Y Laridon

D Grenier

C Doursat

D Huc

N Roland

D Flick

T Lucas

Growth of a single bubble in semi-hard cheese: Comparison between simulation and experiment

Keywords:

This paper proposes a model for bubble growth in semi-hard cheese coupling mechanical behaviour and mass transport. The modelling follows previous work centred on the mechanical aspects, and focuses in this paper on the mass transport phenomena. Data are compared to experimental results obtained on industrial-size cheeses, both under the rind and at core, and a sensitivity study is conducted to discuss the results. The model is in agreement with experiment at core, and underlines the great influence of the carbon dioxide production rate and the amount of cheese material surrounding the bubble on bubble growth. Under the rind, the model yielded poorer agreement, due to the fact that this region in the cheese is less homogeneous, and therefore with more intra-and inter-batch variation on the parameters that were characterized.

Introduction

Whilst bubbles are found in numerous food products [START_REF] Campbell | Creation and characterisation of aerated food products[END_REF], the modelling of bubble growth for foodstuffs was almost exclusively limited to the study of sparkling beverages or cereal products. The studies on the former are inspired by nucleation studies in liquids, while for the latter, the studies are inspired by the works on polymer foaming (e.g. the works of [START_REF] Amon | A study of the dynamics of foam growth: Analysis of the growth of closely spaced spherical bubbles[END_REF]. The dynamics involved for the growth of bubbles in cheese encourages considering the present study as a continuation of works on bread dough proving. In the following, the state of the art in the modelling of dough proving is presented, with its similarities and its limitations for application to the case of cheese ripening. The availability in the literature of values for the input parameters required for this modelling task is discussed for the cheese material and semi-hard cheese in particular.

Bubbles nucleation in aerated products usually requires two conditions: a supersaturated environment and a catalytic site [START_REF] Bisperink | Bubble growth in carbonated liquids[END_REF]. For weakly supersaturated products such as sparkling beverages, nucleation requires a catalytic site, either a pre-existing gas pocket (usually imperfection on the glass-wall) or the presence of an alien substance [START_REF] Jones | Bubble nucleation from gas cavities -A review[END_REF][START_REF] Liger-Belair | Modeling the kinetics of bubble nucleation in champagne and carbonated beverages[END_REF]. For semi-hard cheeses, the cheese material is indeed supersatured with carbon dioxide, though only slightly, and the catalytic site consists of gas pockets existing between curd grains [START_REF] Akkerman | Holes in Dutch type cheese. 1. Conditions allowing eye formation[END_REF]Huc, Mariette, et al., 2014). The complexity of bubble nucleation makes it a subject of its own and exceeds the scope of the present study, which focuses on the next stage, the bubble growth.

Few modelling studies on aerated cereal products went to the extent of characterising the growth in cereal products at the bubble scale [START_REF] De Cindio | Mathematical modelling of leavened cereal goods[END_REF][START_REF] Fan | A model for the oven rise of dough during baking[END_REF][START_REF] Hailemariam | A mathematical model for the isothermal growth of bubbles in wheat dough[END_REF][START_REF] Shah | Proving of bread dough: Modelling the growth of individual bubbles[END_REF], this selection encompassing both modelling studies on proving [START_REF] De Cindio | Mathematical modelling of leavened cereal goods[END_REF][START_REF] Shah | Proving of bread dough: Modelling the growth of individual bubbles[END_REF] and baking. The dough is often thought of as a purely viscous material, but some studies considered viscoelastic models. De [START_REF] De Cindio | Mathematical modelling of leavened cereal goods[END_REF] considered the dough with a linear viscoelastic model very similar to that used in the present study. They showed that, contrary to the viscous models, the use of viscoelastic constitutive equations allowed to reproduce cases that had asymptotic bubble volumes due to the timeindependent elastic properties of their viscoelastic modelling. Some other studies considered non-linear viscoelastic models, such as the Lodge model used by [START_REF] Hailemariam | A mathematical model for the isothermal growth of bubbles in wheat dough[END_REF], that give a better description of the material behaviour, but necessitate a more timeconsuming evaluation of the input parameters (the Lodge model enforces the knowledge of memory function for the material). They found that viscoelasticity of the dough tends to prevent bubbles from collapsing.

Mass transfer was proved to be the main contributing factor to bubble growth during dough proving [START_REF] De Cindio | Mathematical modelling of leavened cereal goods[END_REF][START_REF] Hailemariam | A mathematical model for the isothermal growth of bubbles in wheat dough[END_REF]. Mass transport in the liquid phase can be modelled by Fick's law and applied to the liquid water in the case of dough proving; it should be extended to both the liquid water and liquid fat in the case of cheese [START_REF] Jakobsen | Assessment of carbon dioxide solubility coefficients for semihard cheeses: The effect of temperature and fat content[END_REF]. Exchanges at the bubble-material interface are often described using Henry's law. [START_REF] Shah | Proving of bread dough: Modelling the growth of individual bubbles[END_REF] discussed the influence of the carbon dioxide saturation on the bubble growth, and showed that for subsaturation regime, bubbles were to have an asymptotic size, that was influenced mainly by the initial bubble size and carbon dioxide concentration. For supersaturation regime, they showed that above a critical size, bubble growth would continue indefinitely. No asymptotic bubble size could be observed in the type of cheese studied in the present paper (Huc, Mariette, et al., 2014), evidencing that supersaturation also happens in the semi-hard cheese under study.

Despite the similarity of the driving phenomena between bread dough proving and cheese ripening, some differences have to be underlined. The rate of dough proving in bread manufacture is substantially faster (several hours) than eye formation in semi-hard cheese (several weeks). This discrepancy in the kinetics may be explained by differences in the microbiological phenomena involved. Carbon dioxide production in semi-hard cheese is caused by propionic fermentation, whereas it is caused by alcoholic fermentation for bread dough, and the optimal conditions of carbon dioxide production, in terms of temperature or pH for instance, highly differ between the two processes. The bubble growth kinetics may also be influenced by the mechanical behaviour of the material surrounding the bubbles. Although both materials (dough and cheese) are viscoelastic materials with similar relaxation time spectrum [START_REF] Keentok | The rheology of bread dough made from four commercial flours[END_REF], viscoelasticity of cheese is one to two orders of magnitude higher than that of bread doughs [START_REF] Launay | Biaxial extension of wheat flour doughs: Lubricated squeezing flow and stress relaxation properties[END_REF]. Compared to about 10% for semi-hard cheeses at the end of ripening (Huc, Mariette, et al., 2014), bread dough is a highly porous product: if its porosity is of 20% at the end of mixing, it increases up to 70-90% at the end of baking. The bubbles in cheese are also typically a couple of centimetres in size at the end of ripening, larger than those in bread dough which are typically only a couple of millimetres in size, and for which surface tension effects are not negligible.

Experimental validation of the models focused on dough proving and baking, when implemented, often took place at a macroscopic level, by using overall morphologic descriptors (Chiotellis and Campbell, 2003;[START_REF] Fan | A model for the oven rise of dough during baking[END_REF]. Both characteristic size of bubble and their rate of growth in cheese are compatible with tomographic monitoring, hence offering for the first time the possibility to validate the modelling of bubble growth in food material at the scale of the bubble.

