O Akdogan

N G Akdogan

SmCo-based MFM Probes with High Switching Fields

niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction:

Magnetic force microscope (MFM) is a standard tool for characterizing sub-micrometer magnetic domains [START_REF] Sáenz | Observation of magnetic forces by the atomic force microscope[END_REF][START_REF] Kazakova | Frontiers of magnetic force microscopy[END_REF]. Uncontrolled switching of magnetization at the tip of an MFM probe due to the stray field from the sample under observation and/or an external magnetic field is an issue to be resolved since it may produce unreliable and misleading data. This is especially important when characterizing the domain structure of hard magnetic materials either in their remanent state [START_REF] Ciuta | Some aspects of magnetic force microscopy of hard magnetic films[END_REF] or under an externally applied magnetic field. [START_REF] Mohanty | Variable magnetic field and temperature magnetic force microscopy[END_REF][START_REF] Thielsch | In situ magnetic force microscope studies of magnetization reversal of interaction domains in hot deformed Nd-Fe-B magnets[END_REF] Among commercially available MFM probes, so-called "high coercivity" probes have switching field values of around 0.5 T (e.g. Asylum Research -ASYMFMHC-R2). Several research articles have reported efforts to develop high switching field probes. FePt coated carbon nano-cone probes with switching fields comparable to commercial probes (~0.5 T) were attached to tip-less cantilevers [START_REF] Chen | Iron-platinum-coated carbon nanocone probes on tipless cantilevers for high resolution magnetic force imaging[END_REF]. Switching fields of over 0.5 T has been achieved by coating AFM tips with CoPt layers followed by annealing at 700 o C for 4 h (higher switching fields of around 1 T were achieved when such films were deposited on flat substrates). [START_REF] Liou | Development of high coercivity magnetic force microscopy tips[END_REF] In 2008 [START_REF] Amos | Controlling multidomain states to enable sub-10-nm magnetic force microscopy[END_REF] Amos et al. demonstrated enhanced spatial resolution in standard Si AFM probes coated with an FePt (10 nm) / MgO (8nm) / FePt (10 nm) trilayers, which had coercivities of the order of 1 T. The enhanced spatial resolution was attributed to antiparallel coupling of the two hard magnetic layers. The same group reported similar values of coercivity for FePt based probes made by a rather elaborate procedure involving focused ion beam etching of a Si AFM probe to produce a plateau, followed by deposition of the magnetic layer on the plateau and subsequent FIB etching to produce a sharp tip [START_REF] Amos | Highresolution and high-coercivity FePtL10 magnetic force microscopy nanoprobes to study next-generation magnetic recording media[END_REF]. In 2010, Amos et al. slightly enhanced the coercivity of FePt films deposited on Si probes, to between 1.1 and 1.2 T, by using CrRu and MgO seed layers (CrRu/MgO/FePt) and by optimizing the deposition conditions (temperature and pressure). [START_REF] Amos | Ultrahigh coercivity magnetic force microscopy probes to analyze high-moment magnetic structures and devices[END_REF] They also demonstrated the interest for these probes by imaging perpendicular magnetic recording write head under various excitation conditions. In 2018, Neu et al. used the focus ion beam method to cut triangle-shaped tips out of epitaxial SmCo5 films with a coercivity of 2.5 T, and then attached them to force cantilevers with the aid of a nanomanipulation tool, for MFM imaging. [START_REF] Neu | Epitaxial hard magnetic SmCo5 MFM tips-a new approach to advanced magnetic force microscopy imaging[END_REF] However, the apparent switching field of the tip was observed to be sample dependent, with a value of above 2 T reported when imaging a bulk Sm2Co17 magnet but a reduced switching field value of 0.7 T when a Co/Pt multilayer thin film was imaged. [START_REF] Neu | Epitaxial hard magnetic SmCo5 MFM tips-a new approach to advanced magnetic force microscopy imaging[END_REF] In this study, we exploit our recent development of high coercivity (µ0HC=6.8 T) polycrystalline SmCo films [START_REF] Akdogan | Preparation, Characterization, and Modeling of Ultrahigh Coercivity Sm-Co Thin Films[END_REF][START_REF] Akdogan | Rapid production of highly coercive Sm-Co thin films by triode sputtering[END_REF] to produce ultra-high switching field MFM tips using commercial AFM probes.

Experimental:

Triode sputtering was used to deposit Cr (10 nm) / Sm-Co (20 nm -40 nm) / Cr (10 nm) [START_REF] Akdogan | Preparation, Characterization, and Modeling of Ultrahigh Coercivity Sm-Co Thin Films[END_REF][START_REF] Akdogan | Rapid production of highly coercive Sm-Co thin films by triode sputtering[END_REF] trilayers onto commercial Si AFM probes. The AFM probes are purchased from Olympus; Al coated OMCL-AC200T and bare OMCL-AC200TN. The base pressure in the sputtering chamber was 2x10 -7 mbar and high purity Ar was used to produce the plasma, at a pressure of 1.4x10 -3 mbar. A DC voltage of -900 V was applied to a target of total surface area 9 cm × 9 cm, glued to a Cu base plate. The hard magnetic layers of the probes were deposited from a base alloy target of nominal composition SmCo5 with a 2 cm × 2 cm Sm foil bonded onto the surface of the base target. Cr buffer and capping layers were deposited to prevent interdiffusion with the substrate and oxidation, respectively. Note that though no power was supplied to the substrate holder heater during deposition, the probe temperature rose to a value of about 260°C, due to heating from the plasma, and the transfer of kinetic energy from impinging species. Probes were postdeposition annealed at 550 °C for times in the range 2.5 -30 minutes in a tube furnace under a vacuum of 10 -8 mbar. The AFM probes were mounted on a specially designed sample holder which could be attached to the substrate holder of the sputtering system.

