Leonardo Fl Órez-Valencia

Johan Montagnat
email: montagnat@creatis.insa-lyon.fr

Maciej Orkisz
email: orkisz@creatis.insa-lyon.fr

3D GRAPHICAL MODELS FOR VASCULAR-STENT POSE SIMULATION

Keywords: vascular pathology, stenosis, stent, 3D image, simulation, generalized cylinder, simplex mesh, deformable model

Stents are playing an increasing role in treatment of arterial stenoses and aneurysms. The goal of this work is to help the clinician in the pre-operative choice of the stent's length and diameter. This is done by "plunging" a model of the stent into real vascular 3D image. Two models are used. First, a simple geometrical model, composed of a set of circles or polygons stacked along the vessel's centerline, is used to simulate the introduction and the deployment of the stent. Second, a simplexmesh model with an adapted cylindrical constraint is used to represent the stent surface. Another axially constrained simplex-mesh deformable model is used to reconstruct the 3D vessel wall. We simulate the interaction between the vessel wall and the stent by imposing that the model of the vessel locally fits the shape of the deployed-stent model. Preliminary quantitative results of the vessel reconstruction accuracy are given.

INTRODUCTION

Arterial-wall diseases, namely stenosis and aneurysm, belong to main causes of death in western countries. Aneurysm is a local distension of the artery (fig. 1 left), its rupture can lead to hemorrhage and stroke. Stenosis is an arterial lumen stricture (fig. 1 right), due to atherosclerotic plaque, which can lead to hypoperfusion, ischemia and infarct of organs that the artery irrigates. An increasing number of patients with these pathologies undergo an implantation of an endovascular prosthesis or of a stent. Stents are tubular grids that are deployed within stenotic regions in order to push the vascular wall outwards and thus keep open passageway for blood flow. Endo-prostheses are similar to stents, but covered with a blood-proof tissue. Implanted within an aneurysm, an endo-prosthesis canalizes the blood flow and reduces pressure on the arterial wall, thus preventing rupture. When folded, stents and endo-prostheses are slim and can be inserted into the artery, using a catheter. Deployed by shape-memory effect or by an inflating balloon, they become shorter and should fit the diameter of the artery's healthy part. Appropriate pre-operative choice of stent (endo-prosthesis) dimensions is necessary to avoid loosening, thrombus formation, embolism and branching-vessel obstruction.

The goal of our work is to aid this choice by providing a simulation tool able to represent stent-deployment within the given patient's artery. For the sake of conciseness, "stent" will be used in the sequel to denote both stents and endo-prostheses.

Several recent studies aim at extracting patients' individual vascular 3D geometry and/or at simulating radiological vascular interventions. Virtual vascular project [START_REF] Abdoulaev | The virtual vascular project[END_REF] [START_REF] Zorcolo | A volumetric virtual environment for catheter insertion simulation[END_REF] concentrates on the simulation of insertion of surgical needles and of catheters. In MedIS-VR project virtual endoscopic viewing is used for preoperative aortic stent planning. The vessel centerline is first automatically extracted. A virtual stent is then placed along this path. Its length and its diameters are finally interactively adjusted [START_REF] Stern | Predictive medicine: Computational techniques in therapeutic decision-making[END_REF]. Another project, named Geodesic, brings together automatic and semi-automatic segmentation tools and combines them with mesh generation tools in order to build patient-specific vascular models [START_REF] Wilson | A software framework for creating patient specific geometric models from medical imaging data for simulation based medical planning of vascular surgery[END_REF]. These models are then used for surgical planning, e.g. simulation of revascularization, flow simulation, etc. A similar approach has been followed by the authors of simulation-based medical planning software AS-PIRE [START_REF] Taylor | Predictive medicine: Computational techniques in therapeutic decisionmaking[END_REF] [START_REF] Wang | Improving geometric model construction for blood flow modeling[END_REF]. In [START_REF] Cebral | Patientspecific simulation of carotid artery stenting using computational fluid dynamics[END_REF], the patient-specific vascular geometry is automatically segmented using a tubular deformable mesh model [START_REF] Yim | Vessel surface reconstruction with a tubular deformable model[END_REF]. The stent model is also a cylindrical mesh. Merging both meshes simulates artery stenting.

In our work, we follow a similar approach. However, instead of mesh merging, we attempt to simulate stent/vascular-wall interaction. The vascular lumen 3D image is first acquired using contrast-enhanced magnetic resonance angiography (MRA) technique [START_REF] Douek | Fast mr angiography of the aortoiliac arteries and arteries of the lower extremity: value of bolus-enhanced, whole-volume subtraction technique[END_REF]. This image is then segmented (fig. 2) using Maracas, a software developped in our laboratory [START_REF] Hernandez-Hoyos | Computer-assisted analysis of three-dimensional angiograms[END_REF]. The segmentation method is based on generalized-cylinder model and provides the axial shape of the vessel, planar contours orthogonal to the vessel centerline and estimated values of local vascular lumen radii [START_REF] Hernandez-Hoyos | A Deformable Vessel Model with Single Point Initialization for Segmentation, Quantification and Visualization of Blood Vessels in 3D MRA[END_REF]. Simulation of stent insertion and deployment is carried out using a simplified geomet-rical model "pulled on" the centerline. This model is described in section 2. A more realistic representation of the stent and of the vessel wall surface, as well as interaction between them, is then realized using simplex-mesh deformable model presented in section 3.

GEOMETRICAL MODEL

A generalized cylinder, as used in Maracas for vessels representation, i.e. a stacking of contours orthogonal to the centerline, can also be used to model stents. Let A p (l p ) and A v (l v ) be parametric 3D curves representing the centerlines of the stent and of the vessel respectively, with l p ∈ [0, 1] and l v ∈ [0, 1] the arc-length parameters. The discretized version of each centerline A(l) is a set of vertices {a i }. The discretization of A v (l v ) is not uniform, while A p (l p ) has to be uniformly discretized. The absolute length of the vascular segment L v provided by Maracas is greater than the maximum length L pM of the stent. Let furthermore C p (l p ) be a planar contour orthogonal to A p (l p ) and centered at the centerline point corresponding to the arc-length l p .

The initial model of the stent is constructed by placing predefined-shape contours C p (l p ), cirular or polygonal, equally spaced along a straight axis. It is assumed that the stent does not modify the vessel's axial shape. This implicitly means that the stent is much more flexible than the vessel wall. Fitting the stent to the vessel axial shape can then be seen as a process of sliding the predefined shape C p (l p ) along the centerline A v (l v ), while keeping the contour plane orthogonal to this line (fig. 3). This is obtained by mapping the axis A p (l p ) of the initial straight model of the stent onto the vessel centerline A v (l v ). The stent model is placed between l v0 and an end point at l vM , with l v0 a user-defined delivery point and l vM an end point automatically deduced (for a given radius) from the length/radius relation that characterizes the stent-expansion process [START_REF] Dumoulin | Mechanical behaviour modelling of balloonexpandable stents[END_REF].

This geometrical model is very useful to simulate the stent displacement along the vessel centerline and to interact with the user in order to choose the delivery region and the stent parameters. Once deployed, this model provides the exact length and location of the stent. However, the surface thus constructed is not regular. Contours orthogonal to the centerline may intersect each other, where the centerline is curved (fig. 4 middle). The surface is difficult to represent and appears locally folded. Furthermore, this model is not well-suited for the simulation of stent/vessel-wall interaction. Another model, based on a simplex mesh, is therefore used to simulate this interaction and to display the final result. This model is constrained by the centerline segment

A v (l v ), l v ∈ [l v0 , l vM ].

SIMPLEX MODEL

Simplex meshes are discrete representations of surfaces suited to deformation [START_REF] Delingette | General Object Reconstruction based on Simplex Meshes[END_REF]. In 2simplex meshes used to represent surfaces, each vertex has exactly three neighbors. 2-simplex meshes are topologically dual to triangulations, thus making conversions back and forth easy. In this work, we propose a specific constraint of the simplex meshes deformation framework [START_REF] Montagnat | Globally constrained deformable models for 3D object reconstruction[END_REF] adapted to cylindrical shape deformation [START_REF] Montagnat | A Hybrid Framework for Surface Registration and Deformable Models[END_REF].

Model deformation

A simplex surface is represented by a set of vertices {v i }. For deformation purposes, each vertex undergoes a deformation controlled by an internal (regularizing) energy term and an external (data-driven) term. Discretization of the energy minimization framework leads to the evolutive equation: where v t i denotes the location of vertex v i at iteration t, β ∈ [0, 1] is the external force weight, and γ is a damping parameter. The discretization time step is hidden in the weighting coefficients. The computation of the internal force f int i is specific to the simplex mesh geometry and enforces regularity of the shape curvature (see [START_REF] Montagnat | Globally constrained deformable models for 3D object reconstruction[END_REF][START_REF] Delingette | General Object Reconstruction based on Simplex Meshes[END_REF] for details). The external force ensures an attraction of the model towards the vessel boundary in the image. Both forces are computed as displacement vectors along the simplex-surface normal. Location of the vascular-lumen boundary depends on image acquisition modality. Hence, this force is also modality-dependent and uses either gradient intensity or an iso-value dependent on the maximum intraluminal signal. An empirical study on contrast-enhanced MRA images of vessel-phantoms with stenoses has shown that the real boundary does not correspond to gradient maximum but rather to 45% of the local intra-luminal maximum of intensity. This iso-value is therefore used for the computation of the external forces.

v t+1 i = v t i + γ(v t i -v t-1 i ) + f int i + βf ext i (1)

Model extension to generalized cylinders

Equation 1 defines the local deformation of each surface vertex but it does not take into account the particular shape and expected properties of the modeled object. When dealing with vessels, one expects cylindrical structures with a high bending capability but for which deformations should preserve the generalized cylinder shape (fig. 4 right and fig. 5 left). Biomechanical models [START_REF] Pham | A FEM-based deformable model for the 3D segmentation and tracking of the heart in cardiac MRI[END_REF] are suited for that kind of physical deformations but they are very costly. Instead, we propose an extension of our surface model.