For appropriate validation, the model of transport has also to be fed with values for several input parameters, specific for mass transport and production and appropriate for semi-hard cheeses, such as carbon dioxide diffusivity, solubility and production rate. There are very few papers dedicated to the estimation of carbon dioxide diffusivity in semihard cheese. The notable exception is the study from [START_REF] Acerbi | An appraisal of the impact of compositional and ripening parameters on CO 2 diffusivity in semihard cheese[END_REF], conducted on the same type of semi-hard cheese as in the present paper, measured at various pH, salt content, moisture or cheese age. Numerous studies, however, were carried out on cheese in modified atmosphere packaging, and focused on the exchange of carbon dioxide at the product interface. These studies provide data on interfacial transfers for a wide variety of products, for instance the work on Swiss-type cheese by [START_REF] Blanc | Gas exchanges at the surface of Gruyere cheese during ripening[END_REF]; however, they fail to disentangle the respective influence of diffusion and production within the product by focusing on the interface only. [START_REF] Rodriguez-Aguilera | Gas exchange dynamics in modified atmosphere packaging of soft cheese[END_REF] and [START_REF] Vivier | Study of carbon dioxide release from Feta cheese[END_REF] successfully estimated carbon dioxide production rate, but their study was applied to soft cheeses, for which the fermentation mechanism differs from that of semi-hard cheeses. In semi-hard cheeses, the carbon dioxide production originates from propionic fermentation, but also from proteolysis (to a lesser extent, only 20%, ITFF-ITG/Actilait, 1984). Some studies were conducted to investigate the influence of several manufacturing parameters on bubble growth in cheese [START_REF] Guggisberg | Mechanism and control of the eye formation in cheese[END_REF][START_REF] Schuetz | Quantitative comparison of the eye formation in cheese using radiography and computed tomography data[END_REF], and while they successfully used noninvasive tomographic measures to discuss the effects of these parameters on the bubble volume or aspect, their approach remained empirical and did not provide a mechanistic model. Finally, some studies have estimated the carbon dioxide solubility in cheese, but on the whole cheese block, and only at the end of the ripening [START_REF] Jakobsen | Assessment of carbon dioxide solubility coefficients for semihard cheeses: The effect of temperature and fat content[END_REF][START_REF] Seuvre | Contribution to the Study of Gas-Release during the Maturation of a French Emmental Cheese[END_REF].

The purpose of the present paper was to implement a model of the coupled mechanical behaviour and mass transport in the case of a single bubble growth in semi-hard cheese. The study also aimed at identifying input parameters of the model that were the most influential on bubble growth; this part of the study was based on a compilation of literature data for input parameters but also on the measurement of some specific ones, such as carbon dioxide production rate, concentration of CO 2 in cheese at saturation and mechanical properties of the cheese material. At last, the study aimed at comparing the simulations of this model to experimental data. The bubble growth was monitored by MRI (Magnetic Resonance Imaging) and X-Ray Imaging on industrial cheese blocks during ripening (warm room). The mechanics involved in the bubble growth were already validated on a separate experiment involving known pressure forces (Y. [START_REF] Laridon | Modelling of the growth of a single bubble in semi-hard cheese, with experimental verification and sensitivity analysis[END_REF], allowing the focus of the present study on mass transport and its coupling with mechanical behaviour.

Experimental procedure and data analysis

For experimental validation of the model, cheese blocks were taken from the plant at the end of the brining step and ripened in the lab at two successive controlled temperatures: 12 °C (for 8-10 days) and 20 °C (for 15 days). Temperature was maintained constant using maturing cabinets (Grand Cru, Liebherr, France). For all the followings, the initial time = d 0 refers to the time at which the temperature of the ripening rooms was changed from 12 °C to 20 °C.

At each time chosen in the ripening process (when changing ripening temperature, and then at day 4, 5, 7, 8, 11, 12 and 15, depending on the experimental runs), one cheese block was taken out of the maturing cabinet and images were acquired, either by X-ray or MRI imaging (Table 1), on the whole cheese block, following the method developed by [START_REF] Musse | MRI method for investigation of eye growth in semi-hard cheese[END_REF]. For each experimental campaign, the time-course changes in bubble volume were averaged in two separate regions of interest (ROI) that are under the rind and at core of the cheese blocks. The dimensions of these regions were 16 × 16 × 1.5 cm, the first one being located 1 cm under the rind and the second 4 cm. Since the position of the rind evolves with the inflation of bubbles during ripening, the location of these regions changed with respect to the fixed referential. Bubbles were considered to be in a given ROI if their barycentre belonged to it. Bubbles that had volumes lower than 10 mm 3 and presented no growth during ripening were omitted. Standard deviation was of great magnitude (ranging from 8 to 150% over all the ripening), representative of the high variability in bubble size and growth kinetics within the cheese block. More details about the image analysis can be found in Huc, Mariette, et al., 2014. Experimental conditions are summarised in Table 1.

Model description

Except when stated, values are given at 20 °C and 101,325 Pa.

Geometry

In order to minimise computational cost, modelling was restricted to a single, spherical bubble surrounded by a spherical volume of cheese (representative volume of cheese around one bubble).

Hypotheses

Cheese was considered nearly-incompressible, with a Poisson ratio ν of 0.49. Its composition (other than CO 2 ), mechanical and diffusive properties were considered homogeneous at the scale considered. Carbon dioxide was also considered to be produced homogeneously. Within the bubble, pressure was considered uniform and gases other than carbon dioxide were neglected. Temperature was considered uniform and constant.

Mechanical behaviour

The behaviour of the cheese was modelled with a 5-element generalised Maxwell model described earlier in a dedicated paper [START_REF] Laridon | Identification of broad-spectrum viscoelastic parameters: Influence of experimental bias on their accuracy and application to semihard-type cheese[END_REF]. The mechanical equilibrium was expressed through the conservation of momentum:

∇ = σ g ρ • (1)
where σ is the stress, ρ the density and g the gravitational acceleration.

Stress was calculated as the sum of elastic and viscoelastic parts and the viscoelastic part was described by the Maxwell model.

Mass transport

Mass diffusion was described according to Fick's law; mass conservation writes as follows:

∂ ∂ - = c t D c r Δ CO ch CO 2 2 (2)
where c is the number of moles of carbon dioxide in the water and fat phase of cheese by cubic meter of cheese, later denoted as the apparent carbon dioxide concentration in the domain, D CO ch 2 the apparent diffusivity of carbon dioxide in cheese (Section 4.2) and r CO2 the carbon di- oxide production rate within the cheese domain (Section 4.1). Carbon dioxide mass transport properties (solubility and diffusion) were considered of the same order of magnitude in the fat phase and water phase [START_REF] Jakobsen | Assessment of carbon dioxide solubility coefficients for semihard cheeses: The effect of temperature and fat content[END_REF].