Test samples used to assess the probes were prepared by thermomagnetic patterning (TMP) of an out-of-plane textured NdFeB thick film. NdFeB film is chosen due to the high stray field values.

[3] In the TMP process, a previously magnetized NdFeB film is subjected to reverse applied magnetic field while shining a high-energy pulsed laser onto the film surface through a mask with a given pattern. The magnetization in parts exposed to the laser are reversed due to a drop in coercivity of the NdFeB film with increasing temperature [START_REF] Fujiwara | Characterization of the magnetic properties of NdFeB thick films exposed to elevated temperatures[END_REF], thus the pattern is decorated on the hard magnetic surface. Details of the synthesis of NdFeB thick films and thermomagnetic patterning have been reported elsewhere. [START_REF] Dempsey | High performance hard magnetic NdFeB thick films for integration into micro-electromechanical systems[END_REF][START_REF] Dumas-Bouchiat | Thermomagnetically patterned micromagnets[END_REF] Such test structures were previously used to study specific aspects of MFM imaging of hard magnetic films [START_REF] Ciuta | Some aspects of magnetic force microscopy of hard magnetic films[END_REF] and also to quantitatively study magnetic interactions between a micro-magnet array and individual magnetic microparticles. [START_REF] Ponomareva | A quantitative study of magnetic interactions between a micromagnet array and individual magnetic micro-particles by scanning particle force microscopy[END_REF] The test structure used here consisted of stripes of reversed and non-reversed zones of an out-of-plane magnetized NdFeB film, with stripe widths 3 µm and 7 µm, respectively. A schematic of the TMP process is shown in Figure 1. In a typical test, the following protocol has been applied: a maximum field of +8 T was applied perpendicular to the cantilever of the MFM probe under vacuum in a superconducting coil and then the probe was used to produce an MFM image of the test sample. Following this, the probe was remounted in the superconducting coil, a reverse magnetic field was applied to the MFM probe and then the magnetic structure was rescanned in the MFM. This step was repeated using reverse fields of increasing intensity (at intervals of 1T), and flipping of the MFM contrast associated with the reversed and non-reversed zones of the test sample was observed. The area scanned changed from one measurement to the next by no more than 10µm. MFM imaging was performed with a Veeco NSV microscope. Scan parameters are given in table 1.

Lift

Results and Discussions:

Initially, Cr 10 nm/SmCo 20 nm/Cr 10 nm trilayers were deposited on commercial aluminumcoated Si AFM tips, which were then annealed at 550 o C for 30 min. The Al layer on such probes serves as a reflective layer used to follow the cantilever displacement. Deposition of the trilayer and subsequent annealing did not lead to bending of the cantilever (Fig. Secondly, there is fine contrast within these written patterns, on a scale of a few hundred nanometers, which results from Nd2Fe14B grains in the TMP sample which are misoriented with a respect to the out-of-plane direction. [START_REF] Ponomareva | A quantitative study of magnetic interactions between a micromagnet array and individual magnetic micro-particles by scanning particle force microscopy[END_REF] The overall contrast of the coarse patterns flips between application of a reverse magnetic field on the probe of -4T and -5T, while the contrast of some of the finer features is blurry at fields as low as -1T. This behavior reflects a distribution in switching field values in the SmCo film deposited on the MFM probe. It may also reflect a combination of contributions from very short-range interactions involving SmCo grains of presumably lower coercivity at the tip of the probe and longer-range interactions involving SmCo grains of presumably higher coercivity on the rest of the probe. Blurring of the contrast of the coarse features may be due to a combination of the flipping of lower coercivity grains at the tip of the probe and modifications of phase shift from one MFM to the next, due to slight changes in the tip properties (e.g. contamination with glue used to attach the MFM probe to the superconducting coil sample holder, which would disturb mechanical excitation from the microscope and so the probe oscillation response in phase and amplitude). Although the distribution in coercivities complicates interpretation of phase contrast for sub-micrometer domains, the probes developed here are nevertheless suitable for characterizing changes in high coercivity coarse hard magnetic patterns. MFM images made with a (Cr 10 nm/SmCo 40 nm/ Cr 10 nm) coated probe annealed for 2.5 min, reveals that the shorter annealing time leads to a switching field value of less than 1 T (Figure 6a). We attribute this to incomplete crystallization of the hard magnetic SmCo phase. The influence of longer annealing times was also studied. The RMS phase shift associated with MFM image scans of (Cr 10 nm/SmCo 40 nm/ Cr 10 nm) coated probes annealed for times in the range For comparison, topography and MFM images (Figure 7) has been taken with commercially available high coercivity MFM probe (FePt coated MFM probe (2N/m) ASYMFMHC-R2).

Contrast flip of the coarse patterns has been observed below -1T which agrees with the advertised coercivity value of 0.5 T.

Conclusions:

In conclusion, MFM probes having broad switching field distribution with a maximum value in between 4 T and 5 T have been made by coating bare Si AFM probes with a high coercivity SmCo layer, protected with Cr buffer and capping layers. The maximum switching field value is 10 times higher than commercial high coercivity MFM probes. Our fabrication approach is relatively simple and cheap, compared to other rather elaborate approaches reported in the literature. While the observed distribution in switching field values complicates interpretation of phase contrast for sub-micrometer domains, the probes developed here are suitable for characterizing changes in high coercivity coarse hard magnetic patterns under the influence of an external magnetic field. Efforts will now be made to reduce the distribution in switching field values, so as to improve the spatial resolution of the probes.

