentration inequality (see [START_REF] Boucheron | A Bennett concentration inequality and its application to suprema of empirical processes[END_REF][START_REF] Boucheron | A Bennett concentration inequality and its application to suprema of empirical processes[END_REF].

Lemma 3 (Bousquet's inequality) Let X 1 , . . . , X n be independent identically distributed random variables. Let S be a countable set of functions and define Z = sup s∈S n i=1 s(X i ). Assume that, for all i = 1, . . . , n and s ∈ S, we have Es(X i ) = 0 and s(X i ) ≤ 1 almost surely. Assume also that v = 2EZ + sup s∈S n i=1 E(s(X i )) 2 < ∞. Then we have for all t > 0

P (Z -EZ ≥ t) ≤ exp - t 2 2(v + t 3 )
.

Proof of Theorem 1

Set p ≥ 2, L ∈ N, s ∈ (0, s * ) L , Λ ∈ (0, +∞) L and M > 0. For the sake of readability we denote

h = h n (s, Λ, M ), h = h ( ) n (s, Λ, M ), K = K ( ) h , ξ (t) = ξ (t, h ) and f = f ( ) h .
This proof is decomposed into two parts. We first prove the upper bound (9) and then the lower bound (10).

Proof of the upper bound

Decomposition of the risk. Using the triangle inequality we have:

R p ( mh , m) = E| mh (W ) -m(W )| p 1/p ≤ E| Ȳn -E(Y )| p 1/p + L =1 1 ! E I f -f (W ) p 1/p ≤ E| Ȳn -E(Y )| p 1/p + L =1 c (p) ! E I f -f (W ) 2 1/2
Last lines comes from the hypercontractivity property. Now, using Itô's isometry, we obtain:

R p ( mh , m) ≤ E| Ȳn -E(Y )| p 1/p + L =1 c (p) √ ! E f -f 2 ∆ 1/2 ≤ E| Ȳn -E(Y )| p 1/p + L =1 c (p) √ ! B( ) + V ( ) , ( 12
)
where the bias term B( ) and the stochastic term V ( ) are defined by:

B( ) = E f -f ∆ and V ( ) = E f -E f 2 ∆ 1/2
.

Study of the constant term. Remark that

E| Ȳn -EY | p 1/p = E 1 n n i=1 (Y i -EY i ) p 1/p ≤ C 1,p n 1-1/p (E|Y -EY | p ) 1/p + C 2,p σ Y n 1/2 (13)
where the last line is obtained using Rosenthal's inequality. Here C 1,p and C 2,p denote two positive constants while σ Y stands for the standard deviation of Y (which is finite since both m(W ) and ε belong to L 2 ). Moreover we have

|Y -EY | p = L =1 Θ ! + ε p ≤ (L + 1) p-1 L =1 |Θ | ! p + |ε| p
which implies, using the hypercontractivity property combined with Itô's isometry, that

(E|Y -EY | p ) 1/p ≤ (L + 1) 1-1/p L =1 1 ! (E |Θ | p ) 1/p + (E|ε| p ) 1/p ≤ (L + 1) 1-1/p L =1 c (p) ! EΘ 2 1/2 + (E|ε| p ) 1/p
Using hypercontractivity we obtain

(E|Y -EY | p ) 1/p ≤ (L + 1) 1-1/p M L =1 c (p) √ ! + (E|ε| p ) 1/p (14)
where the right hand side of the above inequality is a finite quantity denoted by s Y (p). Taking together ( 13) and ( 14) we finally obtain

E| Ȳn -EY | p 1/p ≤ C 1,p s Y (p) n 1-1/p + C 2,p σ Y n 1/2 ≤ κ 0 n 1/2 , ( 15
) with κ 0 = C 1,p s Y (p) + C 2,p σ Y .
Study of the bias term. Set ∈ {1, . . . , L} and note that:

E f (t) = E Y ξ (t) (W ) = E (m(W )ξ (t)) = 1 ! E (Θ ξ (t))
Using Itô's isometry we thus obtain:

E f (t) = ∆ f (u)K (t -u) du.
Since f ∈ H (s , Λ , M ), we obtain:

B( ) ≤ b (k, s) Λ h s (16)
Previous inequality follows from Taylor's expansion and classical arguments (see [START_REF] Bertin | Adaptive density estimation on bounded domains[END_REF]. The expression of b (k, s) can be computed:

b (k, s) = 2 m |α|=m i=1 1 0 |k(y)|y α i +γ dy.
Here, α = (α 1 , . . . , α ) ∈ (N ∪ {0}) , |α| = α 1 + . . . + α , and s = m + γ with m ∈ N ∪ {0} and 0 < γ ≤ 1.