The equilibrium at the gas-cheese interface was expressed according to Henry's law:

= c k p i H ch i (3)
where c i is the apparent carbon dioxide concentration at the inter- face, p i the pressure exerted by the gas on the interface and k H ch , Henry's constant for carbon dioxide in cheese.

The atmospheric pressurep atm is used as the pressure of reference in the mechanical aspects.

Coupling of transport phenomena

The carbon dioxide flux at the gas-cheese interface j i can be written as:

= - ∇ n j D c• i CO ch i 2 (4)
where n is the normal vector of the interface. This flux was numerically computed as:

= - - j γ c k p ( ) i i H ch i (5)
where γ is the transfer coefficient at the interface. To ensure that the effect of diffusion in the cheese was limiting, the Biot number for mass transport was set at a high value (10 5 ):

= Bi γL D m CO ch 2 (6)
This implied that γ was set at a high value and assured conditions very close to equilibrium ( → ∞ γ leads to Eq. ( 3)). Characteristic length L was set at 10 mm, as it was the typical length between the gas-cheese interface and the boundaries of the domain (Fig. 1).

Time-course change in the quantity of CO 2 in the gaseous phase, dn dt b , was deduced from the CO 2 flux:

∫ = dn dt j dS b i i Γi (7)
where Γ i is the gas-cheese interface boundary and S i its surface.

Using Eqns. ( 5) and ( 7), and the ideal gas law, the following ODE was solved at the gas-cheese interface:

∫ = - - dn dt γ cdS γk RT S V n b i i H ch i b b Γi (8)
where V b is the volume of the gaseous phase (bubble). The quantity of CO 2 determined with Eq. ( 8) was used to calculate the pressure of the gas at the gas-cheese interface with the cheese domain, following the ideal gas law:

= p n RT V i b b (9)
The pressure calculated using Eq. ( 9) was used as mechanical boundary condition at the gas-cheese interface (Eq. ( 10)), and therefore constitutes the coupling between mechanical behaviour and mass transport. 3.6. Boundary conditions 3.6.1. Mechanical boundary conditions Pressure calculated using Eq. ( 9) was applied at the bubble-cheese interface:

= -- σ n p p • ( ) i i a t m (10)
where σ i is the stress applied to the gas-cheese interface.

The other boundary was left free of stress.

Mass transfer boundary conditions

There was no flux of CO 2 at the outer boundary of the cheese sphere, except at the interface with the bubble, where the flux calculated by Eq. ( 5) was used.

Under these conditions, the growth of a bubble is considered with no mechanical constraint other than that exerted by the cheese material itself (restricted at the boundaries of the cheese sphere which contents a representative volume of cheese around on bubble). Mechanical interactions between adjacent bubbles were not reproduced. All the CO 2 produced in the cheese domain diffuses toward the bubble; competition for CO 2 between adjacent bubbles, or large-scale diffusion between regions of low and high CO 2 content at the cheese block scale were not taken into account.

Initial conditions

It is considered that at a time denoted by t sat the cheese is saturated in CO 2 . All the simulations began at this saturation time, with t defined as = -t t days t ( ) sat . The initial bubble radius r b in the simulation (2.17 mm) was set at the average experimental value observed at that time . The volume of cheese surrounding the bubble was also de- termined in average at that time (4.2 ± 2.0 cm 3 ).

Mechanical initial conditions

Equilibrium at the atmospheric pressure was considered,

= = p t p ( 0) i a t m
(11)

Mass transfer initial conditions

Following the initial saturation hypothesis, initial carbon dioxide concentration within the domain was defined as:

= = c t k p ( 0) H ch atm (12)
The initial CO 2 quantity in the bubble was calculated with the help of ideal gas law:

= = = n t p V t RT ( 0) ( 0) b atm b
(13)

Numerical implementation and calculation

The model was implemented in COMSOL Multiphysics (COMSOL AB, Sweden). Simulations were conducted from the time at which saturation in CO 2 is reached to the end of warm room ripening (15 days).

Estimation of values for input parameters

Carbon dioxide production rate

Carbon dioxide production rates, r CO2 , were measured on the same type of cheese as that used in the experiment, both at core and under rind (same locations as mentioned in Section 2 for the analysis of bubble volume). Cheese blocks used for these measurements were from different batches than those studied according to the protocol defined in Section 2 (data for validation of the model); different batches were used to characterise the inter-batch variability.

For each batch, about 20 cheese discs (2 mm thick at most) were deposited in a sealed bottle onto small grids of large mesh and separated from the other discs by a sustaining stand. The full device is detailed in [START_REF] Huc | Study of the influence of salt content on eye growth in semi-hard cheeses using MRI and CO 2 production measurements[END_REF].

The bottles were initially filled with N 2 at atmospheric pressure, and placed in a temperature-controlled environment at 20 °C. Nitrogen recreated the anaerobic conditions of bacterial growth during cheese ripening and also avoid the growth of moulds.

The pressure of the gas within the bottle, p t ( ), was monitored over duration relevant of the ripening process (warm room). The time-course changes in pressure were attributed to the production of carbon dioxide only. Measurements were assumed to be diffusion-independent, because of the very low thickness of the cheese discs. The number of moles of CO 2 produced at a given time per volume of cheese was deduced from the pressure, using the ideal gas law: cheese volume is presented in Fig. 2. At the very beginning of the experiment (until one day at most), a slight decrease in pressure could be observed and was attributed to the entrance of nitrogen into the cheese.

= - Produced CO ρ m p t p V RT ( (
The carbon dioxide production rate was not constant over time. Production rates were calculated by linear regression of experimental data obtained between days 2 and 4 for the first stage, and between days 7 and 14 for the second stage. Mean values reported in Table 2 were calculated on 5-7 experimental runs, 2 replicates each. Hence the associated standard error was relevant to intra-and inter-batch variability. The time at which the break point between stages I and II was observed, t r , was equal to 5.5 ± 1.9 days, taking into account both under rind and core data.

At the beginning of warm room (until 6 days in Fig. 2) the production rate was 2 times lower than afterwards, both at core and under rind (Table 2). The effect of several factors onto the rate of production was investigated experimentally in the literature, among them the impact of salt content, and moisture content. Cheeses without salt yielded production rates that were up to 2.6 times as much as standard industrial cheeses [START_REF] Huc | Study of the influence of salt content on eye growth in semi-hard cheeses using MRI and CO 2 production measurements[END_REF], whereas for the two moisture contents considered (42 and 46%), there was no impact on the production rate. However, for cheeses that had the highest moisture content, production of carbon dioxide occurred 3 days after the change of room temperature.