Figure 1 .

 1 Figure 1. Schematic of the Thermo-Magnetic patterning (TMP) process used to prepare test

  2 a), however closer examination of the probe revealed that the Al coating had melted (Fig . 2. b, c), and presumably diffused into the Sm-Co layer.

Figure 2 .

 2 Figure 2. a) SEM image of a commercial aluminum coated AFM probe on which a Cr/SmCo/Cr trilayer was deposited, following annealing at 550°C for 30 min, b) and c) close up images of the tip of the probe, revealing that the Al layer had melted.

Figure 3 .

 3 Figure 3. SEM image of a bare Si AFM probe coated with a Cr 10 nm/SmCo 20 nm/ Cr 10 nm trilayer; a, before and b, after annealing at 550°C for 5 min.

Figure 4 . 5 )Figure 5 .

 455 Figure 4. MFM images of a test TMP structure made using a 10 nm Cr/ 40 nm Sm-Co/10 nm Cr coated tip annealed at 550 o C for 5 min. All images are made in the remanent state, following the ex-situ exposure of the tip to a magnetic field of a given direction and intensity.

5 -

 5 30 minutes, following saturation in a field of 8 T, are compared in Figure 6 b. The drop in phase shift for annealing times of 10 minutes and longer is indicative of a drop in the magnetic moment of the probes, possibly due to insufficient protection of the magnetic layer by the Cr buffer, leading to interdiffusion with the Si probe, and/or a change in the spring constant due to annealing.

Figure 6 .

 6 Figure 6. (a) MFM images of a test TMP structure made using a 10 nm Cr/ 40 nm Sm-Co/10 nm Cr coated tip annealed at 550 o C for 2.5 min. (b) RMS phase shift associated with MFM image scans of (Cr 10 nm/SmCo 40 nm/ Cr 10 nm) coated probes annealed (550 o C) for times in the range 5-30 minutes, following saturation in a field of 8 T.

Figure 7 .

 7 Figure 7. Topography and MFM images of the test film by using the commercial coercive FePt Probe magnetized under external field with certain magnetic field values.

Table 1 .

 1 Scan parameters.

		Amplitude	Scan rate	Lines	×	Height	Spring
				points		during first	Constant
						pass
	100 nm < z <	~ 50 nm	0.5 Hz	128 × 512		30 nm -35	9 N/m
	150nm					nm

Acknowledgement

The European Commission is acknowledged for the funding of this work under the Scheme: FP7-NMP-2012-SMALL-6: Small or medium-scale focused research projects. Work program topics addressed: NMP.2012.4.1-3 Development of advanced magnetic materials without, or with reduced use of, critical raw materials (Grant Agreement No. 309729). This work was also supported by TUBITAK (project: 118F319) and Campus France (PHC BOSPHORE project 42620TH)


Philippe Soulier
email: philippe.soulier@u-paris10.fr

Best attainable rates of convergence for the estimation of the memory parameter

The purpose of this note is to prove a lower bound for the estimation of the memory parameter of a stationary long memory process. The memory parameter is defined here as the index of regular variation of the spectral density at 0. The rates of convergence obtained in the literature assume second order regular variation of the spectral density at zero. In this note, we do not make this assumption, and show that the rates of convergence in this case can be extremely slow. We prove that the log-periodogram regression (GPH) estimator achieves the optimal rate of convergence for Gaussian long memory processes

Introduction

Let {Xn} be a weakly stationary process with autocovariance function γ. Its spectral density f , when it exists, is an even nonnegative measurable function such that

γ(k) = Z π -π f (x)e ikx dx .
Long memory of the weakly stationary process {Xt} means at least that the autocovariance function is not absolutely summable. This definition is too weak to be useful. It can be strengthen in several ways. We will assume here that the spectral density is regularly varying at zero with index -α ∈ (-1, 1), i.e. it can be expressed for x ≥ 0 as

f (x) = x -α L(x) ,
where the function L is slowly varying at zero, which means that for all t > 0,

lim x→0 L(tx) L(x) = 1 .
Then the autocovariance function is regularly varying at infinity with index α -1 and non absolutely summable for α > 0. The main statistical problem for long memory processes is the estimation of the memory parameter α. This problem has been exhaustively studied for the most familiar long memory models: the fractional Gaussian noise and the ARFIMA(p, d, q) process. The most popular estimators are the GPH estimator and the GSE estimator, first introduced respectively by Geweke and [START_REF] Geweke | The estimation and application of long memory time series models[END_REF] and [START_REF] Hans | Statistical aspects of self-similar processes[END_REF]. Rigorous theoretical results for these estimators were obtained by Robinson (1995a,b), under an assumption of second order regular variation at 0, which roughly means that there exists C, ρ > 0 such that

f (x) = Cx -α {1 + O(x ρ )} .
Under this assumption, [START_REF] Giraitis | Rate optimal semiparametric estimation of the memory parameter of the Gaussian time series with long range dependence[END_REF] proved that the optimal rate of convergence of an estimator based on a sample of size n is of order n 2ρ/(2ρ+1) . The methodology to prove these results is inspired from similar results in tail index estimation. If F is a probability distribution function on (-∞, ∞) which is second order regularly varying at infinity, i.e. such that F (x) = Cx -α {1 + O(x -αρ )} as x → ∞, then [START_REF] Hall | Best attainable rates of convergence for estimates of parameters of regular variation[END_REF] proved that the best attainable rate of convergence of an estimator of the tail index α based on n i.i.d. observations drawn from the distribution F is of order n 2ρ/(2ρ+1) . In this context, Drees (1998) first considered the case where the survival function F is regularly varying at infinity, but not necessarily second order regularly varying. He introduced very general classes of slowly varying functions for which optimal rates of convergence of estimators of the tail index can be computed. The main finding was that the rate of convergence can be extremely slow in such a case.