The surface is bound to the centerline: each surface vertex v j is associated with the 3 closest centerline vertices {a i-1 , a i , a i+1 } (fig. 5 right). Conversely, we denote E i the set of surface vertices bound to the centerline vertex a i . Each pair of vertices (a i , v j ) is weighted by a coefficient ν ij such that j ν ij = 1. When the model surface undergoes some deformation, the centerline bends accordingly through an external force resulting from the surface forces:

f ext (a i ) = v j ∈E i ν ij f ext (v j ) v j ∈E i ν ij (2)
The centerline is considered as a 1-simplex mesh in IR 3 and the external force ( 2) is used to compute its deformation. Conversely, the centerline bending is reported to the surface as the sum of an axial component (each vertex tends to follow the axis global motion) and a radial component (each vertex tends to align on a circle around the centerline) with:

f axial (v j ) = k=i+1 k=i-1 ν kj f (a k ) f radial (v j ) = k=i+1 k=i-1 v ⊥ j + (1 -ξ) v ⊥ j v j + ξr k n j -v j (3)
where v ⊥ j is the orthogonal projection of v j onto the centerline, n j is the unit normal vector of the centerline in v ⊥ j , r k is the radius (mean distance of the surface vertices to the centerline) in a k and ξ is a radial weight. The surface vertices are thus submitted to the internal and external forces (local forces) plus the axial and radial forces (centerline forces). Let λ ∈ [0, 1] weight the contributions of the local and centerline forces. The equation 1 becomes:

v t+1 i = v t i + γ(v t i -v t-1 i ) + λ f int i + βf ext i + (1 -λ) f axial (v t i ) + f radial (v t i ) (4)

Model initialization

Deformable models are sensitive to their initialization as they usually converge towards a local minimum of their energy functional. In our application however, Maracas accurately extracts centerline points {a i } and roughly estimates a set of radii {r i }. This provides an initialization of the simplex-mesh model close to the vessel boundary and, together with the axial constraint, can be successfully used for 3D segmentation of the vessel. However, the model of the vessel-lumen surface has to interact with a model of the stent in order to converge towards a realistic representation of the stented vessel.

Stent modeling

Stent-deployment simulation using the simplified geometrical model described in the section 2, provides the stent-axis mapping onto the vessel centerline calculated by Maracas, between the bounds l v0 and l vM , for a given stent radius r. The stent surface is represented using a new cylindrical simplex model combining the mapped centerline and the radius r. Once constructed, this model remains static, i.e. it is not submitted to deformations. In the stent area (between l v0 and l vM ), the shape of the vessel-surface model is controlled by the stent shape rather than the external forces extracted from the image data. Therefore, the external force of the vessel-surface model is locally set to zero. Instead, the corresponding section of the vesselsurface model is attracted by the surface of the stent model.

RESULTS AND CONCLUSION

The above described procedure was applied to simulate stent introduction and expansion in real arteries (fig. 6) and in six phantoms (fig. 7). Visualisation of a virtual stent deployed in real MRA data volume is useful for preoperative assessment of the stent's diameter and length. Simulation of the interaction between the vascular wall and the stent generates a realistic view of the luminal shape modification induced by the stent. The simplex-mesh deformable model used for this simulation is applicable to vascular-image segmentation and quantification (fig. 7). Average error of stenosis quantification in the images of phantoms was 7.3%. The largest errors occurred in complex stenoses, where the shape was no more cylindrical, and in 95% stenoses due to the lack of signal. Fig. 7. Simulation of stent/vessel-wall interaction in a physical phantom of stenosed vessel. From left to right: shaded-surface display of an MRA image of the phantom, simplex mesh resulting from its segmentation and "plunged" into the original image, quantification curves, deformed stent model, and shaded-surface display of the vessel with the stent inside.

Our simulation is based on the simplifying assumption that the stent does not modify the axial geometry of the vessel. However, it has been demonstrated [START_REF] Wentzel | Coronary stent implantation changes 3D vessel geometry and 3D shear stress distribution[END_REF] that the stent may locally change the vessel's curvature. As the stent tends to locally stiffen the vessel, the latter may be stretched outside the stented section, thus modifying its actual dimensions. Axial stiffness can be included in our deformable model, in order to simulate this phenomenon. However, this improvement of the model is not straightforward. The link between the weighting coefficient of the stiffness force and the physical properties of the vessels is not obvious. Furthermore, these properties are locally modified by the presence of atherosclerotic plaque that is more or less rigid depending on its composition. Moreover, the vessels within the body are not free, their shape strongly depends on the influence of the surrounding tissues. These open questions leave a wide field for further research combining biomechanical modelling and multi-modality image processing including high-resolution MR, computed tomography and intravascular ultrasound.

Fig. 1 .

 1 Fig. 1. Shaded-surface display of contrastenhanced MRA images: aorta arch with an aneurysm (left), carotid artery with severe stenoses. Pathologies are indicated by arrows.

Fig. 2 .

 2 Fig. 2. Stenosed carotid artery from fig. 1 segmented by Maracas software: vessel centerline and stacking of contours orthogonal to this line.

Fig. 3 .

 3 Fig. 3. Simulation of stent insertion and deployment using a simplified geometrical model: a contracted stent is placed along a catheter and guided until the delivery location (left), once expanded the stent becomes shorter (right).

Fig. 4 .

 4 Fig. 4. Stent representation in a straight (left) and curved-centerline region: the geometrical model generates an irregular folded surface due to intersections of contours (middle), a simplex-mesh model generates a regular surface (right).

Fig. 5 .Fig. 6 .

 56 Fig. 5. Deformed cylinder (left) and notations diagram (right)

ACKNOWLEDGEMENTS

This work has been supported by Rhone-Alpes Region project ADeMo. It falls within the scope of scientific topics of GdR PRC ISIS. The authors are grateful to Professor Philippe C. Douek for providing MRA images.


Surface Simplex Meshes for 3D Medical Image Segmentation

Johan Montagnat, Hervé Delingette, Nicolas Scapel, Nicholas Ayache

To cite this version:


Sylvain Joubaud

Detlef Lohse

Devaraj Van Der Meer

Fluctuation theorems for an asymmetric rotor in a granular gas

Keywords: numbers: 05.40.-a, 05.70.Ln, 45.70.-n

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

Under rather general conditions, the thermodynamic description of non-equilibrium systems is constrained by the consequences of the fluctuation relations [1]. In particular, the fluctuation theorem (FT) provides, for a system in a non-equilibrium steady state, a quantitative symmetry relation between the probability of having a positive fluctuation for the entropy production in a time τ and a corresponding negative one: ln P (σ τ = +a)

P (σ τ = -a) = βa ∀ a for τ ≫ τ c (1)
where τ c represents the largest characteristic time of the system and β a prefactor [2][3][4]16]. In general, the hypotheses used to prove this theorem are not verified experimentally and therefore, it is not clear whether Eq. ( 1) holds or not. Experimental tests of FTs have been mostly performed on stochastic systems in contact with a thermal bath. In such experiments, a FT is valid for the injected work into the system or the total entropy production and the coefficient β is directly related to the temperature of the thermal bath β = 1/k B T [5].

Experimental tests are particularly scarce and inconclusive for another type of systems for which FTs are believed to hold, namely non-thermal dynamical systems. In these systems, the strong fluctuations come from the non-linear interaction of many degrees of freedom of a dissipative system. An interesting example of such a system is a granular gas, which by its nature is already outof-equilibrium due to the dissipative character of the inelastic collisions. Experiments searching for FTs in granular systems have been performed [6], but their interpretation remains unclear [5]. It is a challenge to study FTs in such a system and to test to what extent the fluctuation relations can be satisfied.

In this letter we will study FTs for a rotor immersed in a granular gas. Eshuis et al. [7] showed that such a system not only exhibits Brownian-like dynamics comparable to a thermal system, but also a state in which symmetry is spontaneously broken, and for which there exists no thermal analogue. Here, we will show that FTs are relevant to both states.

Experiment -The rotor is composed of four vanes (25 × 60 mm 2 each, made from a single piece of stainless steel), precisely balanced around an axis which is connected to the container by a low-friction ball bearing. The granular heat bath consists of glass beads of diameter d = 4 mm (density ρ = 2600 kg/m 3 ), which are fluidized by vertical vibrations of the bottom (z = 0 at rest) with amplitude a and frequency f such that the grains interact dissipatively with the vanes. We present the results obtained for a system of N = 1000 particles (500 and 2000 particles have also been used) and for two different heights of the axis above the bottom (h = 51 mm and h = 75 mm). The natural dimensionless control parameter of the granular heat bath is the shaking strength S = 4π 2 f 2 a 2 /(gh), which represents the ratio of the typical kinetic energy injected into the system by the vibrating bottom and the potential energy of the particles at the height of the axis of the rotor. The granular temperature T g is defined as the mean of the kinetic energy fluctuations per particle. The angular position θ of the vanes is measured using an optical angle encoder, with an accuracy of 1.9 • 10 -7 rad, at a sampling rate of 1000 Hz (larger than the typical collision rate, which is about 100 Hz). After the system has reached a steady state, we start to record the position for typically 15 min, which is about 10 3 times the relaxation time of the system. We repeat experiments several times with the same conditions (S and h). An asymmetry is voluntarily introduced in the system by coating the left hand side of each vane with rubber tape. This side will therefore be softer, diminishing its coefficient of normal restitution. As a results, vanes are expected to preferentially rotate in counter-clockwise direction (positive θ). Such a system behaves like a ratchet, as explained in [7][8][9][10][11][12].

The dynamics of the system strongly depends on the experimental parameters (height of the vanes h, shaking strength S, and number of particles). Fig. 1a presents two time series of the angular velocity ω = θ of the rotor: State I has an angular velocity which fluctuates around a non-zero positive mean value (due to the symmetrybreaking coating). In state II, the angular velocity is either positive or negative and there are switches from one state to the other one. This is a state in which, in addition to the symmetry-breaking coating, there is a spontaneously broken symmetry, which can be explained from the development of a convection roll in the granular gas stabilizing the motion of the vanes [7]. These two states will now be studied in detail.

State I -In the first state, the dynamics of the angular velocity ω can be decomposed into a constant positive average value ω and strong fluctuations δω ≡ ω -ω around this mean. To shed more light on the dynamics and statistics, the power spectral density (psd) of δω and the probability density function (pdf) of ω are plotted in Figs. 1b andc. The Lorentzian shape of their psd and their Gaussian distribution indicate that the fluctuations δω are identical to those in the absence of the ratchet effect, as expected for a linear situation. The dynamics is therefore likely to be well described by a first order Langevin equation

I dω dt = -γω + M ∂e + η , (2)
where I (= 7.2 • 10 -5 kg m 2 ) is the angular moment of inertia, γ a viscous drag coefficient, and η a stochastic noise, δ-correlated in time, due to the random collisions between the particles and the rotor. This description is compatible with kinetic theory in the limit in which the mass of the ratchet is much larger than the mass of the particles [11]. The measurement of the variance of the fluctuations, T r /I, and the cut-off frequency of the psd, f c = γ/(2πI) (related to the relaxation time of the system, τ 0 = 1/(2πf c )) provides the value of the viscosity γ ≈ 7.0 • 10 -5 kg m 2 /s and the ratchet temperature T r ≈ 0.115 mJ, which is close to the granular temperature [START_REF] Theoretically | + e)/2. Both are orders of magnitude larger than kBT[END_REF]. The ratchet effect is described by adding a constant torque M ∂e to the Langevin equation [11], which is identified with γ ω (≈ 4.5 • 10 -5 Nm) by time averaging of Eq. ( 2). The subscript ∂e points to the fact that |M ∂e | is an increasing function of |∂e|, the difference in normal restitution coefficient of the coated and uncoated side of each vane. For the symmetric system, i.e., without coating, ω = 0 and thus M ∂e = 0. Based on the Langevin description, the amount of work injected into the rotor during a time τ is W τ = t+τ t M ∂e ω(t ′ )dt ′ = M ∂e ∆θ, where ∆θ ≡ θ(t + τ ) -θ(t), i.e., the symbol ∆ from hereon represents the difference measured over a time delay τ . For W τ we expect a FT of the form ln