Study of the stochastic term V ( ) Set ∈ {1, . . . , L}. We have:

V ( ) 2 = V 1 ( ) + V 2 ( ) n , where V 1 ( ) = ∆ E ε 2 ξ 2 (t) dt and V 2 ( ) = ∆ Var (m(W )ξ (t)) dt.
Since W and ε are independent Lemma 1 implies:

V 1 ( ) = ∆ E ε 2 E ξ 2 (t) dt ≤ µ 2 2 b ,1 h .
Now recall some usual conventions. First we assume that 0! = 1 and, following Di Nunno et al. ( 2009), we define f 0 = E(Y ) while I 0 (•)(W ) stands for the identity application. Using these conventions we have:

V 2 ( ) = ∆ E L k=1 1 k! Θ k ξ (t) 2 dt = L k,k =1 1 k!k ! ∆ E Θ k Θ k ξ 2 (t) dt.
Using Cauchy-Schwarz inequality we obtain:

V 2 ( ) ≤ L k,k =1 (EΘ 4 k ) 1/4 (EΘ 4 k ) 1/4 k!k ! ∆ Eξ 4 (t) 1/2
dt. Now, using Lemma 1 and denoting by convention c 0 (q) = 1 for any q > 0, we obtain:

V 2 ( ) ≤ M 2   ( !)c 2 (4) L k,k =1 c k (4)c k (4)   k 2 ∆ 1 h - Now, define v (k, L) =   ( !)c 2 (4) L k,k =1 c k (4)c k (4) + σ 2   k 2 ∆ 1
Taking the bounds on V 1 ( ) and V 2 ( ) together we obtain

V ( ) 2 ≤ v (k, L)M 2 nh . ( 17
)
Control of the minimax risk Taking together ( 16) and ( 17) and the definition of h , we obtain:

E f ,h -f 2 ∆ ≤ b 2 (k, s) Λ 2 h 2s + v (k, L)M 2 nh ≤ κ (k, s, L)Λ 2 /(2s + ) M 2 n s /(2s + )
where κ (k, s, L) = b 2 (k, s)+v (k, L) . Combining the above result with (12), we obtain the following bound on the risk:

R p ( mh , m) ≤ κ 0 √ n + L =1 c (p)κ (k, s, L) ! Λ 2 /(2s + ) M 2 n s /(2s + ) ≤ κ * max    Λ 2 /(2s + ) M 2 n s /(2s + ) | = 1, . . . , L   
where κ * is a positive constant that depends only on L and s * . Note that last line is valid for n large enough. This ends the proof of the upper bound. Now, let us prove the lower bound.

Proof of the lower bound

Method. We fix s ∈ (0, s * ) L , Λ ∈ (0, +∞) L and M > 0. To prove the lower bound over the space M(s, Λ, M ), we define

= arg max k=1,...,L Λ k/(2s k +k) k M 2 n s k /(2s k +k)
and we follow the strategy developed by [START_REF] Cadre | Minimax regression estimation for Poisson coprocess[END_REF]. In particular Lemma 6.1 of this paper implies (using Itô's isometry combined with Theorem 2.5 in Tsybakov ( 2009)) that the problem boils down to find a finite family of functions {g ω } ω∈W with cardinal |W| ≥ 2 that satisfies the following assumptions:

(i) the null function 0 ∈ {g ω } ω∈W .