Carbon dioxide diffusivity

Diffusivity of carbon dioxide in cheese D CO ch 2 was calculated from carbon dioxide diffusivity in water D CO w 2 [START_REF] Davidson | The determination of diffusion coefficients for sparingly soluble gases in liquids[END_REF]; the same value was applied to the liquid fat phase. Diffusivity of CO 2 in water was then weighted by the volume fraction of water in cheese X w and the volume fraction of non-crystallised fat X fat , and the tortu- osity of the cheese material T :

T = + D X X D ( ) CO ch w f a t CO w 2 2 (15)
Tortuosity was assumed to be at π/2 due to the presence of protein and fat globules. X w was set at 0.42, considering that all the water present in the cheese material is available for CO 2 diffusion, and X fat was set at 0.15, considering that only 55% of the fat phase was not crystallised. This yielded D CO ch 2 = 6.09•10 -10 m 2 s -1 , compared to 1.68•10 -9 m 2 s -1 for D CO w 2 . It should be noted that this assessment method is consistent with the order of magnitude of the values provided by [START_REF] Acerbi | An appraisal of the impact of compositional and ripening parameters on CO 2 diffusivity in semihard cheese[END_REF] on similar cheeses, ranging between 1 and 9•10 -10 m 2 s -1 , depending on the factor studied (salt and moisture content, age, PTA/TN).

Carbon dioxide concentration in cheese and saturation

The values below were considered in cheese material and in pure water.

CO 2 concentration at saturation can be estimated from Henry's constant value reported in the literature. Temperature dependency of k H w is given by:

⎜ ⎟ ⎜ ⎟ = ⎛ ⎝ - ⎛ ⎝ - ⎞ ⎠ ⎞ ⎠ k k T H R T T ( )exp Δ 1 1 H w H w std std (16)
where T std = 25 °C, T is the temperature of the warm room in the ri- pening process (20 °C), and H Δ the standard enthalpy set at 1.995•10 -4 J•mol -1 [START_REF] Sander | Compilation of Henry's law constants, version 3.99[END_REF], R is the ideal gas constant (8.314 J•mol -1 •K -1 ).

Henry's constant equivalent for the cheese considered can be determined from the Henry's constant for CO 2 in pure water, using Eq. ( 18):

= + k X X k ( ) H ch w f a t H w ( 17
)
Total pressure within the bubbles is of the same order of magnitude than the atmospheric pressure, with a slight overpressure (about 2%, [START_REF] Grenier | Monitoring of single eye growth under known gas pressure: Magnetic resonance imaging measurements and insights into the mechanical behaviour of a semi-hard cheese[END_REF], and the bubbles contain mainly carbon dioxide. CO 2 concentration at saturation in pure water s w under the same conditions is given by:

= s p k w atm H w ( 18
)
where k H w is Henry's constant value. This yielded a value of 29.7 mol of CO 2 per m 3 of pure water at 20 °C, which is in accord with the order of magnitude of experimental results obtained on similar cheese by Acerbi, Guillard, Guillaume, and Gontard (2016). CO 2 concentration expected at saturation in the water and fat phase in cheese s ch was found to be equal to 16.87 mol of CO 2 per m 3 of cheese at core, and 16.29 under rind.

CO 2 concentration values were assessed experimentally on the same type of cheese used in this study, following a method similar to that used in [START_REF] Jakobsen | Assessment of carbon dioxide solubility coefficients for semihard cheeses: The effect of temperature and fat content[END_REF]. These measurements showed that, when entering the warm room, CO 2 concentration exceeded or was very close to this theoretical saturation value. Concentration reached a value of 20.6 ± 1.5 and 16.1 ± 1.5 mol of CO 2 per m 3 of cheese at core and under the rind, respectively. However, simulations were still maintained with an initial CO 2 concentration in the domain at the theoretical saturation value in order to ensure mass equilibrium between the gas and liquid phases at the start of the simulations.

Mechanical properties of cheese

Values of the parameters of the Maxwell model were fixed according to the average values that were determined on the same type of semicheese than the one under study (Table 2). See [START_REF] Laridon | Modelling of the growth of a single bubble in semi-hard cheese, with experimental verification and sensitivity analysis[END_REF] for further details about the mechanical test and the data analysis.

Results and discussion

Simulation with reference values of input parameters at the core of the cheese block

Fig. 3 presents the growth of a single bubble simulated with the numerical model using reference values of input parameters characteristic of the core of the cheese block (Table 3); the simulated data are also compared to the experimental one from three runs, averaging the behaviour of about 10 bubbles each, located at the cheese block core. The simulation agreed with the experiment, in a qualitative manner. Despite the attention given to the estimation of input parameters, there was discrepancy between simulation and experiment at short ripening times.

Hence, a sensitivity study was performed in order to characterise the relative influence of the model inputs on the outputs i.e. the bubble volume.

Sensitivity study of the model to input parameters characterised at core

The primary objective of the sensitivity study was to evaluate which input parameter was more prone to adjustment for better fitting of the Table 2 Rate of CO 2 production in cheese (mol•m -3 •s -1 ); standard deviations were calculated on 5 and 7 experimental curves for Stage I and Stage II (respectively), with two replicates each, and were attributed to the intra-and interbatch variability. experimental trend, and the secondary objective was to highlight the parameters that could be modified in the conduction of the process for enhancing bubble growth for instance. Table 3 summarises the upper and lower values around the reference value for each input parameter in this sensitivity study. Each parameter was varied separately, all other parameters being kept at their reference values.

For the mechanical aspects, only the relaxation properties of the last Maxwell element were varied, as they proved to be the most influential parameters on bubble growth (Y. [START_REF] Laridon | Modelling of the growth of a single bubble in semi-hard cheese, with experimental verification and sensitivity analysis[END_REF]; their variations around the reference value (Table 3) corresponded to the inter-batch variability ( ± σ). A ± 75% sweep was used for the diffusivity of CO 2 within the cheese, allowing covering the range of variation reported for this type of measurements in [START_REF] Acerbi | An appraisal of the impact of compositional and ripening parameters on CO 2 diffusivity in semihard cheese[END_REF]. Production rate was swept according to the inter-batch variability estimated in Section 4.1, taking into account the highest standard deviation obtained between stages I and II. The volume of cheese surrounding the bubble was also swept in order to take into account both intra-and inter-batch variability. For further discussion of the results, an additional simulation was run with a non-limiting value (20 cm 3 ).

The variations of initial bubble volumes were representative of extreme behaviour encountered in the experiments.

In order to rank the most influential parameters on bubble growth, sensitivity indices I k were calculated for each input parameter X k as:

= + + - - - - I 100 k V X V X V X V X V X V X X X X X X X ( ) ( ) ( ) ( ) ( ) (
) b k max b k ref b k ref b k min b k ref b k ref k max k ref k ref k min k ref k ref (19)
The sensitivity indices were assessed at the end of the simulations (15 days) and reported in Table 3. Simulated growth of bubbles obtained with the numerical model using some of these values of input parameters were also compared to the experimental data (Fig. 4).

The mechanical parameters of the 5th Maxwell element proved to be the least influential parameters over bubble growth, having sensitivity indices at least 60 times lower than the most influential parameter (Table 3). Values of mechanical parameters typically encountered under rind did not even oppose much more resistance to the bubble growth (data not reported). Initial bubble volume proved to have little influence over the bubble growth (Table 3) though it must be noted that the smallest bubbles benefited from a higher surface-to-volume ratio, and subsequently yielded higher volumes at the end of ripening. Henry's constant had also a small impact, confirming that the rough estimate taken from the literature for this study did suffice.