P (W τ ) P (-W τ ) = ln P (∆θ) P (-∆θ) = M ∂e ∆θ T r , τ ≫ τ c ( 3
)
where P (∆θ) is the pdf of ∆θ. The expression in the left hand side of the equation is called the symmetry func- For state I (red symbols) the pdf is well fitted by a Gaussian (red line). For state II (blue symbols) there is a pronounced double peak; the blue line is the result of Eqs. ( 6) and (7).

tion. Due to the Gaussianity of the distribution of ω, the pdfs of ∆θ are gaussian too, and the symmetry functions are expected to be linear with ∆θ for different values of τ . This is indeed found in experiment (Fig. 2a). The slope, Σ(τ ), is found to decrease with τ and reaches the constant value M ∂e /T r expected from the FT Eq. (3) for τ > 3 s, which is approximatively 3 times the relaxation time of the system. Thus we find that two independent measurements -one from the pdf of Fig. 1b and one using the FT Eq. ( 3)-lead to the very same value of the rotor temperature T r , hereby confirming the validity of the FT Eq. ( 3). We now turn to the fluctuations of the total (trajectory dependent) entropy production in a time span τ , ∆s tot , as defined by Seifert in [13]. We follow [5] and first write the dissipated heat Q τ [18] as the injected work W τ minus the difference of the kinetic energy of the rotor E k = 1 2 Iω 2 at the beginning and end point. The entropy change ∆s m in a time span τ is now defined as the dissipated heat divided by the ratchet temperature In the limit when τ is larger than the relaxation time of the system Σ(τ ) is equal to M ∂e /Tr (dashed line), as expected from the steady state FT Eq. ( 3). (c) The symmetry function ln(P (∆stot)/P (-∆stot)) versus the total entropy production ∆stot for different values of τ . As predicted by the detailed FT, the data lie on a straight line with slope one (dashed line). In (a) and (c), the deviations from the linear regime for large fluctuations are due to a lack of statistics in a region where the pdfs are at least two orders of magnitude smaller than their maximum value.

T r ∆s m ≡ Q τ ≡ W τ -∆E k . (4)
First we note that for the symmetric system (i.e., without the symmetry-breaking coating) the dissipated heat is equal to ∆E k,sym = T r ∆s m,sym , which is small but not strictly zero. This contribution has to be subtracted from the entropy change Eq. ( 4) in order to obtain the entropy created by the presence of the external torque M ∂e . As argued in [5], it is this last entropy that may be identified as the total entropy production, i.e., ∆s tot = ∆s m -∆s m,sym . This quantity is interesting because it should satisfy a detailed FT: The symmetry functions for ∆s tot are expected to be equal to ∆s tot itself for all values of τ when the system is in a steady state [13,14]. From the Langevin equation ( 2), the total entropy takes the following form

T r ∆s tot = M ∂e ∆θ -I ω ∆ω. ( 5
)
In Fig. 2c we present the symmetry functions for different integration times τ : We find that the pdfs are Gaussian (not shown) and, most importantly, that the symmetry functions lie on a straight line with slope one within experimental errors, whatever the time delay τ is. So indeed a detailed FT holds for the total entropy production.

State II -In state II the dynamics is completely different due to the coupling to a spontaneous convection roll in the granular heat bath: The time series of Fig. 1a, shows fluctuations around two preferred velocities, corresponding to the two rolls. As a result, in the pdf of ω (Fig. 1c) two distinct peaks appear. Their different heights reflect that the system obtains a finite mean angular velocity ω by residing preferably around the positive peak -in contrast to the shift of the entire pdf in state I. From Fig. 1b we find that in both states the psds are almost exactly the same and very well fitted by a Lorentzian. This indicates that the fluctuations in each of the two peaks -and therefore the short time dynamics-are very similar to those of state I.

This suggests the separation of the dynamics into two parts: On a short time scale, ω fluctuates around a mean value, with dynamics described by a Langevin equation similar to Eq. ( 2), independent of the direction of the roll. On a much larger time scale, a coupling between the granular gas and the vanes induces collective motion, in which every now and then, through a particularly strong fluctuation, the mean value of ω quickly switches between positive and negative, reversing both the sense of rotation of the vanes and the roll. These reversals happen randomly in time. The system stays few seconds in each state as can be seen in Fig. 1a. Due to the asymmetric coating of the rotor, reversals are easier to realize when the vanes are rotating in the clockwise direction than anti-clockwise, explaining why the system has a preference for the latter [19].

Guided by the above considerations, we write the pdf of ω as the sum of two Gaussian distributions with the same variance σ 2 = T r /I; one with mean value ω 0 + ω ∂e and the other with -ω 0 + ω ∂e :

P (ω) = A p + exp - I 2T r (ω -(ω 0 + ω ∂e ) 2 + p -exp - I 2T r (ω -(-ω 0 + ω ∂e ) 2 . (6)
Here ω ∂e stands for the shift due to the symmetry breaking coating (i.e., ±ω 0 denote the locations of the peaks without coating). The factors p ± represent the weight of each peak (the probability of each direction of the convection roll) and the normalization factor A = I/(8πT r ) 1/2 has been chosen such that p + + p -= 1 implies P (ω)dω = 1. What can we learn from FTs for this very non-Gaussian system? It seems reasonable to assume that if a FT exists, it will be for the work done on short timescales, when the system resides in either of the two peaks of the pdf of ω. And indeed, in Fig. 3a (dashed blue line) we find that the symmetry function for ∆θ is linear in ∆θ for τ = 0.01 s. Since in this limit ω ≈ ∆θ/τ , we may expect a similar symmetry relation for ω, which we plot in Fig. 3b. Indeed, ln [P (ω)/P (-ω)] ∝ ω, which is truly remarkable in view of the peculiar shape of its distribution (Fig. 1c). Computing the symmetry function from the double Gaussian of Eq. ( 6) we find that it can only be linear if the weights of the peaks obey

p + p - = exp 2I ω ∂e ω 0 T r , (7)
which then directly leads to the symmetry relation ln

P (ω) P (-ω) = 2Iω ∂e T r ω . (8)
From the slope of the experimental symmetry function (Fig. 3b), which, with T r = 0.062 mJ leads to ω ∂e = 0.372rad/s rad/s, combined with ω 0 = 1.8 rad/s from the pdf of the symmetric system, we compute p + and p -and plot the resulting theoretical pdf in Fig. 1c. The agreement with the experimental pdf is good, except for angular velocities close to zero. We now return to the symmetry function of ∆θ in Fig. 3a): For the smallest value (τ = 0.01 s) we find a symmetry relation. This is because the changes ∆θ are dominated by the fluctuations, and the reversals are just rare events without large consequences for the value of ∆θ. For the intermediate value τ = 3 s, there is no valid symmetry relation. This stands to reason because when the integration time τ increases, the probability that it contains at least one reversal increases as well. ∆θ is then likely to include a reversal and there is no reason to consider the system to be in a steady-state, which is a prerequisite for a symmetry relation to hold. There are two distinct regimes: One at small |∆θ|, dominated by reversals, and another at large |∆θ| ω 0 τ , a distance which in the given time interval is unlikely to be reached when the trajectory includes one or more reversals (dotted red line in Fig. 3a). For very large τ , typically many reversals are included in each ∆θ and we can consider the system to be in a steady state again. Indeed, for τ = 10 s the pdf is Gaussian (not shown) due to the central limit theorem, and the symmetry function tends to become linear (Fig. 3a, solid black line). However, the experimental slope is smaller than the slope M ∂e /T r one would expected based upon Eq. ( 3). This may be because the entropy production depends on two parts: one is the work M ∂e ∆θ and a second one coming from another variable responsible for the reversals, which is not included here (cf. the theoretical work in [15]).

In conclusion, we have investigated the validity of FTs in two different states of an asymmetric rotor experiment in a granular gas. The first state, with a Gaussian pdf for the angular velocity, is found to be well described by a first order Langevin equation and therefore analogous to a Brownian system of temperature T r . This is reflected in the observation that the FT is valid at large τ for the injected work and for all τ for the total entropy production. In the second state the pdf of ω is doublepeaked due to convection rolls developing in the granular gas. Here symmetry relations are found for very large and very small τ , of which the latter of leads to a good prediction of the ratio of the height of the two peaks.

FIG. 1 .

 1 FIG. 1. (color online) (a) Typical time series of the angular velocity ω for state I [S = 2.78 and h = 75 mm (top, red)] and for state II [S = 2.15 and h = 51 mm (bottom, blue)]. (b) Corresponding power spectral densities of the fluctuations δω, which for both cases virtually overlap and are fitted well by a Lorentzian. (c) Corresponding probability distribution functions of ω.For state I (red symbols) the pdf is well fitted by a Gaussian (red line). For state II (blue symbols) there is a pronounced double peak; the blue line is the result of Eqs. (6) and(7).

FIG. 2 .

 2 FIG. 2. (color online) State I [S = 2.78 and h = 75 mm]: (a) The symmetry function ln(P (∆θ)/P (-∆θ)) versus ∆θ for different values of the time interval τ . All are linear with ∆θ, such that we can compute their slopes Σ(τ ). (b) Σ(τ ) as a function of τ .In the limit when τ is larger than the relaxation time of the system Σ(τ ) is equal to M ∂e /Tr (dashed line), as expected from the steady state FT Eq. (3). (c) The symmetry function ln(P (∆stot)/P (-∆stot)) versus the total entropy production ∆stot for different values of τ . As predicted by the detailed FT, the data lie on a straight line with slope one (dashed line). In (a) and (c), the deviations from the linear regime for large fluctuations are due to a lack of statistics in a region where the pdfs are at least two orders of magnitude smaller than their maximum value.

FIG. 3 .

 3 FIG. 3. (color online) State II [S = 2.15 and h = 51 mm]: (a) Symmetry function of ∆θ for three different integration times. (b) Symmetry function of ω.

The authors thank S. Ciliberto for helpful discussions. The work is part of the research program of FOM, which is financially supported by NWO.


Peter Ruckdeschel
email: peter.ruckdeschel@itwm.fraunhofer.de

Helmut Rieder
email: helmut.rieder@uni-bayreuth.de

Fisher information of scale

Keywords: one-dimensional scale, M-estimators, Fisher information bound, L 2 -differentiability, LAN, absolute continuity of measures and functions 2000 MSC: 62F12, 62F35

Motivated by the information bound for the asymptotic variance of M-estimates for scale, we define Fisher information of scale of any distribution function F on the real line as the supremum of all x φ ′ (x) F(dx) 2 φ 2 (x) F(dx), where φ ranges over the continuously differentiable functions with derivative of compact support and where, by convention, 0/0 := 0. In addition, we enforce equivariance by a scale factor. Fisher information of scale is weakly lower semicontinuous and convex. It is finite iff the usual assumptions on densities hold, under which Fisher information of scale is classically defined, and then both classical and our notions agree. Fisher information of scale finite is also equivalent to L 2 -differentiability and local asymptotic normality, respectively, of the scale model induced by F.