(ii) for any ω ∈ W, the function

g ω ∈ H (s , Λ , M ) (iii) there exists κ * > 0 such that for ω = ω , g ω -g ω ∆ ≥ 2κ * φ n (s, Λ, M ) (iv) there exists 0 < α < 1/8 such that n |W| ω∈W g ω 2 ∆ ≤ 2α log(|W|).
Under these assumptions, the lower-bound (10) holds.

Notation. Now we consider 0 < α < 1/8,

κ 2 * = 1 32 ψ 2 [-1,1] 2 L c 2 1 and c 1 = min 1, (2Lλ * ) -1 , α log(2)(8 • 2 L M 2 ) -1 1/2 .
Here, we construct a finite set of functions used in the rest of the proof. We consider the function ψ : R → R defined, for any u ∈ R by

ψ(u) = exp(-1/(1 -u 2 ))I (-1,1) (u).
Note that, since the function ψ is infinitely differentiable with compact support, we have:

λ * = max 0<s<s * sup x =y ψ ( s ) (x) -ψ ( s ) (y) |x -y| s < +∞.
We consider the bandwidth

h = M 2 Λ 2 n 1 2s +
and we set R = 1/(2h). We assume, without loss of generality, that R is an integer and nh ≥ 1. Let R = {0, . . . , R -1} and define, for any r = (r 1 , . . . , r ) ∈ R, the function φ r : ∆ → R by:

φ r (y) = i=1 ψ   y i -x (r) i h   .
where x (r) i = (2r i + 1)h. Finally, for any w : R → {0, 1} we define:

g w = ρ n r∈R w(r)φ r where ρ n = c 1 Λ h s = c 1 M √ nh .
Proof of (ii). Set w : R → {0, 1}. The following property can be readily verified:

g w 2 ∆ = |w| ψ 2 [-1,1] ρ 2 n h where |w| = r∈R w(r) ≤ R = 1 2 h .
This implies that

g w 2 ∆ ≤ ψ 2 [-1,1] 2 ρ 2 n ≤ c 2 1 M 2 nh ≤ !M 2 . ( 18
)
Moreover note that, for any y ∈ ∆ and α = (α 1 , . . . , α ) such that |α| = s , we have:

D α φ r (y) = 1 h |α| i=1 ψ (α i ) y i -x r i h
which implies that, for any z ∈ ∆ we have

|D α φ r (y) -D α φ r (z)| ≤ ψ -1 ∞ h |α| i=1 ψ (α i ) y i -x r i h -ψ (α i ) z i -x r i h ≤ λ * h s i=1 |y i -z i | s -s ≤ λ * h s |y -z| s -s
This also implies, since the function ψ vanishes outside (-1, 1), that

|D α g w (y) -D α g w (z)| ≤ (2 λ * ) ρ n h s |y -z| s -s ≤ c 1 (2 λ * ) Λ |y -z| s -s ≤ Λ |y -z| s -s . ( 19
)
Using ( 18) and ( 19), we deduce that g w belongs to H (s , Λ , M ) and then (ii) is fulfilled.

Proof of (i) and (iii). Using Lemma 2.9 of Tsybakov ( 2009), there exists a set W ⊂ {w : R → {0, 1}} such that the null function belongs to W, log 2 |W| ≥ R /8 and

∀w = w ∈ W, r∈R |w(r) -w (r)| ≥ R /8.
Let w, w ∈ W such that w = w . We have

g w -g w 2 ∆ =ρ 2 n r∈R (w(r) -w (r)) 2 φ r 2 =ρ 2 n r∈R |w(r) -w (r)|h ψ 2 ≥ρ 2 n h ψ 2 R /8 ≥ 1 8 ψ 2 [-1,1] 2 L c 2 1 Λ 2 h 2s ≥4κ 2 * φ 2 n (s, Λ, M ).
Then Assumptions (i) and (iii) are fulfilled.

Proof of (iv). Using (18), we deduce that using the definition of c

1 n |W| ω∈W g ω 2 ∆ ≤c 2 1 M 2 h - ≤ α 8 (2h) -log(2) ≤α log |W|.
Then Assumption (iv) is fulfilled.