Besides the cheese volume surrounding the bubble, parameters related to the mass transport or production showed the greatest influence on the bubble volume.

Simulated results were consistent with the experimental data, with an overestimation at short ripening times (Fig. 3). However, this overestimation could be also balanced if considering lower cheese volume available around the bubble (Fig. 4a), lower production rate (Fig. 4b) or lower diffusivity (Fig. 4c).

The carbon dioxide production rate in cheese proved to be nearly as important for the bubble growth as the cheese volume (Table 3). High sensitivity of bubble growth to carbon dioxide production, combined to the high inter-batches variability of this parameter (about 30%), argued in favour of measuring it specifically on a cheese block from the same batch, if accurate validation of the model is required.

Carbon dioxide diffusivity in cheese was the only parameter that was not specifically determined on the cheese material under study but was deduced from the diffusivity in pure water (Section 4.2). Simulated bubble volume obtained with the lower value of diffusivity proved to reproduce results that were a better fit to experimental data, especially at the beginning of the ripening (Fig. 4c). Lower diffusivity than expected can be explained by the fact that all the water is not available for the carbon dioxide to diffuse or that the tortuosity of the material has been underestimated. Indeed, tortuosity was set at π/2 under the hy- pothesis that fat and protein were spherical, but because of the cheese structure complexity, the tortuosity value may be higher. This result, together with the high-ranking value of this input parameter in the sensitivity study also encourages further effort for experimental characterisation of carbon dioxide diffusivity in cheese. Let us remind from the introduction section that there is only one paper providing such data in the literature [START_REF] Acerbi | An appraisal of the impact of compositional and ripening parameters on CO 2 diffusivity in semihard cheese[END_REF].

In the experiment, the initial rate of bubble growth was low despite of CO 2 saturation (Fig. 4) and low levels of diffusivity were assumingly incriminated [START_REF] Huc | Study of the influence of salt content on eye growth in semi-hard cheeses using MRI and CO 2 production measurements[END_REF]. Results from the present study also showed a contribution of the lower CO 2 production for ripening times lower than 5-6 days (Fig. 2).

The relatively small effect of D CO ch 2 on bubble growth (nearly 6 times lower than the most influential parameter) was well illustrated in Fig. 4c, despite the wide variations of this input parameter. The system behaved as if the amount of CO 2 produced in cheese was limiting, which was consistent with the great effect of the increase in cheese volume surrounding the bubble (Fig. 4a). Suppl. Mat. 1 gives the evolution of the screening index over the simulation time and allows to comprehend the evolution of the influence of each parameter over time.

Considerations about the spatial variability of the model parameters

For this study, two regions of interest (ROI) were considered in the cheese block: at core and under the rind. The two ROI considered experimentally were 3 cm from each other. Assuming diffusivity of CO 2 of the order of 10 -10 m 2 s -1 , this yielded a characteristic diffusion time of about 100 days between these two regions, which was significantly higher than the average stay in warm room (about 15 days). Therefore, the two ROI could be considered independently from each other, from a mass transport point of view.

In order to further validate the model, simulations were conducted for parameter values specific of the cheese region under the rind. In particular, production of CO 2 was set at its value measured for cheese material sampled under rind (Table 2), and the other parameters were set in accordance with the lower X w and X fat values found under the rind, typically at 0.40 and 0.15, respectively. At last, the cheese volume surrounding the bubble was set at 7.6 cm 3 (average experimental value). Following the results at core, the diffusivity was set at its lowest value.

Compared to the simulations at core, simulations under the rind yielded better agreement in the first half of the ripening time. However, the simulated bubble volume was too high for the second half of the ripening time, at nearly twice the experimental value at the end (Reference curve in Fig. 5). Similarly to the results at core (Section 5.2), two simulations were conducted by varying V ch and r CO2 to cover the intra-and inter-batch variability. While the highest values (representing the most favourable conditions for bubble growth) yielded poor agreement with experiment, the least favourable condition (lowest V ch and r CO2 ) yielded good agreement with experiment. This adds up to the conclusions made previously about the need of measurement of these properties for this specific food material.

Another explanation might come from the fact that it was assumed saturation in CO 2 under the rind was reached at the beginning of the warm room. Inter-batch variability of this parameter was not characterised under rind, and a lower carbon dioxide concentration when entering the warm room would mean that the simulations would have to be started whenever the actual saturation was reached. Given that the production rates are lower under rind (Table 2) and depending on the actual carbon dioxide concentration at = d 0, a better fit could be achieved.

Conclusions

The growth of bubbles in cheese was investigated both experimentally and by simulation. Bubble volumes in cheese blocks during ripening were assessed experimentally with the help of X-Ray imaging and MRI with a dedicated image processing method. Special care was taken on the determination of the model parameters. One third of the input parameters were assessed from the literature. The other two thirds were assessed from dedicated measurements performed on the cheese material under study. A sensitivity study was conducted on the bubble growth model in order to rank the input parameters that were the most influential and brought some teachings for improving their estimate.

The model proved to reproduce the average bubble growth in cheese in a qualitative manner. This was yet a satisfactory result, keeping in mind that not all phenomena taking place at the cheese block scale (the scale of experimental data used for validation) were taken into account with such a simplistic model, relevant of the bubble scale.

Production of carbon dioxide, r CO2 , and cheese volume surrounding the bubble, V ch , were the most influential input parameters on the bubble growth. The experimental data also showed to be quite variable between batches, with a standard deviation ranging from 30% for the production rate to 50% for the cheese volume. This conclusion argued in favour of high number of repetitions (for instance more than three runs retained for the monitoring of bubble growth) in order to attain a more complete validation of the model. The only input parameter that was not determined experimentally and that revealed of high influence on bubble growth, D CO ch 2 , was adjusted to fit the experimental sets of data with better agreement. Lower diffusivity of carbon dioxide than expected was explained by either an underestimation of tortuosity (initially assumed for spherically-shaped and regularly arranged obstacles) or an overestimation of the space available for diffusion (it was assumed for the first-hand estimation of diffusivity that the whole fraction of water was available for carbon dioxide diffusion). This highlighted the need of the measurement of this property in cheese, still lacking in the literature.

Finally, the mechanical behaviour of cheese or initial bubble volume proved not to have any effect on bubble growth.

Fig. 1 .

 1 Fig. 1. Geometry used for modelling.

Fig. 2 .

 2 Fig.2. Carbon dioxide production under rind (left) and at core (right) at 20 °C. Two replications are shown for each graph, i.e. one bottle associated to one cheese block. Note that the scale of the y-axis differs between the two Figures (production of CO 2 in cheese samples under rind was decreased by 1/3 at least compared to the core).

Fig.

  Fig. Experimental averaged and simulated bubble volume at core. Runs involved a cheese block coming from a different batch, each being separated by several months.

Fig. 4 .

 4 Fig. 4. Experimental and simulated bubble volumes at the core of the cheese block. Effect of the variations of input parameters of the model, (a) V ch , (b) r CO2 , (c) D CO ch 2 , all other parameters being set at fixed value. Runs involved a cheese block coming from a different batch, each being separated by several months.