In the literature on estimating the memory parameter, the possibility that the spectral density is not second order regularly varying has not yet been considered. Since this has severe consequences on the estimations procedures, it seems that this problem should be investigated. In this note, we parallel the methodology developped by [START_REF] Drees | Optimal rates of convergence for estimates of the extreme value index[END_REF] to deal with such regularly varying functions. Not surprisingly, we find the same result, which show that the absence of second order regular variation of the spectral density has the same drastic consequences.

The rest of the paper is organised as follows. In Section 2, we define the classes of slowly varying functions that will be considered and prove a lower bound for the rate of convergence of the memory parameter. This rate is proved to be optimal in Section 3. An illustration of the practical difficulty to choose the bandwidth parameter is given in Section 4. Technical lemmas are deferred to Section 5.

Lower bound

In order to derive precise rates of convergence, it is necessary to restrict attention to the class of slowly varying functions referred to by [START_REF] Drees | Optimal rates of convergence for estimates of the extreme value index[END_REF] as normalised. This class is also referred to as the Zygmund class. Cf. (Bingham et al., 1989, Section 1.5.3) Definition 1. Let η * be a non decreasing function on [0, π], regularly varying at zero with index ρ ≥ 0 and such that limx→0 η * (x) = 0. Let SV (η * ) be the class of even measurable functions L defined on [-π, π] which can be for expressed x ≥ 0 as

L(x) = L(π) exp  - Z π x η(s) s ds ff ,
for some measurable function η such that |η| ≤ η * .

This representation implies that L has locally bounded variations and η(s) = sL ′ (s)/L(s). Usual slowly varying functions, such as power of logarithms, iterated logarithms are included in this class, and it easy to find the corresponding η function.

Examples are given below. We can now state our main result.

Theorem 1. Let η * be a non decreasing function on [0, π], regularly varying at 0 with index ρ ≥ 0 and such that limx→0 η * (x) = 0. Let tn be a sequence satisfying

lim n→∞ η * (tn)(ntn) 1/2 = 1 . (1) Then, if ρ > 0, lim inf n→∞ inf αn sup L∈SV (η * ) sup α∈(-1,1) Eα,L[η * (tn) -1 | αn -α|] > 0 , (2)
and if ρ = 0 lim inf n→∞ inf αn sup L∈SV (η * ) sup α∈(-1,1) Eα,L[η * (tn) -1 | αn -α|] ≥ 1 , (3)
where Pα,L denotes the distribution of any second order stationary process with spectral density x -α L(x) and the infimum inf αn is taken on all estimators of α based on n observations of the process.

Example 1. Define η * (s) = Cs β for some β > 0 and C > 0. Then any function

L ∈ SV (η * ) satisfies L(x) = L(0) + O(x β
), and we recover the case considered by [START_REF] Giraitis | Rate optimal semiparametric estimation of the memory parameter of the Gaussian time series with long range dependence[END_REF]. The lower bound for the rate of convergence is n β/(2β+1) .

Example 2. For ρ > 0, define η * (s) = ρ/ log(1/s), then exp

( Z 1/e x η * (s) s ds ) = exp {ρ log log(1/x)} = log ρ (1/x).
A suitable sequence tn must satisfy ρ 2 / log 2 (tn) ≈ ntn. One can for instance choose tn = log 2 (n)/(nρ 2 ), which yields η * (tn) = ρ/ log(n){1 + o(1)}. Note that η(s) = ρ/ log(s) belongs to SV (η * ), and the corresponding slowly varying function is log -ρ (1/x). Hence, the rate of convergence is not affected by the fact that the slowly varying function vanishes or is infinite at 0.

Example 3. The function L(x) = log log(1/x) is in the class SV (η * ) with η * (x) = {log(1/x) log log(1/x)} -1 . In that case, the optimal rate of convergence is log(n) log log(n).

Even though the slowly varying function affecting the spectral density at zero diverges very weakly, the rate of convergence of any estimator of the memory parameter is dramatically slow.

Proof of Theorem 1 Let ℓ > 0, tn be a sequence that satisfies the assumption of Theorem 1, and define αn = η * (ℓtn) and

ηn(s) =  0 if 0 ≤ s ≤ ℓtn, αn if ℓtn < s ≤ π, Ln(x) = π αn exp  - Z π x ηn(s) ds ff . Since η * is assumed non decreasing, it is clear that Ln ∈ SV (η * ). Define now f - n (x) = x -αn Ln(x) and f + n = (f - n ) -1 . f - n can be written as f - n (x) =  (ℓtn/x) αn if 0 < x ≤ ℓtn, 1 if ℓtn < x ≤ π.
Straighforward computations yield