Proof of Theorem 2

We have using ( 12) and ( 15) that

E| m(W ) -m(W )| 2 1/2 ≤ κ 0 n -1/2 + L =1 1 √ ! E f ( ) ĥ -f 2 ∆ 1/2 . ( 20
)
Let ∈ {1, . . . , L}. Let h ∈ H . We have

f ( ) ĥ -f ∆ ≤ f ( ) h -f ∆ + f ( ) ĥ ∨h - f ( ) h ∆ + f ( ) ĥ - f ( ) ĥ ∨h ∆ ≤ f ( ) h -f ∆ + B( , h) + M ( , ĥ ) + B( , ĥ ) + M ( , h) + 2M ( , ĥ ∨ h) ≤ f ( ) h -f ∆ + 4(B( , h) + M ( , h)).
Then we have

E f ( ) ĥ -f 2 ∆ 1/2 ≤ E f ( ) h -f 2 ∆ 1/2 + 4M ( , h) + 4 EB 2 ( , h) 1/2 . ( 21
)
Note that using (17), we have

E f ( ) h -f 2 ∆ 1/2 ≤ C E f ( ) h -f ∆ + M ( , h) . ( 22
)
In the following, we will demonstrate that

EB 2 ( , h) 1/2 ≤ C max h ≤h E f ( ) h -f ∆ + M ( , h) + O log n n . (23)
Combining ( 21) with ( 22) and ( 23), we obtain that

E f ( ) ĥ -f 2 ∆ 1/2 ≤ C max h ≤h E f ( ) h -f ∆ + M ( , h) + O log n n .
Theorem 2 is then a direct consequence of the above inequality and (20).

Proof of (23) Now let us control B( , h) for h ∈ H . We have

B( , h) = max h ≤h f ( ) h - f ( ) h∨h -M ( , h ) -M ( , h ∨ h ) + ≤ max h ≤h { f ( ) h -E f ( ) h -M ( , h )} + + { f ( ) h -E f ( ) h -M ( , h)} + + 2 max h ≤h E f ( ) h -f . Then EB 2 ( , h) 1/2 ≤ 2 max h ≤h E f ( ) h -f + 2 #(H ) max h ≤h A (h ) 1/2 (24)
where

A (h ) = E f ( ) h -E f ( ) h 2 -M ( , h ) 2 + . We have f ( ) h -E f ( ) h = L k=1 U k, + U
where

k!U k, (t) = 1 n n i=1 Θ i,k ξ i, (t, h ) -E (Θ i,k ξ i, (t, h ))
and

U (t) = 1 n n i=1 ε i ξ i, (t, h ).
Using these notations we have:

A (h ) ≤ 2 L+1 L k=0 E { U k, 2 -T (k, )} 2 + + E { U 2 -T ( )} 2 + ,
where

T ( ) = µ 4 1 + 4 log(1/(h ) ) n(h ) and T (k, ) = c 4 (k)M k! 1 + 4 log(1/(h ) ) n(h ) . Now note that Eε 4 i 1/4 = µ 4 < ∞ and for k ∈ {0, . . . , L} EΘ 4 i,k 1/4 ≤ c 4 (k)M k!. Using Lemma 2 with U = U (respectively U = k!U k, ), T = T ( ) (respectively T = T (k, )) and χ i = ε i (respectively χ = Θ i,k ), we deduce that for all for h ≤ h A (h ) ≤ Cn -1 . This implies that #(H ) max h ≤h A (h ) ≤ C(log n)n -1 . ( 25
)
Now ( 24) and ( 25) entail (23).

Proof of Theorem 3.

Let

s = (s 1 , . . . , s L ) ∈ [τ, s * ) L , Λ ∈ (0, +∞) L , M > 0 and f ∈ M s,Λ,M . Define for ∈ {1, . . . , L} k = 1 2s + log n log n .
For n large enough, we have h = e -k ∈ H . Using ( 16) and ( 7), Theorem 2 implies that

R 2 ( m, m) ≤ C L =1 max h ≤h (h ) s + M ( , h ) + Υ 2 log n n 1/2 ≤ C L =1    h s + log n nh 1/2    + Υ 2 log n n 1/2 ≤ C φn (s) + Υ 2 log n n 1/2 ≤ C φn (s).
where C may depend on s, Λ and M . Since C does not depend on m ∈ M(s, Λ, M ), this ends the proof.