Fig. 5 .

 5 Fig. 5. Experimental and simulated bubble volumes under the rind. Runs involved a cheese block coming from a different batch, each being separated by several months.

Table 1

 1 Experimental configurations.

		Run 1	Run 2	Run 3
	Imaging technique	X-Ray	MRI	MRI
	Replications	3	3	1
	Days in cold room	10	8	8
	Days in warm room	15	15	15

Table 3

 3 Values of input parameters and the associated sensitivity indices of the bubble volume to each variation of each input parameter for the screening study including the reference set of values. I k was determined on day 15. 5 and α 5 : mechanical parameters for the longest relaxation time (see[START_REF] Laridon | Identification of broad-spectrum viscoelastic parameters: Influence of experimental bias on their accuracy and application to semihard-type cheese[END_REF].

							Reference	Lower value	Upper value	Variation (%)	I k
	λ 5 (s)				9324	6428	12,230	± 31	0.6
	α 5						0.04	0.03	0.05	± 25	2.5
	D CO ch	2 (10 -10 m 2 s -1 )		6.1	1.5	10.7	± 75	26
	r CO2 (mol•m -3 •s -1 )	Stage I	1.65•10 -5	1.17•10 -5	2.13•10 -5	± 29	111
						Stage II	3.56•10 -5	2.74•10 -5	4.37•10 -5	± 23
	k H ch (mol•m -3 •Pa -1 )		1.67•10 -4	1.25•10 -4	2.08•10 -4	± 25	0.3
	V t ( b	=	0)	(10 -9 m 3 )		42.6	5.4	96	-87 + 125	4.2
	V ch (10 -6 m 3 )		4.2	2.2	6.2	± 47	156
	D CO ch	2 : carbon dioxide diffusivity in the cheese.
	r CO2 : carbon dioxide production rate.
	k H ch : solubility of carbon dioxide in cheese.
	V

λ b : bubble volume.

V ch : cheese volume surrounding the bubble. I k : sensitivity index (see Section 5.2).

Acknowledgements

This study was carried out with financial support from VALORIAL and the Regional Councils of Bretagne, Pays de Loire and Basse-Normandie. The first author, Yannick Laridon, is the holder of a grant from INRA and Irstea.

Declaration of Competing Interest

The authors declared that there is no conflict of interest.

Appendix A. Supplementary material

Supplementary data to this article can be found online at https:// doi.org/10.1016/j.foodres.2019.108858.


Clment Dombry

Discrete approximation of stable white noise -Application to spatial linear filtering

Keywords: stable white noise, stable fractional noise, linear filtering

Motivated by the simulation of stable random fields, we consider the issue of discrete approximations of independently scattered stable noise. Two approaches are proposed: grid approximations available when the underlying space is R d and shot noise approximations available on more general spaces. Limit theorems stating the convergence of discrete random noises to stable white noise are proved. These results are then applied to study moving average spatial random fields with heavy-tailed innovations and related limit theorems. A second application deals with discrete approximation for Brownian Lvy motion on the sphere or on the euclidean space.

Motivations

Stable integration is a basic tool in the theory of stable random process. Indeed, if W denotes a stable random noise on the measurable space (E, E) with control measure m and (f t ) t∈T a kernel such that f t ∈ L α (E, E, m) for any t ∈ T , then the random process defined by

X t = E f t (x)W (dx), t ∈ T
is a stable process, and the path properties of the process can be deduced from the properties of the kernel (see for instance [START_REF] Samorodnitsky | Stable Non-Gaussian Random Processes: Stochastic Models with Infinite Variance[END_REF] chapter 10). For example, the first examples of self-similar stationary increments (SSSI) stable process were constructed in such a way: using a moving average kernel, Taqqu & Wolpert [START_REF] Taqqu | Infinite variance self similar processes subordinate to a Poisson measure[END_REF] and Maejima [START_REF] Maejima | On a class of self-similar processes[END_REF] constructed the linear fractional stable motion; the harmonisable fractional stable motion was proposed by Cambanis & Maejima [START_REF] Cambanis | Two classes of self-similar stable processes with stationary increments[END_REF] as the stable integral of the harmonisable kernel. More recently, Cohen and Samorodnitsky [START_REF] Cohen | Random rewards, fractional Brownian local times and stable self-similar processes[END_REF] propose a new class of SSSI stable process defined as the stable integral of a random kernel, the random kernel being the local time of a fractional Brownian motion. Generalizing the notion of selfsimilarity for random field in higher dimension, operator scaling stable random fields were defined and constructed by Bierm & Scheffler [START_REF] Bierm | Operator scaling stable random fields[END_REF] as stable integrals of high-dimensional kernel with suitable scaling properties.

For the purpose of simulation of these processes, we need a general theory of discrete approximation of stable random measures. An approach based on Lepage's series also called shot noise series was developped: generalized shot noise series were introduced for simulation in [START_REF] Rosinski | On Path Properties of Certain Infinitely Divisible Processes Stochastic Processes and their Applications[END_REF], further developments were done in [START_REF] Rosinski | On Series Representations of Infinitely Divisible Random Vectors[END_REF] and [START_REF] Rosinski | Series representations of Lvy processes from the perspective of point processes[END_REF] and a general framework was developed in [START_REF] Cohen | A general framework for simulation of fractional fields[END_REF]. In this paper, we propose a new approach for the discrete approximation of stable random noise. We propose and developp two alternatives: in the case when W is a stable white noise on R d , a grid approximation can be used (see Theorem 3.1); for more general spaces, an approximation of the stable white noise by a Poisson random measure is discussed (see Theorem 3.2).

We apply our results on grid approximation of stable random noise on R d to the study of stable noise obtained by linear filtering from i.i.d. random fields in the domain of attraction of stable distributions. This extends the results of Kokoszka & Taqqu [8,[START_REF] Kokoszka | New classes of self-similar symmetric stable random fields[END_REF][START_REF] Kokoszka | Fractional ARIMA with stable innovations[END_REF][START_REF] Kokoszka | Infinite variance stable moving averages with long memory[END_REF] for one dimensional sequences such as ARMA or FARIMA with stable inovations to spatial random fields. We prove a limit theorem for the suitably rescaled random noise (see Theorem 4.1). Two cases occur: if the coefficients of the random filter decrease quickly, the limit random field is stable white noise and the dependence vanishes in the limit; if the coefficients decrease slowly like a power function, the limit random field is fractional stable random noise with long range dependence.

Another application of the Poisson approximation for stable white noise is given in the framework of stable Lvy motion on the sphere S d or on the euclidean space R d . These processes mimick the simple covariance structure of standard Brownian motion on R to more general metric spaces (see [START_REF] Kubo | Brownian motion parametrized with metric space of constant curvature[END_REF]). As a direct application of our results, we give a Donsker's type Theorem for stable Lvy motion on the sphere or on the euclidean space.