Z π 0 {f - n (x) -f + n (x)} 2 dx = 8ℓtnα 2 n (1 + O(α 2 n )) = 8ℓn -1 (1 + o(1)). (4)
The last equality holds by definition of the sequence tn. Let P - n and P + n denote the distribution of a n-sample of a stationary Gaussian processes with spectral densities f - n et f + n respectively, E - n and E + n the expectation with respect to these probabilities,

dP + n dP - n
the likelihood ratio and An = {

dP + n dP - n
≥ τ } for some real τ ∈ (0, 1). Then, for any estimator αn, based on the observation (X1, . . . , Xn), 4) and (Giraitis et al., 1997, Lemma 2), we obtain that there exist constants C1 and C2 such that

sup α,L Eα,L[| αn -α|] ≥ 1 2 `E+ n [| αn -αn|] + E - n [| αn + αn|] ≥ 1 2 E - n » ½A n | αn + αn| + dP + n dP - n ½A n | αn -αn| - ≥ 1 2 E - n [{| αn + αn| + τ | αn -αn|}½A n ] ≥ τ αnP - n (An). Denote ǫ = log(1/τ ) and Λn = log(dP + n /dP - n ). Then P - n (An) = 1 -P - n (Λn ≤ -ǫ). Applying (
E - n [Λn] ≤ C1ℓ , E - n [(Λn -mn) 2 ] ≤ C2ℓ .
This yields, for any η > 0 and small enough ℓ,

P - n (An) ≥ 1 -ǫ -2 E[Λ 2 n ] ≥ 1 -Clǫ -2 ≥ 1 -η .
Thus, for any η, τ ∈ (0, 1), and sufficiently small ℓ, we have lim inf

n→∞ inf L∈SV (η * ) inf α∈(-1,1) Eα,L[η * (tn) -1 | αn -α|] ≥ τ (1 -η) lim n→∞ η * (ℓtn) η * (tn) = τ (1 -η)ℓ ρ .
This proves (2) and (3). ⊓ ⊔

Upper bound

In the case η * (x) = Cx ρ with ρ > 0, [START_REF] Giraitis | Rate optimal semiparametric estimation of the memory parameter of the Gaussian time series with long range dependence[END_REF] have shown that the lower bound (2) is attainable. The extension of their result to the case where η * is regularly varying with index ρ > 0 (for example to functions of the type x β log(x)) is straightforward. We will restrict our study to the case ρ = 0, and will show that the lower bound (3) is asymptotically sharp, i.e. there exist estimators that are rate optimal up to the exact constant. Define the discrete Fourier transform and the periodogram ordinates of a process X based on a sample X1, . . . , Xn, evaluated at the Fourier frequencies xj = 2jπ/n, j = 1, . . . , n, respectively by

dX,j = (2πn) -1/2 n X t=1
Xte -itx j , and IX,j = |dX,j| 2 .

The frequency domain estimates of the memory parameter α are based on the following heuristic approximation: the renormalised periodogram ordinate IX,j/f (xj), 1 ≤ j ≤ n/2 are approximately i.i.d. standard exponential random variables. Although this is not true, the methods and conclusion drawn from these heuristics can be rigourously justified. In particular, the Geweke and Porter-Hudak (GPH) and Gaussian semiparametric estimator have been respectively proposed by [START_REF] Geweke | The estimation and application of long memory time series models[END_REF] and [START_REF] Hans | Statistical aspects of self-similar processes[END_REF], and a theory for them was obtained by Robinson (1995b,a) in the case where the spectral density is second order regularly varying at 0.

The GPH estimator is based on an ordinary least square regression of log(I X,k ) on log(k) for k = 1, . . . , m, where m is a bandwith parameter:

( α(m), Ĉ) = arg min α,C m X k=1 {log(I X,k ) -C + α log(k)} 2 .
The GPH estimator has an explicit expression as a weighted sum of log-periodogram ordinates:

α(m) = -s -2 m m X k=1 ν m,k log(I X,k ), with ν m,k = log(k) -m -1 P m j=1 log(j) and s 2 m = P m k=1 ν 2 m,k = m{1 + o(1)}.
Theorem 2. Let η * be a non decreasing slowly varying function such that limx→0 η * (x) = 0. Let Eα,L denote the expectation with respect to the distribution of a Gaussian process with spectral density x -α L(x). Let tn be a sequence that satisfies (1) and let m be a non decreasing sequence of integers such that

lim n→∞ m 1/2 η * (tn) = ∞ and lim n→∞ η * (tn) η * (m/n) = 1 . ( 5
)
Assume also that the sequence m can be chosen in such a way that

lim n→∞ log(m) R π m/n s -1 η * (s) ds mη * (m/n) = 0 . (6)
Then, for any δ ∈ (0, 1), lim sup

n→∞ sup |α|≤δ sup L∈SV (η * ) η * (tn) -2 Eα,L[( α(m) -α) 2 ] ≤ 1. ( 7
)
Remark 1. Since η * is slowly varying, it is always possible to choose the sequence m in such a way that (5) holds. Condition (6) ensures that the bias of the estimator is of the right order. It is very easily checked and holds for all the examples of usual slowly varying function η * , but we have not been able to prove that it always holds.

Since the quadratic risk is greater than the L 1 risk, we obtain the following corollary.