Motivation and Definition

If F is any distribution function on R, the real line, and φ : R → R a suitable scores function such that φ dF = 0, an M-estimate of scale S n may formally be defined by n ∑ i=1 φ

x i S n = 0 .

(1.1)

The estimand refers to the scale model (F σ ) 0<σ <∞ induced by F = F 1 , where F σ (x) = F(x/σ ).

Taylor expanding φ (x/s) = φ (x/σ ) -(sσ )φ ′ (x/σ ) x/σ 2 + • • •, we formally obtain

√ n (S n -σ ) = σ n -1/2 ∑ n 1 φ (x i /σ ) n -1 ∑ n 1 φ ′ (x i /σ ) x i /σ + • • • (1.2)
such that under observations x 1 , . . . , x n i.i.d.∼ F σ and assuming sufficient regularity, in particular consistency, √ n (S n -σ ) will as n → ∞ be asymptotically normal with mean zero and variance

V (φ , F σ ) = σ 2 V 1 (φ , F) , V 1 (φ , F) := φ 2 (x) F(dx) x φ ′ (x) F(dx) 2 . (1.3)
If φ is differentiable with continuous derivative of compact support, both φ (x) and x φ ′ (x) are bounded, so the integrals in (1.3) are well-defined for any distribution F on the Borel σ -algebra B of R. As in the theory of generalized functions (Rudin (1991, Ch. 6)), regularity conditions are shifted to the test functions whenever possible. The usual information bound for asymptotic variance would say that V (φ , F σ ) ≥ I -1 s (F σ ) and, hopefully, the lower bound will also be achieved.

This leads us to the following definition of I s1 (F). The extension to I s (F σ ) for the scale transforms F σ of F matches (1.3).

Definition 1.1. Fisher information of scale, for any distribution F on the real line, is defined by

I s1 (F) := sup φ ∈C c1 x φ ′ (x) F(dx) 2 φ 2 (x) F(dx) , (1.4)
where C c1 denotes the set of all differentiable functions φ : R → R whose derivative is continuous and of compact support, and 0/0 := 0 by convention. For the scale transforms F σ of F we define

I s (F σ ) := σ -2 I s1 (F) , 0 < σ < ∞ . (1.5) Remark 1.2. Since the map φ → φ σ , where φ σ (x) := φ (σ x) and φ ′ σ (x) = σ φ ′ (σ x)
, defines a one-to-one correspondence on C c1 , we obtain scale invariance of I s1 ,

I s1 (F σ ) = I s1 (F) , 0 < σ < ∞ . (1.6)
So extension (1.5) is needed to obtain scale equivariance. In the scale model, as opposed to location, it matters whether a given distribution F is considered element F = F 1 or, for example, element F = F .5 (in the scale model generated by F 2 ).

Motivated by the information bound, Definition 1.1 is instrinsically statistical. It does not a priori use the assumption of, and suitable conditions on, densities. These properties rather follow from the definition in case I s is finite. Another advantage is that Definition 1.1 implies certain topological properties (convexity and lower continuity) of I s .

The definition parallels Huber (1981, Def. 4.1) in the location case,

I l (F) := sup φ φ ′ (x) F(dx) 2 φ 2 (x) F(dx) , (1.7)
where φ , subject to φ 2 dF > 0, ranges over the (smaller) set C 1 c of all continuously differentiable functions which themselves are of compact support. I l is shift invariant. Huber (1981, p. 79), states vague lower semicontinuity and convexity of I l . By Huber (1981, Thm. 4.2), I l (F) is finite iff F is absolutely continuous with an absolutely continuous density

f such that f ′ /f ∈ L 2 (F), in which case I l (F) = ( f ′ /f ) 2 dF.
Remark 1.3. The latter result, by arguments of the proof to Theorem 2.2 below, still obtains if definition (1.7) is based on C c1 . Only vague lower semicontinuity of I l would be weakened to weak continuity (which, however, makes no difference in the setup of normed measures). The convention 0/0 := 0 could replace the side condition φ = 0 a.e. F in (1.7) as well.

The non-suitability of C 1 c , and suitability of C c1 instead, is the tribute to the scale model, for which the functions x → x φ ′ (x) need to be dense in L 1 (F 0 ) with respect to the punctuated (substochastic) measure F 0 introduced in (2.1) below.

Fisher information of scale has been treated by [START_REF] Huber | Robust Estimation of a Location Parameter[END_REF]Huber ( , 1981) ) not in the previous generality but only under suitable assumptions on densities and, in an auxiliary way, has been reduced to the location case by symmetrization and the log-transform, Huber (1981, Sec. 5.6).

Main Results

Proposition 2.1. I s1 is weakly lower semicontinuous and convex.

Zero observations do not contain any information about scale. Removing the mass of any distribution F at zero, we define the punctuated, possibly substochastic measure F 0 by

F 0 := F -F({0})1 0 , (2.1)
where 1 0 denotes Dirac measure at 0. In terms of distribution functions, denoting by 1 [0,∞) the indicator function, we have

F 0 (x) = F(x) -(F(0) -F(0-))1 [0,∞) (x).
Theorem 2.2. For any distribution F on the real line,

I s1 (F) is finite iff i) F 0 is absolutely continuous with a density f such that ii) x → x f (x) is absolutely continuous, and iii) x → Λ(x) := -[ x f (x)] ′ /f (x) ∈ L 2 (F 0 ), in which case I s1 (F) = Λ 2 dF 0 = x =0 [ 1 + x f ′ (x)/f (x) ] 2 F(dx) .

Consequences for the Scale Model

For the scale transforms F σ of F, I s1 (F σ ) = I s1 (F) and I s (F σ ) = σ -2 I s1 (F) by (1.6) and (1.5), respectively. In particular, I s1 (F σ ) and I s (F σ ) are finite iff I s1 (F) is finite. Also conditions i) and ii) of Theorem 2.2 are simultaneously fulfilled for a density f of F 0 and the density f σ (x) = σ -1 f (x/σ ) of the punctuation F σ ,0 of F σ . In the finite case, since

[x f σ (x)] ′ /f σ (x) in condition iii) of Theorem 2.2 is just Λ(x/σ ), this theorem yields I s1 (F σ ) = Λ 2 (x/σ ) F σ ,0 (dx), which is Λ 2 (x) F 0 (dx) = I s1 (F σ )
; that is, (1.6) again. Therefore, in the finite case,

I s (F σ ) = Λ 2 σ dF σ ,0 , 0 < σ < ∞ . (3.1)
the representation of I s (F σ ) in terms of the usual score function Λ σ ,

Λ σ (x) := 1 σ Λ x σ = ∂ ∂ σ log f σ (x) = - 1 σ 1 + x σ f ′ ( x σ ) f ( x σ ) . (3.2)
As an analogue to a lemma due to [START_REF] Hájek | Local asymptotic minimax and admissibility in estimation[END_REF] in the location case, Swensen (1980, Ch.2, Sec.3) for an absolutely continuous F has shown that conditions i)-iii) of Theorem 2.2 even imply L 2 -differentiability (Rieder, 1994, Def. 2.3.6) of the scale model,

dF σ +t -dF σ (1 + 1 2 tΛ σ ) = o(t) as t → 0 (3.3)
at σ = 1 and, by invariance, at any 0 < σ < ∞. By definition, L 2 -differentiability already entails that Λ 2 σ dF σ < ∞. Setting Λ(0) := 0, we may extend his result to F({0}) > 0.

Proposition 3.1. Assume that I s1 (F) < ∞ . Then the scale model (F σ ) 0<σ <∞ is L 2 -differentiable with derivative Λ σ at every 0 < σ < ∞ .

L 2 -differentiability of a parametric model implies an expansion of the log-likelihhods, see e.g. Rieder (1994, Thm. 2.3.5); in our case, for each h ∈ R,

log dF n σ +h/ √ n /dF n σ = 1 √ n ∑ n i=1 h τ Λ σ (x i ) -1 2 h τ I s (F σ )h + o F n σ (n 0 ) ; (3.4)
that is, the scale model is locally asymptotically normal (LAN). LAN is the basis of asymptotic optimality results as Hájek's Asymptotic Convolution Theorem and the Local Asymptotic Minimax Theorem, see e.g. Rieder (1994, Thm.'s 3.2.3, 3.3.8) and van der Vaart (1998, Thm.'s 8.8, 8.11). Le Cam (1986, 17.3 Prop. 2) even shows that, in the i.i.d. setup, LAN is equivalent to L 2 -differentiability. Thus we obtain the following result.

Proposition 3.2. The following statements are equivalent:

i) I s (F σ ) < ∞ at some 0 < σ < ∞.
ii) The scale model is L 2 -differentiable at some 0 < σ < ∞.

iii) The scale model has the LAN property (3.4) at some 0 < σ < ∞.

By invariance, the validity of each statement at one σ implies its validity at any other 0 < σ < ∞.

Acknowledgements

We thank two referees for their helpful comments.

Appendix A. Proofs and Absolute Continuity

Proof of Proposition 2.1 The sup over a family of l.s.c., resp. convex, functions being l.s.c., resp. convex, it suffices to show that, for each φ ∈ C c1 , the reciprocal function V -1 1 (φ , • ) from (1.3), is weakly l.s.c. and convex. In this proof only, we pay a price for the simplifying convention 0/0 := 0.

Let F n → F weakly. Then φ 2 dF n → φ 2 dF. First assume φ 2 dF > 0. Then φ 2 dF n > 0 eventually, and V -1 1 (φ , F n ) → V -1 1 (φ , F). Secondly suppose that φ 2 dF = 0. If also x φ ′ dF = 0, then Huber (1981, Lemma 4.4). Secondly, let

Thirdly, let both φ 2 dF j be zero. Then, if also both x φ ′ dF j = 0, we get

Lemma A.1. For any finite measure F on B, the class

Proof On the basis of Lusin's theorem, Rudin (1974, Thm. 3.14), it suffices to approximate the indicator of bounded intervals (a, b].

For ε ↓ 0 one may choose functions

Concerning denseness of D c1 in L 1 (F 0 ), we may assume that a > 0. Drawing on the functions g ε define

and, in the second case, φ n (x) = φ (nx), where 2 φ (x) = 1 + cos(|x| ∧ π).

4

Absolute Continuity From real analysis, e.g., Rudin (1974, Ch.8), we recall: An R-valued measure on the Borel σ -field B of the real line is dominated by λ , the Lebesgue measure, iff its distribution function is absolutely continuous. A function f : R → R is absolutely continuous, if for any ε > 0 there is a δ > 0 such that for any finite collection of disjoint segments

the derivative f ′ exists a.e. λ , and

, implying bounded variation on R, and the limit f (a) → 0 as a → -∞ require further conditions, respectively. These are obviously satisfied in the location case for absolutely continuous densities f such that

If f and g are absolutely continuous, so is their product f g on any compact [a, b]. Thus, integration by parts holds: Rieder (1994, Lemma C.2.1).