The paper is organised as follows. In section 2, we remind the reader of general results on random noise, stable white noise, Poisson random measures and their convergence. In section 3, the convergence of discrete random measure to stable white noise is proven for two different models: grid approximations and Poisson approximation. In section 4, two applications are exposed: linear filtering of i.i.d. random field and Donsker's type theorem for Lvy stable motion on the sphere or on the euclidean space. Section 5 is devoted to the proof of our results. Technical results on convergence of deterministic functions are gathered in an appendix.

Stable random noises and Poisson random measures

A random noise on (E, E) is a generalized random field (W [f ]) f ∈F indexed by a linear subspace F of the space of real-valued measurable functions on E, verifying the linearity property: for all a 1 , a 2 ∈ R and f 1 , f 2 ∈ F.

W [a 1 f 1 + a 2 f 2 ] = a 1 W [f 1 ] + a 2 W [f 2 ] a.s. ( 1
)
Using the Cramer-Wold's device, this implies that the finite-dimensional distributions of the process (W [f ]) f ∈F are determined by its one-dimensional distributions. The set of functions F is thought as the set of integrands. To emphasize the analogy with usual integration, we use the notation

W [f ] = E f (x)W (dx). If 1 A ∈ F for some measurable set A ⊆ E, the value W (A) = W [1 A
] is thought as the measure of A. However, each realisation of W does not define an usual signed measure because it needs not to have finite variations. That's why some author (including myself) prefer the term random noise to random measure, this last term being reserved for measure-valued random variables. We recall in this section the definition and properties of stable white noises (including Gaussian white noises) and Poisson random measures.

Independently scattered α-stable random noises

Let (E, E, m) be a measured space with m a σ-finite measure. For α > 0, we denote by

L α = L α (E, E, m) the space of measurable functions f : E → R such that E |f | α dm < ∞. For f ∈ L α , let ||f || L α = E |f | α dm 1/α . If α ≤ 1, this defines
a norm and L α is a Banach case. This is no longer the case if 0 < α < 1.

The stable distribution of index α ∈ (0, 2] and parameters σ ≥ 0 (scale), ν ∈ [-1, 1] (skewness) is denoted by S α (σ, ν). For the sake of simplicity, we will always assume that ν = 0 if α = 1. The distribution S α (σ, ν) is defined by its Fourier transform λ(θ

) = exp -σ α |θ| α 1 -iνε(θ) tan πα 2 , θ ∈ R (2)
where ε(a) = +1 if a > 0, ε(a) = -1 if a < 0 and ε(0) = 0. The α-stable random noise W α on (E, E) with control measure m and skewness function v : (E, E) → [-1, 1] is the stable random process defined on some probability space (Ω, F, P) and indexed by L α , verifying the linearity property 1 and with one dimensional marginal distributions given by

W α [f ] ∼ S α (σ f , ν f ), f ∈ L α where σ f = E |f | α dm 1/α and σ α f ν f = E νε(f )|f | α dm. (3)
Evaluating the process W α at functions (f i ) 1≤i≤k with pairwise disjoint supports yields independent random variables (W α [f i ]) 1≤i≤k : we say that the random noise M α is independently scattered or white to qualify this property.

In the Gaussian case α = 2, the parameter ν is irrelevant and we retrieve the usual Wiener integral, which is an isometry from L 2 (E) onto some Gaussian subspace of L 2 (Ω, F, P). In the case 1 < α < 2, the stable integral induces an isometry from L α (E) onto some subspace of stable random variables on (Ω, F, P) equipped with the covariation norm (see [START_REF] Samorodnitsky | Stable Non-Gaussian Random Processes: Stochastic Models with Infinite Variance[END_REF]).

Poisson random measures

Poisson random measures are measure-valued random variables and they induce a random noise on their sets of integrands. In the following presentation, we focus on the point of view of random noises, a general reference is [START_REF] Kallenberg | Random Measures[END_REF].

Let (E, E, n) be a measured space with n a σ-finite measure. We denote by E 0 the class of measurable sets with finite n-measure. A random measure N on (E, E) is called a Poisson random measure with intensity n if and only if for any finite collection (A i ) 1≤i≤n of pairwise disjoint sets in E 0 , the random variables (N (A i )) 1≤i≤n are independent with N (A i ) following a Poisson distribution with mean n(A i ). The (a priori random) space L 1 (E, E, N ) of integrable functions with respect to N is almost surely equal to

F N = f : E → R | |f | ∧ 1 ∈ L 1 (E, n)
and N induces a random noise on F N . We note N [f ] = E f dN for f ∈ F N . This random noise is characterized by the linearity property (1) and the one dimensional marginals given by their characteristic functions

E [exp(iθN [f ])] = exp E (e iθf -1)dn , f ∈ F N , θ ∈ R.

Convergence of random measures

In this paper, we are mainly concerned with convergence of random noises (µ h ) h>0 to stable random noise W , either independently scattered (Theorem 3.1) or fractional (Theorem 4.1). We shall now give a precise definition of convergence of random noises.

For h > 0, let µ h be a random noise with integrand space F µ h and let W be some random noise with integrand space F W . Let furthermore F be a subspace included in all the F µ h 's, h > 0 and in F W . Definition 2.1 We say that µ h converges weakly to W on F as h → 0 and write

µ h F =⇒ W as h → 0, if the finite dimensional marginal distributions of (µ h [f ]) f ∈F weakly converge to those of (W [f ]) f ∈F .
Using the linearity property of the random noises and the Cramer-Wold's device, we easily see that convergence of finite dimensional distributions is equivalent to convergence of one dimensional distributions, which is in turn equivalent to convergence of characteristic functions. Although straightforward, this will be of constant use and is stated for future reference in the following proposition: Proposition 2.1 The following statements are equivalent, where convergence is meant as h → 0:

1. µ h converges weakly to W on F, 2. for any f ∈ F, µ h [f ] converges weakly to W [h], 3. for any f ∈ F and θ ∈ R, E [exp(iθµ h [f ])] converge to E [exp(iθW [f ])].
To illustrate this definition, we give a first example of weak convergence of random noises.

Example : Let (E, d) be some metric space and E the borelian σ-algebra. Let m and (m h ) h>0 be finite measures on E and ν and ν h be measurable functions E → [-1, 1]. Define W (resp. W h ) as the α-stable random measure with control measure m (resp. m h ) and skewness function ν (resp. ν h ). Let C b (E) denotes the space of bounded continuous functions on E. Note that from the assumption that m is finite,

C b (E) ⊂ F W = L α (E, E, m), and similarly C b (E) ⊂ F W h = L α (E, E, m h ) for all h > 0.
Then the following two statements are equivalent, where convergences are meant as h → 0:

1. the random noises W h converge weakly to W on C b (E), 2. the control measures m h converge weakly to m and the skewness measures ν h dm h converge weakly to νdm.

The proof is direct once we recall that the α-stable distributions S α (σ h , ν h ) converge to S α (σ, ν) (with σ > 0)if and only if (σ h , ν h ) → (σ, ν).