Corollary 1. Let δ ∈ (0, 1) and η * be a non decreasing slowly varying function such that limx→0 η * (x) = 0 and such that it is possible to choose a sequence m that satisfies (6). Then, for tn as in (1),

lim inf n→∞ inf αn sup L∈SV (η * ) sup α∈(-δ,δ) Eα,L[η * (tn) -1 | αn -α|] = 1 . ( 8
)
Remark 2. This corollary means that the GPH estimator achieves the optimal rate of convergence, up to the exact constant over the class SV (η * ) when η * is slowly varying. This implies in particular that, contrarily to the second order regularly varying case, there is no loss of efficiency of the GPH estimator with respect to the GSE. This happens because in the slowly varying case, the bias term dominates the stochastic term if the bandwidth parameter m satisfies (5). This result is not completely devoid of practical importance, since when the rate of convergence of an estimator is logarithmic in the number of observations, constants do matter.

Example 4 (Example 2 continued). If L(x) = log ρ (1/x) L(x), where L ∈ SV (Cx β ) for some ρ >, β > 0 and C > 0, then 5), ( 6) and log(n)( α(m) -α) converges in probability to ρ.

P m k=1 ν m,k log(L(x k )) ∼ ρm log -1 (xm). Choosing m = log 1+δ (n) yields (

Proof of Theorem 2. Define

E k = log{x α k I k /L(x k )}.
The deviation of the GPH estimator can be split into a stochastic term and a bias term:

α(m) -α = -s -2 m m X k=1 ν m,k E k -s -2 m m X k=1 ν m,k log(L(x k )). (9)
Applying Lemma 2, we obtain the following bound:

E hn m X k=1 ν m,k log(E k ) o 2 i ≤ C(δ, η * ) m. ( 10
)
The bias term is dealt with by applying Lemma 1 wich yields

˛m X k=1 ν m,k log(L(x k )) ˛≤ mη * (xm){1 + o(1)}, (11)
uniformly with respect to |η| ≤ η * . Choosing m as in ( 5) yields (7). ⊓ ⊔

Bandwidth selection

In any semiparametric procedure, the main issue is the bandwidth selection, here the number m of Fourier frequencies used in the regression. Many methods for choosing m have been suggested, all assuming some kind of second order regular variation of the spectral density at 0. In Figures 123below, the difficulty of choosing m is illustrated, at least visually. In each case, the values of the GPH estimator are plotted against the bandwidth m, for values of m between 10 and 500 and sample size 1000.

In Figure 1 the data is a simulated Gaussian ARFIMA(0,d,0). The spectral density f of an ARFIMA(0,d,0) process is defined by f (x) = σ 2 |1e ix | -2d /(2π), where σ 2 is the innovation variance. Thus it is second order regularly varying at zero and satisfies f (x) = x -α (C + O(x 2 )) with α = 2d. The optimal choice of the bandwidth is of order n 4/5 and the semiparametric optimal rate of convergence is n 2/5 . Of course, it is a regular parametric model, so a √ n consistent estimator is possible if the model is known, but this is not the present framework. The data in Figure 2 comes from an ARFIMA(0,d,0) observed in additive Gaussian white noise with variance τ 2 . The spectral density of the observation is then

σ 2 2π |1 -e ix | -2d + τ 2 2π = σ 2 2π |1 -e ix | -2d  1 + τ 2 σ 2 |1 -e ix | 2d
ff .

It is thus second order regularly varying at 0 and the optimal rate of convergence is n 2d/(4d+1) , with optimal bandwidth choice of order n 4d/(4d+1) . In Figures 1 and2 the outer lines are the 95% confidence interval based on the central limit theorem for the GPH estimator of d = α/2. See Robinson (1995b).

A visual inspection of Figure 1 leaves little doubt that the true value of d is close to .4. In Figure 2, it is harder to see that the correct range for the bandwidth is somewhere betwen 50 and 100. As it appears here, the estimator is always negatively biased for large m, and this may lead to underestimating the value of d. Methods to correct this bias (when the correct model is known) have been proposed and investigated by [START_REF] Hurvich | The local whittle estimator of long-memory stochastic volatility[END_REF] and [START_REF] Hurvich | Estimating long memory in volatility[END_REF], but again this is not the framework considered here.

Finally, in Figure 3, the GPH estimator is computed for a Gaussian process with autocovariance function γ(k) = 1/(k + 1) and spectral density log |1e ix | 2 . The true value of α is zero, but the spectral density is infinite at zero and slowly varying. The plot d(m) is completely misleading. This picture is similar to what is called the Hill "horror plot" in tail index estimation. The confidence bounds are not drawn here because there are meaningless. See Example 4.

There has recently been a very important literature on methods to improve the rate of convergence and/or the bias of estimators of the long memory parameter, always under the assumption of second order regular variation. If this assumption fails, all these methods will be incorrect. It is not clear if it is possible to find a realistic method to choose the bandwidth m that would still be valid without second order regular variation. It might be of interest to investigate a test of second order regular variation of the spectral density.