Proof of Theorem 2.2 First assume I s1 (F) < ∞. On C c1 define T (φ ) := -x φ ′ dF, which operator is well defined, because φ 2 dF = 0, in view of Definition 1.1, entails that x φ ′ dF = 0.

Evaluated on C c1 , T has operator norm I s1 (F) . C c1 being dense in L 2 (F), T may be extended to L 2 (F) keeping its norm. By Riesz-Fréchet there exists some g ∈ L 2 (F), whose norm equals the operator norm of T , such that T (φ ) = φ g dF for all φ ∈ L 2 (F), hence

Inserting φ n from Lemma A.1, both choices, we obtain that, in addition to g 2 dF = I s1 (F),

In particular, the integrals in (A.1) and (A.2) may be restricted to R \ {0} . Define the function

Then, if φ -∞ denotes the constant value of φ ∈ C c1 left to the support of φ ′ , φ g dF = (φφ -∞ ) g dF 0 and φ (x) -φ -∞ = 0 =y≤x φ ′ (y) λ (dy). Due to compact support of φ ′ , and g ∈ L 2 (F 0 ), the product g(x) φ ′ (y) is in L 1 (F 0 (dx) ⊗ λ (dy)), and so x φ ′ dF 0 = -x>y =0 g(x) φ ′ (y) F 0 (dx) λ (dy) = y f (y) φ ′ (y) λ (dy) by Fubini; thus,

By denseness of D c1 in L 1 (F 0 ), Lemma A.1, the LHS determines F 0 . As pointwise and dominated convergence x h ′ ε = g ε → 1 (a,b] has been established in that proof, also f dλ on the RHS is completely determined by (A.4)

This completes the identification of g under F, and i)-iii) are proved.

Conversely, assume i)-iii). By ii), m(x) = x f (x) is absolutely continuous. Differentiability of m at x = 0 implies that of f , and

Thus, m and its measure m ′ dλ = -Λ dF 0 are of bounded variation on R.

By Hölder inequality, |m(y)-m(x)| 2 ≤ |F(y)-F(x)| Λ 2 dF 0 , so m(x) for x → ∞ is a Cauchy sequence. But lim x→∞ m(x) must be zero since otherwise f (x) ∼ 1/x for x → ∞ would not integrate. The same holding for x → -∞, we obtain m ′ dλ = 0. (A.8)

For φ ∈ C c1 , the function φφ -∞ and corresponding measure φ ′ dλ have bounded variation on R. Thus integration by parts in the general form of Rieder (1994, Lem. C.2.1) yields φ ′ m dλ = -φ m ′ dλ , such that x φ ′ dF = φ ′ m dλ = -φ m ′ dλ = φ Λ dF 0 . (A.9)

Applying Cauchy-Schwarz, we get

where Λ 2 dF 0 is finite by iii). It follows that I s1 (F) < ∞.

Proof of Proposition 3.1 We decompose √ dF σ +t -√ dF σ (1 + 1 2 tΛ σ ) into the following sum,

The first summand is o(t) by [START_REF] Swensen | Asymptotic inference for a class of stochastic processes[END_REF]. The second is 0, since F σ ({0}) = F({0}) and Λ σ (0) = 0.


Péter Elek

László Márkus
email: markus@cs.elte.hu

Tail behaviour of β-TARCH models

Keywords: ARCH-type model, conditional heteroscedasticity, extreme value theory, tail behaviour

It is now common knowledge that the simple quadratic ARCH process has regularly varying tail even when generated by a normally distributed noise, and the tail behaviour is well-understood under more general conditions as well. Much less studied is the case of β-ARCH-type processes, i.e. when the conditional variance is a 2β-power function with 0 < β <1. Being a little more general and allowing for asymmetry, we consider threshold β-ARCH models, driven by noises with Weibull-like tails. (Special cases include the Gaussian or the Laplace distributions.) We show that the generated process has approximately Weibull like tail, too, albeit with different exponent: 1-β times that of the noise, in the sense that the tail can be bounded from both sides by Weibull distributions of this exponent but slightly different constants. The proof is based on taking an appropriate auxiliary sequence and then applying the general result of Klüppelberg and Lindner (2005) for the tail of infinite MA sequences with light-tailed innovations.

Introduction

In this paper we examine the tail behaviour of the stationary distribution of certain ARCH-type models defined by the equation

X t = ω + α + X + t-1 2β + α -X - t-1 2β 1/2 Z t , (1)

ACCEPTED MANUSCRIPT

where we apply the usual notations x + = max (x, 0) and x -= -min (x, 0) . The model parameters ω, α + and α -satisfy min (α + , α -) ≥ 0, max (α + , α -) > 0 and ω > 0, and Z t is an i.i.d. sequence with zero mean and finite variance.

An important feature of this process is that -if the autocorrelation function is defined at all -it is an uncorrelated but not an independent sequence because its conditional variance is changing over time as a function of the lagged values (conditional heteroscedasticity). If β = 1 and α + = α -> 0, we obtain the well-known ARCH (autoregressive conditionally heteroscedastic) model [START_REF] Engle | Autoregressive conditional heteroskedasticity with estimates of the variance of the United Kingdom inflation[END_REF], where the conditional variance has a quadratic functional form. Since this process can reproduce the stylised facts (e.g. uncorrelatedness, conditional heteroscedasticity, nonnormality) of financial time series in an easily estimable way, it has become a basic tool in financial econometrics in the past two decades, and gave rise to various generalisations. For instance, in order to model the fact that the variance of stock returns responds more strongly to negative shocks than to positive ones, [START_REF] Glosten | On the relation between the expected value and the volatility of the normal excess return on stocks[END_REF] defined the TARCH (threshold ARCH) process by allowing α + = α -in the equation with β = 1.

(Hence α + < α -generally holds in financial applications.) For a broad overview of the various generalisations of ARCH models and some of their properties we refer the reader to [START_REF] Terasvirta | An introduction to univariate GARCH models[END_REF].

Due to the popularity of the quadratic ARCH models in finance, their probabilistic properties are quite much studied and well understood. It is a well-known fact for β = 1 (see e.g. [START_REF] Embrechts | Modelling Extremal Events for Insurance and Finance[END_REF]) that not all choices of (ω, α + , α -) and of the distribution of Z t permit a stationary solution of equation (1). For instance, if Z t is normally distributed, the quadratic ARCH model (i.e. the case α + = α -) has a stationary solution if and only if α + = α -< 2 exp (δ) ≈ 3.562, where δ is the Euler-constant.

(A different choice for the noise distribution yields a different domain of stationarity.) Also, much is known about the tail behaviour of the stationary distribution if β = 1.

It was proven two decades ago [START_REF] Goldie | Implicit renewal theory and tails of solutions of random equations[END_REF] that the simple ARCH process has regularly varying tail (roughly speaking: polynomially decaying tail) even when Z t is normally distributed. This phenomenon is often summarised as: "light-tailed input can cause heavy-tailed output". More generally, [START_REF] Borkovec | The tail of the stationary distribution of an autoregressive process with ARCH(1) errors[END_REF] proved that the AR(1) model driven by a quadratic ARCH(1) innovation has regularly varying

ACCEPTED MANUSCRIPT

tail for a very general class of noise distributions. Using the concepts of extreme value theory (EVT) it follows that the stationary distribution of quadratic ARCH processes belongs to the maximum domain of attraction of the Frechet extreme value distribution or, equivalently, their tail can be approximated by a generalised Pareto distribution (GPD) with shape parameter ξ > 0. (For an introduction into EVT we refer the reader to [START_REF] Embrechts | Modelling Extremal Events for Insurance and Finance[END_REF].)

The 0 < β < 1 case -where the conditional variance is increasing slower than a quadratic function of the lagged values -is very different from the usual β = 1 parameter choice, and is much less studied in the literature. The model may then be called the β-TARCH process and was analysed e.g. by [START_REF] Guegan | Probabilistic properties of the β-ARCH-model[END_REF]. It follows relatively easily from the drift condition for Markov chains [START_REF] Meyn | Markov Chains and Stochastic Stability[END_REF] that in the 0 < β < 1 case the X t process defined by ( 1) is stationary irrespective of the choice for the parameters and for the distribution of Z t (provided that the latter has a finite second moment and some basic conditions for its density are fulfilled). Moreover, if the mth moment of Z t is finite, the mth moment of the stationary distribution of X t will be finite, too (see [START_REF] Guegan | Probabilistic properties of the β-ARCH-model[END_REF], or in a more general setting [START_REF] Elek | A light-tailed conditionally heteroscedastic time series model with an application to river flows[END_REF]). Hence, if all moments of Z t is finite and its distribution has infinite support, the distribution of X t may only belong to the maximum domain of attraction of the Gumbel law and, equivalently, the shape parameter of the GPD fitted to it may only be zero -if the distribution belongs to the maximum domain of attraction of an extreme value law at all.

This result already yields that the β-TARCH model is lighter tailed than the usual, quadratic specification: for light-tailed Z t noises the tail of X t decays faster to zero than a polynomial function. The finding, however, does not determine the exact tail behaviour: the maximum domain of attraction of the Gumbel law contains many different types of distributions (e.g. normally, exponentially or lognormally decaying ones). In this paper we give a more precise estimate for the tail decay by showing that X t has approximately Weibull-like tail provided that Z t has a Weibull-like distribution. Our research is motivated by the fact that β-TARCH models proved useful to model conditional heteroscedasticity in areas where the quadratic ARCH model was considered too heavy-tailed, such as in the analysis of water discharge series of rivers with large

ACCEPTED MANUSCRIPT

catchments. (See e.g. [START_REF] Elek | A light-tailed conditionally heteroscedastic time series model with an application to river flows[END_REF] or in a broader context [START_REF] Szilágyi | Hybrid, Markov chain-based model for daily streamflow generation at multiple catchment sites[END_REF].)

Throughout the paper we will use the notations FX (u) = 1 -F X (u) for the survival function and f X (u) for the density function of the random variable X.

Tail behaviour

To examine the tail behaviour of X t let us introduce an assumption on the tail of

Z t :
Assumption 1. Z t is an i.i.d. sequence with a symmetric, absolutely continuous probability distribution. Moreover, there exist u 0 > 0, γ > 0, κ > 0, K 1 > 0 and K 2 such that its probability density satisfies

f Zt (u) = K 1 |u| K2 exp (-κ|u| γ ) (2)
for every |u| > u 0 , and f Zt (u) is bounded away from zero on [-u 0 , u 0 ] .