Proof of Lemma 1.

We have

E|ξ (t, h)| 2r 1/r ≤ c 2 (2r)E|ξ (t, h)| 2 ≤ c 2 (2r) ! ∆ K ( ) h (t, u) 2 du ≤ c 2 (2r) !2 k 2 [0,1] h - ≤ b ,r h -. Moreover we have E |ξ (t, h)| r I |ξ (t,h)|>ϕ ≤ E|ξ (t, h)| 2r 1/2 (P (|ξ (t, h)| > ϕ)) 1/2 ≤ (b ,r h -) r/2 E|ξ (t, h)| 2q ϕ 2q 1/2 ≤ (b ,r ) r/2 (b ,q ) q/2 h -(r+q)/2 ϕ -q .

Proof of Lemma 2

Since h is fixed, we simplify the notation and use in the proof ξ i, (t, h) = ξ i, (t) and ξ (t, h) = ξ (t). Now, we have for k ≥ 1:

U = η0 (t) + η 1 (t) + η 2 (t) + η 3 (t)
where

η0 (t) = η 0 (t) -Eη 0 (t) η 0 (t) = 1 n n i=1 χ i I |χ i |≤ϕ(n) ξ i, (t)I |ξ i, |≤ψ(n) η 1 (t) = Eη 0 (t) -E(χξ (t)) η 2 (t) = 1 n n i=1 χ i I |χ i |>ϕ(n) ξ i, (t) η 3 (t) = 1 n n i=1 χ i I |χ i |≤ϕ(n) ξ i, (t)I |ξ i, (t)|>ψ(n) .
where for any = 1, . . . , L:

ψ(n) = n α with α = α(τ, ) = τ + 2τ + 4 and ϕ(n) = n β with β = 1/2 -α > 0.
Note that both α and β are positive numbers.

Control of η 1

We have using Cauchy-Schwarz, Markov inequality

|η 1 (t)| = -E χξ (t)I |χ|>ϕ(n) -E χξ (t)I |χ|≤ϕ(n) I |ξ (t)|>ψ(n) ≤ E |χξ (t)|I |χ|>ϕ(n) + E |χξ (t)|I |χ|≤ϕ(n) I |ξ (t)|>ψ(n) (26)
Note that we have

E |χξ (t)|I |χ|>ϕ(n) ≤ E |χ| 2 I |χ|>ϕ(n) E|ξ (t)| 2 1/2 ≤ b ,1 h -a 2 (P (|χ| > ϕ(n))) 1/4 ≤ b ,1 a 2 (E|χ|) 8/β ) 1/4 ≤ b ,1 a 2 (a 4/β ) 2/β h -(ϕ(n)) -2/β ≤ Cn -1 . ( 27
)
Moreover since h ∈ H , using Lemma 1 with q = 8τ +6 τ , we have

E |χξ (t)|I |χ|≤ϕ(n) I |ξ (t)|>ψ(n) 2 ≤a 2 1 E|ξ (t)| 2 I |ξ (t)|>ψ(n) ≤ (b ,2 )(b ,q ) q/2 (ψ(n)) -q h -(2+q)/2 ≤ Cn -2 . ( 28
)
Now using ( 26), ( 27) and ( 28), we finally obtain

E η 1 2 ∆ ≤ Cn -1 . 5.5.2 Control of η 2 Define µ n (t) = E χI |χ|>ϕ(n) ξ (t) .
We have

E η 2 2 ∆ ≤ 2(A + B)
where using Lemma 1 with q = 4/β

A = 1 n ∆ E χI |χ|>ϕ(n) ξ (t) -µ n (t) 2 dt ≤ 1 n ∆ E χ 4 I |χ|>ϕ(n) Eξ 4 (t)
1/2 dt ≤ n -1 a 2 4 P (|χ| > ϕ(n)) 1/4 b ,2 h - ≤ a 2 4 b ,2 P (|χ| > ϕ(n)) 1/4 ≤ a 2 4 b ,2 (a q/2 ) q/4 (ϕ(n)) -q/4 ≤ Cn -1 and following ( 27)

B = 1 n ∆ µ 2 n (t)dt ≤ Cn -1 .