Technical results

Lemma 1. Let η * be a non decreasing slowly varying function on (0, π] such that lims→0 η * (s) = 0. Let η be a measurable function on (0, π] such that |η| ≤ η * , and define h

(x) = - R π x η(s) s ds and h * (x) = R π x η * (s) s ds. Then, for any non decreasing sequence m ≤ n, ˛m X k=1 νm(k)h(x k ) ˛≤ mη * (xm) + O(log 2 (m)η * (xm) + log(m)h * (xm)), (12)
uniformly with respect to |η| ≤ η * . Proof. Since η is slowly varying, the function h is also slowly varying and satisfies limx→0 η(x)/h(x) = 0. Then,

m X k=1 h(x k ) = n 2π Z xm 0 h(s) ds + n 2π m X k=1 Z x k x k-1 {h(x k ) -h(s)} ds = n 2π Z xm 0 h(s) ds + n 2π m X k=1 Z x k x k-1 Z x k s η(t) t dt ds .
Thus, for |η| ≤ η * and η * increasing,

Fig. 3. GPH "horror" plot ˛m X k=1 h(x k ) - n 2π Z xm 0 h(s) ds ˛≤ η * (xm) n 2π m X k=1 Z x k x k-1 Z x k s dt t ds = η * (xm) ( m X k=1 log(x k ) - n 2π Z xm 0 log(s) ds ) = η * (xm) ( m X k=1 log(k) -m log(m) + m ) = O(η * (xm) log(m)) .
By definition, it holds that xh ′ (x) = η(x). Integration by parts yield

n 2π Z xm 0 h(s) ds = mh(xm) - n 2π Z xm 0 η(s) ds . (13)
Thus,

m X k=1 h(x k ) = mh(xm) - n 2π Z xm 0 η(s) ds + O(log(m)η * (xm)) . (14)
Similarly, we have:

m X k=1 h(x k ) log(x) = mh(xm) log(xm) - n 2π Z xm 0 {η(s) log(s) + h(s)} dx + O(log(m){log(xm)η * (xm) + h * (xm)}) . ( 15
)
By definition of νm(k), we have:

νm(k) = log(k) - 1 m m X j=1 log(j) = log(x k ) -log(xm) + 1 + O " log(m) m « .
Hence, applying ( 13), ( 14) and ( 15), we obtain This yields (12). ⊓ ⊔ Lemma 2. Let η * be a non decreasing slowly varying function such that limx→0 η * (x) = 0. Let X be a Gaussian process with spectral density f (x) = x -α L(x), where α ∈ [-δ, δ] and L ∈ SV (η * ). Let γ = 0, 577216... denote Euler's constant. Then, for all n and all k, j such that 0

m X k=1 νm(k)h(x k ) = m X k=1 log(x k )h(x k ) -{log(xm) -1 + O(log(m)/m)} m X k=1 h(x k ) = n 2π
< x k , xj ≤ π/2, |E[log(E k )] -γ| + ˛E[log 2 (E k )] - π 2 6 ˛≤ C(δ, η * ) log(1 + k)k -1 , |E[log(E k ) -γ)(log(Ej) -γ)]| ≤ C(δ, η * ) log 2 (j)k -2 .

Proof of Lemma 2

It is well known (see for instance [START_REF] Hurvich | The mean squared error of Geweke and Porter-Hudak's estimator of the memory parameter of a long-memory time series[END_REF], [START_REF] Moulines | Broadband log-periodogram regression of time series with long-range dependence[END_REF], [START_REF] Soulier | Moment bounds and central limit theorem for functions of Gaussian vectors[END_REF]) that the bounds of Lemma 2 are consequences of the covariance inequality for functions of Gaussian vectors of (Arcones, 1994, Lemma 1) and of the following bounds. For all n and all k, j such that 0

< |x k | ≤ |xj| ≤ π/2, |cov(d X,k , dX,j)| + |cov(d X,k , dX,j) -f (x k )δ k,j | ≤ C(δ, η * ) p f (x k )f (xj) log(j) k -1 .
Such bounds have been obained when the spectral density is second order regularly varying. We prove these bounds under our assumptions that do not imply second order regular varition. Denote Dn(x) = (2πn) -1/2 P n t=1 e -itx . Then

cov(d X,k , dX,j) = Z π -π f (x)Dn(x k -x)Dn(xj + x)dx.
Recall that by definition of the class SV (η * ), there exists a function η such that |η| ≤ η * and L(x) = L(π) exp{-R π x s -1 η(s)ds}. Since only ratio L(x)/L(π) are involved in the bounds, without loss of generality, we can assume that L(π) = 1. We first prove that for all k such that

x k ≤ π/2, ˛Z π -π " f (x) f (x k ) -1 « |Dn(x k -x)| 2 dx ˛≤ C(δ, η * ) log(k)k -1 . ( 16
)
Since L ∈ SV (η * ), the functions x ǫ L(x) and x ǫ L -1 (x) are bounded for any ǫ > 0 and

sup x∈[0,π] x δ (L(x) + L -1 (x)) ≤ C(η * , δ) , (17)
sup

α∈[-1+δ,1-δ] Z π -π f (x) dx ≤ C(η * , δ) . (18)
Since η * is increasing, for all 0 < x < y ≤ π/2, it holds that

|f (x) -f (y)| = |x -α L(x) -y -α L(y)| ≤ Z y x |α -η(s)|s -α-1 L(s) ds ≤ Z y x (1 + η * (π))s -α-1 L(s) ds . Since α ∈ [-1 + δ, 1 -δ], x -α-1 L(x) is decreasing. Hence |f (x) -f (y)| ≤ C(η * , δ)x -1 f (x)(y -x) . (19)
Define Fn(x) = |Dn(x)| 2 (the Fejer kernel). We have

sup π/2≤|x|≤π |Fn(x k -x)| = O(n -1 ) , Z π -π Fn(x) dx = 1 , (20)
Z π -π Dn(y + x)Dn(z -x) dx = (2πn) -1/2 Dn(y + z) . (21)
From now on, C will denote a generic constant which depends only on η * , δ and numerical constants, and whose value may change upon each appearance. Applying (17), ( 18) and (20), we obtain