According to this assumption, Z t has a Weibull-like tail with exponent γ. The Gaussian (γ = 2) or the Laplace (γ = 1) distributions are obtained as special cases. [START_REF] Guegan | Probabilistic properties of the β-ARCH-model[END_REF] showed under the assumption min (α + , α -) > 0 that if β > (γ -1)/γ, X t has no exponential moment (i.e. it is heavier tailed than the exponential distribution) while if β < (γ -1)/γ, X t has a moment generating function defined around the neighbourhood of zero. This finding already suggests that X t may possess (approximately) a Weibull-like tail with exponent γ(1 -β). Assuming a normally distributed noise (i.e. γ = 2), α + = α -and 1/2 < β < 1, [START_REF] Robert | Extremes of alpha-ARCH models[END_REF] argued that this is indeed the case: under his assumptions X t has Weibull-like tail with exponent 2 (1 -β) . Although the proof of his conjectures seems to be incomplete, 2

2 He derives a functional equation for the logarithm of the moment generating function L Y (s) = E (exp (sXt)) of Yt = log X 2 t and estimates the tail of Yt based on the behaviour of L Y (s) around ∞. During the calculations he assumes (see Appendix 1 of his paper) that if a function g satisfies

g(x) -g(αx) = O(1/x) as x → ∞, then g(x) = O(1/x)
. However, this is not the case: if e.g.

g(x) = sin (2π log x/ log α) then g(x) -g(αx) = 0.
ACCEPTED MANUSCRIPT some of his ideas are useful to prove that X t has approximately Weibull-like tail even if we consider the more general case, i.e. α + = α -, γ = 2 and 0 < β < 1.

Theorem 1. Assume that X t satisfies equation (1), Assumption 1 holds, and ω > 0, min (α + , α -) > 0, 0 < β < 1. Then, using the notation α = max (α + , α -) , the survival function of the stationary distribution of X t satisfies

exp - α -γ/2 κγβ -β 1-β 2 u γ(1-β) + O u γ(1-β)/2 ≤ FXt (u) ≤ exp - (α + ω) -γ/2 κγβ -β 1-β 2 u γ(1-β) + O u γ(1-β)/2 . (3) Clearly, since (K -ǫ) z < Kz + O z 1/2 < (K + ǫ) z as z → ∞ for any K > 0
and ǫ > 0, the exponent and the bounds for the multiplier of u γ(1-β) are exact in the above given Weibull like approximation, but (even if the constants were the same in the upper and lower bound) there still remains room for a deviation from an asymptotically exact Weibull limit distribution.

Proof. We may assume without loss of generality that α = α + ≥ α -. Let Y t = log(X 2 t ), U t,1 = log α + Z 2 t , and U t,2 = log α -Z 2 t . Furthermore, let us introduce the functions h 1 (y) = log (ω/α + + exp (βy)) , h 2 (y) = log (ω/α -+ exp (βy))

and the random variables

V t,i = h i (Y t-1 ) -βY t-1 (i = 1, 2) . Then Y t = h 1 (Y t-1 ) + U t,1 = βY t-1 + U t,1 + V t,1 if Z t-1 > 0, Y t = h 2 (Y t-1 ) + U t,2 = βY t-1 + U t,2 + V t,2 if Z t-1 ≤ 0.
Since h i (y) ≥ βy (i = 1, 2) , V t,i ≥ 0 a.s. Moreover, since Z t is a symmetrically distributed i.i.d. sequence, Y t can be written as

Y t = βY t-1 + U t + V t ,
where

U t = U t,1 χ {Zt-1>0} + U t,1 χ {Zt-1≤0} is an independent 1/2-1/2 mixture of U t,1
and U t,2 , and is itself an i.i.d. process.

Similarly V t = V t,1 χ {Zt-1>0} + V t,1 χ {Zt-1≤0}
is an independent 1/2-1/2 mixture of V t,1 and V t,2 .

A C C E P T E D M A N U S C R I P T ACCEPTED MANUSCRIPT

Let us introduce the auxiliary autoregressive sequence

Y * t = βY * t-1 + U t = ∞ i=0 β i U t-i .
As we use log in the definition of U t , the convergence of the sum cannot be taken for granted. However, the distribution of Z t is absolutely continuous (at zero as well) and it has finite variance, hence E U 2 t < ∞, therefore Y * t has a stationary distribution and E (Y * t ) 2 < ∞. The importance of this process comes from the fact that it is stochastically smaller than Y t , i.e. FY * t (u) ≤ FYt (u) . To see this, let Ŷ0 = Y 0 and define Ŷt recursively as

Ŷt = β Ŷt-1 + U t . (4)
Using that V t ≥ 0 a.s., we can prove by induction that Ŷt ≤ Y t :

Ŷt = β Ŷt-1 + U t ≤ βY t-1 + U t ≤ Y t . (5)
Since Ŷt tends in distribution to the stationary distribution of Y * t as t → ∞, Y * t is indeed stochastically smaller than Y t . Therefore, by examining the tail behaviour of Y * t we can obtain a lower bound for the tail of Y t as well. (A similar majorisation technique is used in a different context in Lemma 2.2. of [START_REF] Brachner | Extremes of autoregressive threshold processes[END_REF].)

To determine the tail of Y * t , we will apply the framework of [START_REF] Klüppelberg | Extreme value theory for moving average processes with light-tailed innovations[END_REF] who examined the tail behaviour of linear moving average processes with increments lighter tailed than the exponential distribution. Let ∞ -∞ c i W t-i be the examined process and assume that the probability density of the i.i.d. sequence W t satisfies

f (u) = ν(u) exp (-ψ(u)) , u ≥ u 0 (6)
for some u 0 , and

ψ(u) is C 2 , ψ ′ (u 0 ) = 0, ψ ′ (∞) = ∞ and ψ ′′ is strictly positive on [u 0 , ∞].
Moreover, using the notation φ = 1/ √ ψ ′′ , we assume that the function ν is flat for φ, i.e.

lim u→∞ ν (u + xφ (u)) ν (u) = 1
uniformly on bounded x-intervals, and also that φ is self-neglecting, i.e. flat for itself.

(Roughly speaking, these assumptions require that ψ(u) should be strictly convex and ν(u) should behave approximately as a constant as u → ∞.)

ACCEPTED MANUSCRIPT

Furthermore, following the original notations of [START_REF] Klüppelberg | Extreme value theory for moving average processes with light-tailed innovations[END_REF], define q(τ ) = ψ ′-1 (τ ) and

Q(τ ) = ∞ i=-∞ c i q(c i τ ), σ 2 i (τ ) = c 2 i q ′ (c i τ ), σ 2 ∞ (τ ) = ∞ i=-∞ σ 2 i (τ ).
It follows from the conditions that Q is a strictly increasing function. Then, provided that c i is a summable sequence of non-negative real numbers, not all zero, and assuming that the two conditions below hold:

lim m→∞ lim sup τ →∞ |j|>m σ 2 j (τ ) σ 2 ∞ (τ ) = 0, (7)
lim m→∞ lim sup τ →∞ |j|>m σ j (τ ) σ 2 ∞ (τ ) = 0, (8)
the following theorem is true:

Theorem 2. [START_REF] Klüppelberg | Extreme value theory for moving average processes with light-tailed innovations[END_REF] Under the above conditions, as u → ∞,

P ∞ i=-∞ c i W t-i > u ∼ 1/ √ 2π Q -1 (u) σ ∞ (Q -1 (u)) exp - u u0 ci Q -1 (v) + ρ Q -1 (v) dv where ρ (τ ) = o (1/σ ∞ (τ ))
. It is also true that 1/σ ∞ (τ ) = o (τ ) so the first term in the integral is the leading term.

In our case, this theorem will be used with the choice U t = W t so the conditions of the theorem should be checked first. Here,

f Ut (u) = 1 2 K 1 exp (K 2 u) exp -κα -γ/2 + e γu 2 + 1 2 K 3 exp (K 4 u) exp -κα -γ/2 - e γu 2 = ν(u) exp (-ψ(u))
with appropriate constants K 1 > 0, K 3 > 0 and K 2 , K 4 . To satisfy the necessary assumptions with u 0 = 0 in (6), ψ(u) can be defined as

ψ (u) = κα -γ/2 e γu 2 -eκα -γ/2 /2 if u ≥ 2/γ, ψ (u) = eκα -γ/2 (γ/2) 2 u 2 /2 if u < 2/γ.

ACCEPTED MANUSCRIPT

because τ → ∞ implies that j → ∞ in the summation condition.

Therefore the sum in (10) (and so the second term in ( 9)) is o τ -θ . As a consequence, using the notations

A = 2γ -1 (1 -β) -1 and B = β (1 -β) -1 log β + 1 -log τ 0 , we obtain for τ → ∞ Q (τ ) = 2γ -1 ∞ j=0 β j log β j τ + 1 -log τ 0 + o τ -θ = 2γ -1 (1 -β) -1 log τ + β (1 -β) -1 log β + 1 -log τ 0 + o τ -θ = A (log τ + B) + o τ -θ . Trivially, Q -1 (u) → ∞ and hence exp -o Q -1 (u) -θ = 1 + o Q -1 (u) -θ = 1 + o (1) (11)
as u → ∞, therefore

Q -1 (u) = exp A -1 u -B exp -o Q -1 (u) -θ = exp A -1 u -B (1 + o (1)) . (12)
Using again (11) and then (12) we obtain a better estimate for Q -1 (u) :

Q -1 (u) = exp A -1 u -B 1 + o Q -1 (u) -θ = exp A -1 u -B 1 + o exp -θA -1 u = κγα -γ/2 β -β 1-β 2 exp γ (1 -β) 2 u + o exp γ (1 -β) (1 -θ) 2 u .
Let us also check conditions ( 7)-( 8). We obtain that q ′ (τ

) = 2γ -1 /τ 0 if τ < τ 0 and q ′ (τ ) = 2γ -1 /τ if τ ≥ τ 0 , hence σ 2 i (τ ) = 2γ -1 β 2i /τ 0 if β i τ < τ 0 and σ 2 i (τ ) = 2γ -1 β i /τ if β i τ ≥ τ 0 . Thus σ 2 ∞ (τ ) = 2γ -1   j: β j τ ≥τ0 β j /τ + j: β j τ <τ0 β 2j /τ 0   ∼ 2γ -1 (1 -β)
-1 /τ, so ( 7)-( 8) are easily seen for the c i = β i sequence, consequently Theorem 2 can be applied. It also follows that ρ (τ ) = O τ 1/2 and u 0 = 0 in that Theorem. If we

ACCEPTED MANUSCRIPT

choose θ > 1/2, we obtain

FY * t (u) = exp -log Q -1 (u) -log σ ∞ Q -1 (u) - u 0 Q -1 (v) + O e γ(1-β) 4 v dv = exp - κα -γ/2 β -β 1-β 1 -β e γ(1-β) 2 u + O e γ(1-β) 4 u . (13)
Taking into account that Y * t is stochastically smaller than Y t = log X 2 t , the lower bound is obtained for FXt (u) in (3).

To show the upper bound for the tail, we can first prove with a similar majorisation technique as used in the proof of FY * t (u) ≤ FYt (u) (equations ( 4)-( 5)) that the increase of either α + or α -does not make the tail of Y t any lighter. Therefore, we can assume that α = α + = α -and get an upper bound for the tail of this restricted model.

In this case, let us introduce for each t a random variable U * * t ≥ 0 such that U t ≤ U * * t a.s. and f U * * t (u) = Kf Ut (u) for all u > 0 with an appropriate K > 0. (Such a variable can easily be constructed.) Define also h(y) = βyχ {y≥0} + log (1 + ω/α) .