Control of η 3

Define ν n (t) = E χI |χ|≤ϕ(n) ξ (t)I |ξ (t)|>ψ(n) .

We have

E η 3 2 ∆ ≤ 2(A + B)
where

A = 1 n ∆ E χI |χ|≤ϕ(n) ξ (t)I |ξ (t)|>ψ(n) -ν n (t) 2 dt and B = 1 n ∆ ν 2 n (t)dt
Note that using similar arguments as above with q = (8τ + 4 )/τ ,

A ≤ 1 n ∆ E χ 4 E ξ 4 (t)I |ξ (t)|>ψ(n) 1/2
dt ≤ a 2 2 b ,4 (b ,q ) q/4 (ψ(n)) -q/2 h -q/4 ≤ Cn -1 and following (28) B ≤ Cn -1 .

Control of η0

We have to bound

E { η0 2 -T } 2 + ≤ +∞ 0 P η0 2 -T > √ u du ≤ 2 +∞ 0 uP ( η0 2 > u + T ) du.
Note that, using duality arguments, there exists a countable set S of functions s ∈ L 2 (∆ ) such that s 2 ≤ 1 and η0 2 = sup

X i,s = E n i=1 X i (•) ∆ ≤   E n i=1 X i (•) 2 ∆   1/2 ≤ 1 4 ∆ Eχ 2 ξ 2 (t)dt 1/2 ≤ a 2 b ,2 4h /2
Combining the previous results we have:

v ≤ θ n + θ n with θ n = a 2 2 b ,2 4h . Define T = (1 + δ) a 2 b ,2 2 √ nh = (1 + δ) θ n n .
We have:

P ( η0 2 > u + T ) ≤ P Z -EZ > √ nu 2 + δ 2 θ n .
Define:

a = δ √ nθ n 2 , b = δ 2 θ n /4, c = √ n 3 , d = 2θ n + 2 θ n (1 + δ/6).
Using Bousquet's inequality we have:

P Z -EZ > √ nu 2 + δ 2 θ n ≤ C n (u)D n (u)
where This implies that:

C
E { η0 2 -T } 2 + ≤ Cn -1 .
Combining results of Sections 5.5. 1, 5.5.2, 5.5.3 and 5.5.4, we obtain (11)

  n (u) = expnu 2 4(cu + d) and D n (u) = expau + b cu + d . (29)Since adbc > 0, we have, D n (u) ≤ D n (0), that is:D n (u) ≤ expb d ≤ expδ 2 θ n 4(2θ n + 2 √ θ n (1 + δ/6)) .Since for n large enough (1 + δ/6) ≤ √ θ n we have D n (u) ≤ Ch . Moreover we have doing the change of variables v =

  Note that we have both E(X i,s ) = 0 and X i,s ∞ ≤ 1. Now, let us control:

	n
	s∈S ∆ 2ϕ(n)ψ(n) s(t)η 0 (t)dt n Z = 2Z √ n Z = sup = s∈S n i=1 X i,s s(t)X i (t)dt and, for s ∈ S, we have: where X i,s = ∆ with X i (t) = g i (t) -Eg i (t) and g i (t) = 1 2 √ n s∈S i=1 X i,s + n sup s∈S EX 2 1,s . Using Cauchy-schwarz's inequality and Fubini's theorem we obtain: EX 2 1,s ≤ ∆ s 2 (t)dt ∆ EX 2 i (t)dt ≤ ∆ Eg 2 i (t)dt We have ∆ Eg 2 i (t)dt ≤ a 2 2 b ,2 4nh and E sup n χ v = 2E sup s∈S i=1

i I {|χ i |≤ϕ(n)} ξ i, (t)I {|ξ i, (t)|≤ψ(n)} .

Acknowledgements

The authors have been supported by Fondecyt projects 1171335 and 1190801, and Mathamsud 18-MATH-07.