Z π/2≤|x|≤π ˛f -1 (x k )f (x) -1 ˛Fn(x -x k ) dx ≤ Cn -1 (f -1 (x k ) + 1) ≤ Ck -1 . The integral over [-π/2, π/2] is split into integrals over [-π/2, -x k /2] ∪ [2x k , π/2], [-x k /2, x k /2] and [x k /2, 2x k ]. If x ∈ [-π/2, -x k /2] ∪ [2x k , π/2], then Fn(x -x k ) ≤ Cn -1 x -2
. Hence, applying Karamata's Theorem (cf. (Bingham et al., 1989, Theorem 1.5.8)), we obtain:

Z -x k /2 -π/2 + Z -x k /2 -π/2 f (x)Fn(x) dx ≤ Cn -1 Z π/2 x k /2 x -α-2 L(x) dx ≤ Cn -1 x -α-1 k L(x k ) ≤ Ck -1 f (x k ) , Z -x k /2 -π/2 + Z -x k /2 -π/2 Fn(x) dx ≤ Cn -1 Z ∞ x k /2 x -2 dx ≤ Cn -1 x -1 k ≤ Ck -1 . For x ∈ [-x k /2, x k /2], Fn(x k -x) ≤ n -1 x -2 k .
Thus, applying again Karamata's Theorem, we obtain:

Z x k /2 -x k /2 f (x)Fn(x -x k ) dx ≤ Cn -1 x -2 k Z x k /2 -x k /2 x -α L(x) dx ≤ Cn -1 x -2 k x -α+1 k L(x k ) ≤ Ck -1 f (x k ) , Z x k /2 -x k /2 Fn(x -x k )dx ≤ Cn -1 x -1 k ≤ Ck -1 .
Applying ( 19) and the bound R x k -x k /2 |x|Fn(x)| dx ≤ Cn -1 log(k), we obtain:

Z 2x k x k /2 |f (x) -f (x k )|Fn(x -x k ) dx ≤ Cx -α-1 k L(x k /2) Z 2x k x k /2 |x -x k |Fn(x -x k )| dx ≤ Cf (x k )k -1 log(k) .
This proves ( 16). We now prove that all k, j such that 0 where a slight modification of the argument is necessary. On this interval, it still holds that |E n,k,j (x)| ≤ n -1 x -2 k . Moreover, x δ L(x) can be assumed increasing on [0, x k /2], and we obtain:

< x k = |xj| ≤ π/2, ˛Z π -π " f (x) f (x k ) -1 « Dn(x k -x)Dn(xj -x) dx + ˛Z π -π " f (x) f (x k ) -1 « Dn(x k -x)
Z x j /2 -x k /2 x -α L(x) dx ≤ x δ k L(x k ) Z x j /2 -x k /2
x -α-δ dx ≤ Cx -α+1 k L(x k ) .

The rest of the argument remains unchanged.

• To obtain the bound in the case xj < 0 < x k , the interval [-π, π]

Fig. 1 .Fig. 2 .

 12 Fig. 1. GPH estimator for ARFIMA(0,.4,0)

  {log(xm)log(s)} ds + O(log(m){η * (xm) log(xm) + h * (xm)}) .Finally, since |η| ≤ η * and η * is non decreasing, we obtain mη * (xm).

  Dn(xj -x) dx ≤ C(δ, η * ) log(k ∨ |j|)(k ∧ |j|) -1 . (22) Define E n,k,j (x) := Dn(x k -x)Dn(xj -x). Since 0 ≤ x k , xj ≤ π/2, for π/2 ≤ |x| ≤ π, we have |E n,k,j (x)| ≤ Cn -1 . Hence, as above, Z π/2≤|x|≤π |f -1 (x k )f (x) -1| dx ≤ Cn -1 (x α k L -1 (x k ) + 1) ≤ Ck -1 .We first consider the case k < j and we split the integral over[-π/2, π/2] into integrals over [-π/2, -x k /2] ∪ [2xj , π/2], [-x k /2, x k /2], [x k /2, (x k + xj)/2], [(x k + xj)/2, 2xj],denoted respectively I1, I2, I3 and I4.• The bound for the integral over [-π/2, -x k /2] ∪ [2xj, π/2] is obtained as above (in the case k = j) since |E n,k,j | ≤ Cn -1 x -2 . Hence |I1| ≤ Ck -1 . • For x ∈ [-x k /2, x k /2], |E n,k,j (x)| ≤ Cn -1 x -2k , hence we get the same bound:|I2| ≤ Ck -1 . • To bound I3, we note that on the interval [x k /2, (x k + xj)/2], |E n,k,j (x)| ≤ Cn 1/2 (j -k)|Dn(x -x k )|,and n 1/2 |x -x k ||Dn(x -x k | is uniformly bounded. Hence, applying (19), we obtain|I3| ≤ C(j -k) -1 x -1 k xj ≤ Ck -1 .• The bound for I4 is obtained similarly: |I4| ≤ Ck -1 log(j).• To obtain the bound in the case xj< x k , the interval [-π, π] is split into [-π, -π/2]∪[π/2π] [-π/2, -x k /2]∪[2x k , π/2], [-x k /2,xj/2], [xj/2, (x k +xj)/2] and [(x k + xj )/2, 2x k ]. The arguments are the same except on the interval [-x k /2, xj/2]

  is split into [-π, -π/2] ∪ [π/2, π] [-π/2, 2xj] ∪ [2x k , π/2], [2xj , -x k /2], [-x k /2, x k /2] and [x k /2, 2x k ]and the same arguments are applied.⊓ ⊔