It follows from α + = α -that h i (y) ≤ h(y) (i = 1, 2) and thus it can be shown again with a majorisation argument that FY * * t (u) ≥ FYt (u) holds for the stationary distribution of the model defined by

Y * * t = h Y * * t-1 + U * * t .
Indeed, let Ỹ0 = Y 0 and define Ỹt recursively as

Ỹt = h Ỹt-1 + U * * t .
Using U t ≤ U * * t we can prove by induction that Y t ≤ Ỹt :

Y t ≤ h (Y t-1 ) + U t ≤ h Ỹt-1 + U t ≤ h Ỹt-1 + U * * t = Ỹt .
Since the distribution of Ỹt tends to the stationary distribution of Y

Y * * t = βY * * t-1 + U * * t + log (1 + ω/α) = ∞ i=0 β i U * * t-i + log (1 + ω/α) 1 -β . ( 14
)

ACCEPTED MANUSCRIPT

The result of [START_REF] Klüppelberg | Extreme value theory for moving average processes with light-tailed innovations[END_REF] again gives that the tail of

∞ i=0 β i U * * t-i
has the same form as the tail of Y * t (equation ( 13)). Completing the above reasoning with the effect of the last constant term in ( 14) it is easy to derive the upper bound for the tail of X t in (3).

In some applications (e.g. [START_REF] Elek | A light-tailed conditionally heteroscedastic time series model with an application to river flows[END_REF]) the α -= 0 restriction is used.

Then the upper bound in (3) certainly holds and a slightly weaker lower bound can also be proven easily: Proposition 1. Assume the assumptions of Theorem 1 but allow α -= 0. Then for every δ > 0 there exists a K > 0 such that exp -Ku (1+δ)γ(1-β) ≤ FXt (u) . with a possibly different K > 0, and transforming Y t to X t gives the statement of the proposition.

YFY

  Let us use the same notations as in the proof of Theorem 1 and let Y * * * α 0 Z 2 t . It is easily shown that Y * * * t ≤ Y t stochastically in this case.Moreover, as Z t is symmetrically distributed, for every n ∈ Z + with probability 2 -n + t-i + β n Y * * * t-n .Therefore, using the notations q = FY * * * t-n (0) andY t,n = n-1 i=0 β i U + t-i , FY * * * t (u) ≥ q2 -n FYt,n (u) .Moreover, similarly to the derivation of the tail of Y * t , it follows again from Theorem 2 K > 0. Choosing n such that 1/ (1 -β n ) < 1 + δ we obtain

A C C E P T E D M A N U S C R I P T ACCEPTED MANUSCRIPT

Then it is a matter of routine to check that the resulting ν(u) function is flat for φ (u) and that φ(u) is self-neglecting (see also Example 2.4. (c) in [START_REF] Klüppelberg | Extreme value theory for moving average processes with light-tailed innovations[END_REF]), so the tail of Y * t can in principle be approximated using c i = β i for i ≥ 0 and c i = 0 for i < 0. (Conditions ( 7)-(8) will be checked, see below.) Using the notation

and hence

Then

For any 0 < θ < 1 (we shall choose θ appropriately from this range later), the sum in the second term can be written as j: β j τ <τ0

β 1-θ j e θ β j τ /τ 0 1+θ -β j eτ /τ 0 θ log β j eτ /τ 0 .

(10)

For 0 < θ < 1 the function g(x) = x θ log x is bounded on (0, e], hence both β j eτ /τ 0 θ log β j eτ /τ 0 and e θ β j τ /τ 0 1+θ are bounded if 0 < β j τ /τ 0 < 1. This way we have for τ → ∞ j: β j τ <τ0

ACCEPTED MANUSCRIPT

Conclusions

In this paper we showed that the tail of a β-TARCH model can be approximated by

Weibull-like distributions with exponent γ (1 -β) if the generating noise has Weibulllike tail with exponent γ. We conjecture that the lower bound in Theorem 1 gives the precise asymptotics for the tail of the model but we could not prove this conjecture yet.

It is a natural question to ask how the tail behaviour is modified when AR-or MA-terms are added to the simple uncorrelated β-TARCH model, as e.g. in [START_REF] Elek | A light-tailed conditionally heteroscedastic time series model with an application to river flows[END_REF]. This question is not yet settled but we conjecture that the more general ARMA-β-TARCH model has approximately a Weibull-like tail with the same exponent as the more restricted model examined in this paper.


Adam Ose

Adam Osȩkowski

Weak type inequalities for conditionally symmetric martingales

Keywords: Martingale, square function, weak type inequality

Let f be a conditionally symmetric martingale and let S(f ) be its square function. We prove that

where

In addition, the constant C p is shown to be the best possible even for the class of dyadic martingales.

Introduction

Square function inequalities appear in many areas of mathematics, for example in harmonic analysis, potential theory and both classical and noncommutative probability, where they play an important role: see e.g. [START_REF] Burkholder | Explorations in martingale theory and its applications[END_REF], [START_REF] Dellacherie | Probabilities and Potential B: Theory of martingales[END_REF], [START_REF] Pisier | Noncommutative martingale inequalities[END_REF] and [START_REF] Stein | The development of the square functions in the work of A. Zygmund[END_REF]. It is therefore of interest to establish sharp versions of such estimates. The primary objective of this paper is to determine the best constants in some weak-type estimates for the martingale square function under the assumption of conditional symmetry.

We start with introducing the background and notation. Let (Ω, F , P) be a probability space, filtered by (F n ) n≥0 , a nondecreasing family of sub-σ-fields of F . Let f = (f n ) n≥0 be an adapted martingale taking values in a separable Hilbert space H with scalar product

•, • and norm | • |. Then df = (df n ) n≥0 , the difference sequence of f , is given by df 0 = f 0 and df n = f n -f n-1 . We define the square function of f by

S(f ) = ∞ k=0 |df k | 2 1/2

Manuscript

Click here to view linked References

A C C E P T E D M A N U S C R I P T

ACCEPTED MANUSCRIPT

We will also use the notation

S n (f ) = n k=0 |df k | 2 1/2 for n = 0, 1, 2, . . . and write ||f || p = sup n ||f n || p , ||f || p,∞ = sup n sup λ>0 λ(P(|f n | ≥ λ)) 1/p
for the strong and weak p-th norm of f .

A martingale f is conditionally symmetric, if for any n ≥ 1 the conditional distributions of df n and -df n given F n-1 coincide. For example, this is the case if f is a dyadic martingale. To recall what it means, let (h n ) n≥0 be the system of Haar functions on [0, 1]. Then f is dyadic if for some a 0 , a 1 , a 2 , . . . ∈ H we have

f n = n k=0 a k h k for n ≥ 0.
The problem of comparing the sizes of f and S(f ) is classical and goes back to 30's and the works of Khintchine, Littlewood, Marcinkiewicz, Paley and Zygmund (clearly, the concept of a martingale was not used there; the results concerned the partial sums of Rademacher and Haar series). Consider the inequality

a p ||S(f )|| p ≤ ||f || p ≤ A p ||S(f )|| p , (1)
to be valid for all conditionally symmetric martingales f . As shown by [START_REF] Burkholder | Martingale transforms[END_REF], for any 1 < p < ∞ there are finite universal a p and A p such that the double inequality above holds. It follows from the results of [START_REF] Burkholder | Extrapolation and interpolation of quasi-linear operators on martingales[END_REF] that the right inequality above holds also for 0 < p ≤ 1 with some absolute A p . What about the optimal values of a p and A p ? Let ν p be the smallest positive zero of the confluent hypergeometric function and let µ p be the largest positive zero of the parabolic cylinder function of parameter p. [START_REF] Wang | Sharp square-function inequalities for conditionally symmetric martingales[END_REF] showed that a p = ν p for p ≥ 2, A p = ν p for 0 < p ≤ 2 and A p = µ p for p ≥ 3 are the best choices, even if we restrict ourselves in (1) to dyadic martingales. For the remaining values of parameter p, the optimal constants are not known. When p = 1, the left inequality in (1) does not hold with any universal a 1 < ∞. However, [START_REF] Bollobás | Martingale inequalities[END_REF] established the weak type inequality

||S(f )|| 1,∞ ≤ exp(-1/2) + 1 0 exp(-s 2 /2)ds ||f || 1 = 1.4622 . . . ||f || 1
and [START_REF] Osȩkowski | On the best constant in the weak type inequality for the square function of a conditionally symmetric martingale[END_REF] proved it is sharp. The purpose of this paper is to prove the following related result.

Theorem 1.1. For any conditionally symmetric martingale f we have

||f || p,∞ ≤ C p ||S(f )|| p , 1 ≤ p ≤ 2, (2)
where

C p p = 2 1-p/2 π p-3/2 Γ((p + 1)/2) Γ(p + 1) 1 + 1 3 2 + 1 5 2 + 1 7 2 + . . . 1 -1 3 p+1 + 1 5 p+1 -1 7 p+1 + . . . .
The constant C p is the best possible, even for the class of real dyadic martingales.

Unfortunately, our approach works only for 1 ≤ p ≤ 2 does not allow to obtain the best constants in the weak type estimates for p ∈ (0, 1) ∪ (2, ∞). The proof is based on Burkholder's technique: in the next section we introduce a special function and study its properties, which will be exploited in Section 3, where we establish Theorem 1.1.

ACCEPTED MANUSCRIPT

A special function and its properties

We will use the notation H = R × (0, ∞), S = R × (-1, 1) and S + = (0, ∞) × (-1, 1). For 1 ≤ p ≤ 2, introduce a harmonic function A = A p : H → R, given by the Poisson integral

A(α, β) = 1 π ∞ -∞ β 2 π log |t| p (α -t) 2 + β 2 dt.
It is easy to see that the function A satisfies lim

(α,β)→(z,0) A(α, β) = 2 π p | log |z|| p , z = 0. ( 3
)
Consider a conformal mapping ϕ given by ϕ(z) = ie πz/2 , or, in the real coordinates,

ϕ(x, y) = -e πx/2 sin π 2 y , e πx/2 cos π 2 y , (x, y) ∈ R 2 .
It can be easily verified that ϕ maps S onto H. Let A = A p be defined on the strip S by A(x, y) = A(ϕ(x, y)). Then the function A is harmonic on S, since it is a real part of an analytic function. By (3), we can extend A to the continuous function on the closure S of S by A(x, ±1) = |x| p . One easily checks that for (x, y) ∈ S,

A(x, y) = 1 π R cos π 2 y 2 π log |s| + x p (s -sin( π 2 y)) 2 + cos 2 ( π 2 y) ds (4)
for |y| < 1. Substituting s := 1/s and s := -s above, we see that A satisfies A(x, y) = A(-x, y) = A(x, -y) for (x, y) ∈ S.

(5)

Finally, let U = U p : [0, ∞) × R → R be given by U (x, y) = x p for |y| > 1 and

U (x, y) = c p R A(ux, y) exp(-u 2 /2)du otherwise; here c p = R |u| p exp(-u 2 /2)du -1 = 2 (p+1)/2 Γ p+1 2 -1
. Clearly, U is continuous and, by (5), we have

U (x, y) = 2c p ∞ 0 A(ux, y) exp(-u 2 /2)du on S + . (6)
Let us study the propertes of the function U which will be needed later.

Lemma 2.1. We have

U (0, 0) = C -p p .
Proof. This is straightforward: since π 2 /8 = ∞ k=0 (2k + 1) -2 , we have

U (0, 0) = c p √ 2πA(0, 0) = 2 -p/2 Γ p+1 2 √ π ∞ -∞ 2 π log |s| p s 2 + 1 ds ACCEPTED MANUSCRIPT = 2 1+p/2 π p+1/2 Γ p+1 2 ∞ 0 |log s| p s 2 + 1 ds = 2 1+p/2 π p+1/2 Γ p+1 2 ∞ -∞ |s| p e s e 2s + 1 ds = 2 2+p/2 π p+1/2 Γ p+1 2 ∞ 0 s p e -s ∞ k=0 (-e -2s ) k ds = 2 2+p/2 Γ(p + 1) π p+1/2 Γ p+1 2 ∞ k=0 (-1) k (2k + 1) p = C -p p .
The proof is complete.

Lemma 2.2. (i) The function U satisfies the differential equation

U x (x, y) + xU yy (x, y) = 0 on S + . (7)
(ii) The function U is superharmonic on S + .

Proof. By Fubini's theorem, we may take the derivatives inside the integral while computing the partial derivatives of U on S + . (i) Since A is harmonic, we have, for (x, y) ∈ S + ,

xU yy (x, y) = x R A yy (ux, y) exp(-u 2 /2)du = - R xA xx (ux, y) exp(-u 2 /2)du
and the claim follows from the integration by parts: the above is equal to

- R A x (ux, y)u exp(-u 2 /2)du = -U x (x, y).
(ii) By the previous part, the assertion can be rewritten in the form

xU xx (x, y) -U x (x, y) ≤ 0 on S + .
Since U x (0+, y) = 0, we will be done if we show that U xxx ≤ 0 or, by (6), A xxx ≤ 0 on S + . To this end, fix x > 0, ε ∈ (0, x) and introduce the function

f ε (h) = 2|h| p-2 h -|h -ε| p-2 (h -ε) -|h + ε| p-2 (h + ε), h ∈ R.
One easily verifies that f ε is odd and

f ε ≥ 0 on [0, ∞). (8)
We write

2A x (x, y) -A x (x -ε, y) -A x (x + ε, y) = p π ∞ -∞ f ε x + 2 π log |s| cos( π 2 y) (s -sin( π 2 y)) 2 + cos 2 ( π 2 y) ds,
split the integral into two, over the nonpositive and nonnegative halfline, and, finally, substitute s = ±e r . As the result, we get

2A x (x, y) -A x (x -ε, y) -A x (x + ε, y) = p π ∞ -∞ f ε x + 2 π r g y (r)dr, ( 9
)
where the function g y is given by

g y (r) = cos( π 2 y)e r (e r -sin( π 2 y)) 2 + cos 2 ( π 2 y) + cos( π 2 y)e r (e r + sin( π 2 y)) 2 + cos 2 ( π 2 y)
.

ACCEPTED MANUSCRIPT

Note that g y is even and nonincreasing on [0, ∞): indeed, for r > 0,

(g y ) ′ (r) = cos( π 2 y)e r (1 -e r ) [(e r -sin( π 2 y)) 2 + cos 2 ( π 2 y)] 2 + cos( π 2
y)e r (1 -e r ) [(e r + sin( π 2 y)) 2 + cos 2 ( π 2 y)] 2 ≤ 0. Thus, by ( 8), the integral in ( 9) is nonnegative and, since ε ∈ (0, x) was arbitrary, the function A x is concave on S + . Lemma 2.3. (i) For any (x, y) ∈ [0, ∞) × R we have

x p ≤ U (x, y) ≤ x p + U (0, 0)1 {|y|<1} . ( 10
) (ii) We have U x (x, y) ≥ 0 on (0, ∞) × R and U y (x, y) ≤ 0 on (0, ∞) × ((0, ∞) \ {1}).
Proof. We may assume that |y| < 1, since otherwise the claim is obvious, both in (i) and (ii). The lower bound in (10) follows from Jensen's inequality: we have

A(x, y) = ∞ -∞ 2 π log |s| + x p • 1 π cos( π 2 y) (s -sin( π 2 y)) 2 + cos 2 ( π 2 y) ds ≥ 1 π ∞ -∞ cos( π 2 y) 2 π log |s| + x (s -sin( π 2 y)) 2 + cos 2 ( π 2 y) ds p = x p
(to see the latter equality, make the substitution s := 1/s) and it suffices to apply ( 6). Now we turn to (ii). Similar argument gives that the function A(•, y) is convex and hence the same is valid for U . Therefore, by part (ii) of Lemma 2.2 we have U yy ≤ 0 and (ii) follows. Indeed, U x ≥ 0 by the first part of that lemma and U y ≤ 0 for y > 0, since the function U (x, •) is even (see ( 5) and ( 6)). Thus we have shown (ii); moreover, we see that it suffices to establish the upper bound in (10) for y = 0. Using an elementary inequality |a

+ b| p + |a -b| p ≤ 2|a| p + 2|b| p for a, b ∈ R yields 2A(x, 0) = A(x, 0) + A(-x, 0) = 1 π R 2 π log |s| + x p + 2 π log |s| -x p s 2 + 1 ds ≤ 1 π R 2 2 π log |s| p + 2|x| p s 2 + 1 ds = 2A(0, 0) + 2|x| p .
It suffices to use (6) to get the claim.

In the lemmas below, we will use the following notation. For a ∈ H, let a ′ = a/|a| if a = 0 and a ′ = 0 otherwise; furthermore, a * = a for |a| ≤ 1 and a * = a/|a| otherwise.

* | ≤ |a + b|. Proof. If both |a|, |b| do not exceed 1, the claim is obvious. If |a| > 1 ≥ |b|, |a + b| 2 -|a * + b * | 2 = |a| 2 -1 + 2 a, b (1 -|a| -1 ) ≥ |a| 2 -1 -2|a|(1 -|a| -1 ) ≥ 0 and similarly for |b| > 1 ≥ |a|. Finally, if |a| > 1 and |b| > 1, then |a + b| 2 -|a * + b * | 2 = |a| 2 + |b| 2 + 2 a, b (1 -(|a||b|) -1 ) -2 ≥ |a| 2 + |b| 2 -2|a||b| ≥ 0, as desired.

ACCEPTED MANUSCRIPT

The key ingredient of the proof of this estimate is the fact that the process (U (S n (f ),

|f n |)) n≥0 is an (F n )-submartingale. (12)
Indeed, for n ≥ 0 the variable U (S n (f ), |f n |) is integrable by the condition S(f ) ∈ L p and (10). Moreover, by the conditional symmetry, 2E(U (S n+1 (f ),

|f n+1 |)|F n ), n ≥ 0, equals E U ((S 2 n (f ) + |df n+1 | 2 ) 1/2 , |f n + df n+1 |) + U ((S 2 n (f ) + |df n+1 | 2 ) 1/2 , |f n -df n+1 |) F n .
This is not smaller than 2U (S n (f ), |f n |): apply ( 11 Sharpness. Suppose that γ p is the optimal constant in (2) for real-valued dyadic martingales. Arguing as in [START_REF] Osȩkowski | On the best constant in the weak type inequality for the square function of a conditionally symmetric martingale[END_REF], this yields a corresponding weak type inequality

P(|B τ | ≥ 1) ≤ γ p p Eτ p/2 , ( 13
)
where B is a standard Brownian motion and τ is any stopping time of B. On the other hand, let η = inf{t : |B t | = 1} and consider the process (U ( √ η ∧ t, B η∧t )) t≥0 .

By ( 7) and Itô's formula, it is a martingale with expectation equal to U (0, 0). By (10) and exponential integrability of η, this martingale converges almost surely and in L 1 to η p/2 , which, by Lemma 2.1, yields C -p p = Eη p/2 and, consequently, 1 = P(|B η | ≥ 1) = C p p Eη p/2 . By (13), this implies γ p ≥ C p and completes the proof.

Lemma 2. 4 .

 4 For any a, b ∈ H we have |a * + b

  ) with x = S n (f ), y = f n and d = df n+1 . Thus (12) follows and, using (10) and Remark 2.1, we getU (0, 0) ≤ EU (S 0 (f ), |f 0 |) ≤ EU (S n (f ), |f n |) ≤ ES p n (f ) + U (0, 0)P(|f n | < 1). This completes the proof, by virtue of Lemma 2.1.

Acknowledgement

I would like to thank Professor Stanis law Kwapień for bringing this problem to my attention. The author was supported in part by MNiSW Grant N N201 397437.

ACCEPTED MANUSCRIPT

Lemma 2.5. For any (x, y) ∈ [0, ∞) × H and any d ∈ H we have 2U (x, |y|) ≤ U (x 2 + |d| 2 ) 1/2 , |y + d| + U (x 2 + |d| 2 ) 1/2 , |y -d| .

(11)

Proof. It is convenient to split the proof into three parts.

Case 1: |y| ≥ 1. Then the estimate is trivial: indeed, by Lemma 2.3,

, where x t + = (x 2 + t 2 |d| 2 ) 1/2 and y t ± = y ± td. We have that ψ ′ (t)/|d| equals

(when |y + td| = 0, the differentiation is allowed since U y (x, 0) = 0). We will prove that this is nonnegative, which will clearly yield the claim. It suffices to show that

To this end, note that if we square both sides, the estimate becomes This follows from the fact that U yy is nonpositive and concave: by Lemma 2.2, we have x 2 U yyyy (x, y) = U xx (x, y) + U yy (x, y) ≤ 0 for (x, y) ∈ S + . The case (y t + ) ′ = -(y t -) ′ is dealt with in the same manner.

Case 3: |y| < 1, d > 1 -|y|. This can be reduced to the previous case. Set y + = (y + d) * , y -= (y -d) * and ỹ = (y + + y -)/2, d = (y + -y -)/2, x = (x 2 + |d| 2 -d2 ) 1/2 . By Lemma 2.4, |ỹ| ≤ |y| and | d| ≤ |d|, so x ≥ x. Now, using Lemma 2.3 (ii) and the fact that ỹ, d satisfy the assumptions of Case 2, we may write 2U (x, y) ≤ 2U (x, ỹ) ≤ U ((x 2 + d2 ) 1/2 , y + ) + U ((x 2 + d2 ) 1/2 , y -) and the latter sum is precisely the right hand side of (11).

Remark 2.1. The choice x = 0, y = 0 in (11) gives U (0, 0) ≤ U (d, d) for all d ≥ 0.

Proof of Theorem 1.1

Proof of (2). We may assume that S(f ) ∈ L p , since otherwise there is nothing to prove. By homogeneity, it suffices to show that for any n ≥ 0, P(|f n | ≥ 1) ≤ C p p ES p n (f ).