Discrete approximations of white stable noises

We introduce two methods for approximating an independently scattered stable random noise on (E, E). The first one, available when the underlying space is E = R d and the control measure is the Lebesgue measure, relies on a grid approximation hZ d ⊆ R d when the span h of the grid goes to zero. The second one, avalaible on a general space E, relies on the approximation of the stable random noise by suitable Poisson random measures or shot noise, when the intensity λ goes to infinity.

Grid approximation of white stable noise on R d

In this section, E = R d and W α denotes the independently scattered α-stable random noise on R d with Lebesgue intensity and constant skewness ν. We propose a discrete approximation of W α based on the grid approximation hZ d ⊆ R d with span h > 0.

The construction uses a family ξ = {ξ k , k ∈ Z d } of real random variables satisfying the following assumption: H α the ξ k 's are i.i.d. random variables in the normal domain of attraction of the stable distribution S α (σ, ν).

We suppose furthermore ν = 0 if α = 1. Recall that the normal domain of attraction of the stable distribution S α (σ, ν, 0) consists in the random variables Y such that k ] = 2σ 2 . The stable distribution S 2 (σ, 0) is then the normal distribution with mean 0 and variance 2σ 2 (the skewness parameter ν is irrelevant in this case). In the case 0 < α < 2, from the known characterization of the domain of attraction of stable distributions (see [START_REF] Feller | An Introduction to Probability Theory and its Applications[END_REF]), the ξ k 's must satisfy the following tail assumptions as x → +∞:

n -1/α n i=1 Y i =⇒ S α (σ, ν) as n → ∞
P(ξ k ≥ x) ∼ px -α and P(ξ k ≤ -x) ∼ qx -α , (4)
where the constants σ, ν, p, q satisfy

σ α = (p + q) ∞ 0 t -α sin tdt, σ α ν tan(πα/2) = (p -q) ∞ 0 t -α (1 -cos t)dt.
Note that equation ( 4) is equivalent to the fact that ξ -τ belongs to the normal domain of attraction of the α-stable distribution S α (σ, ν) for some τ ∈ R, but we assume here furthermore that τ = 0.

We propose the discrete approximation of W α obtained by replacing W α (dx) by

γ h ξ k 1 h(k+I d ) (x)dx on each cell h(k + I d ), k ∈ Z d . Here I d = [0, 1) d and γ h = σ -1 h ( 1 α -1)d .
More formally, let µ h = µ h (ξ) be the random measure on R d absolutely continuous with respect to Lebesgue measure with random density

dµ h dx (x) = γ h k∈Z d ξ k 1 h(k+I d ) (x). (5)
The random signed measure µ h defined by ( 5) induces a random noise on the integrands set

F µ h = f ∈ L 1 loc (R d ) ; R d f dµ h converges a.s. . (6)
Note that in this definition, only the semi-convergence of the integral R d f dµ h or equivalently of the series

γ h x∈Z d ξ k h(k+I d ) f (x)dx is required almost surely.
The following scaling relation is worth noting:

µ h [f (c.)] = c -α -1 d µ ch [f (.)], c > 0 (7)
whenever these quantities are well-defined.

Our first result precise the integrands sets F µ h . We need to introduce the space

D α (R d ) of locally integrable function f ∈ L 1 loc (R d ) such that f (x) = o(|x| -η ) for some η > α -1 d.
Proposition 3.1 Let 0 < α ≤ 2 and suppose that ξ satisfies assumption H α . For h > 0, consider the random measure µ h defined by equation [START_REF]Theory and Applications of Long-Range Dependence[END_REF]. Then,

1. If 1 ≤ α ≤ 2, L α (R d ) ⊂ F µ h . 2. If 0 < α < 1, D α (R d ) ⊂ F µ h .
We then consider weak convergence of the random measures (µ h ) h>0 . To unify the notation, we introduce

F α = L α (R d ) if 1 ≤ α ≤ 2, and F α = D α (R d ) if 0 < α < 1.
Theorem 3.1 Let 0 < α ≤ 2 and suppose that ξ satisfies assumption H α . Then as h → 0, µ h converge weakly to W α on F α .

Remark: In the case 1 ≤ α ≤ 2, we obtain convergence on L α (R d ) the full natural set of integrands for W α . In the case 0 < α < 1, we have to restrict on the smaller subspace D α (R d ) because L α is not included in F µ h (see the remark after Lemma 5.2 in the Appendix.)

Remark: We prove in fact a slightly stronger result

µ h [f h ] ⇒ W α [f ] as h → 0 if f h → f in F α .

See Proposition 5.1 for a precise statement including the definition of convergence in

D α (R d ) if 0 < α < 1.
This diagonal convergence can be used to prove the following interesting corollary.

Let C(R d ) denote the space of continuous functions on R d . Corollary 3.1 Let 0 < α ≤ 2 and suppose that ξ satisfies assumption H α .

Then the random signed measure μh on R d defined by

μh = σ -1 h α -1 d k∈Z d ξ k δ hk converges weakly to W α on F α ∩ C(R d ) as h → 0.
These results can be used to propose simulation of α-stable processes represented as stable integrals. Another approach developed in [START_REF] Cohen | A general framework for simulation of fractional fields[END_REF] is to use the representation of stable integral in Lepage's series. For the purpose of simulation, it is important to provide error bounds or speed of convergence for the method. We now consider the speed of the convergence of the random noise µ h to the stable integrals W α in Theorem 3.1. Better rates of convergence are expected if the random variables ξ satisfy H α with distribution S α (σ, ν) rather than in the domain of attraction of the stable distribution. In this case, the discrete approximation µ h [f ] has the same distribution as μh

[f ] = R d ( ψh f )(x)W α (dx),
where ψh f is the discrete approximation of f obtained by replacing f by its mean value on each cell h(k

+[0, 1[ d ), k ∈ Z d (see equation 19 below). In this representation ξ k corresponds to σh -d/α W α [h(k +[0, 1[ d )] ∼ S α (σ, ν).
As a consequence, the random variable can be represented on the same space and we can focus on L p convergence and give L p error bounds. Furthermore, during the simulation, the kernel f must sometimes be replaced by some approximated kernel f M . For example, when f has infinite support, f M might be the restriction of f on a bounded domain. Or f M might be a piecewise constant approximation of f so that the mean value of f M on h(k

+ [0, 1[ d ) reduces to f M (hk).
Proposition 3.2 Let 0 < α ≤ 2 and f M ∈ F α be an approximation of the kernel f ∈ F α and let

μh [f M ] = R d ( ψh f M )(x)W α (dx).
Then, for every 0 < p < α,

E μh [f M ] -W α [f ] p ≤ C α,p || ψh f M -f || p L α
and C α,p is the absolute p-th moment of the distribution S α (1, ν).

In the case α = 2, the result holds for all p > 0 Remark: Note that ψh f M → f M in F α as h → 0 and explicit error bounds can be given under suitable regularity condition on f M .

Shot noise approximation of white stable noise on a general space

In this section, (E, E, m) is a general measured space with m a σ-finite measure and W α denotes the independently scattered α-stable random noise on E with control measure m and constant skewness ν. We propose a discrete approximation of W α based on shot noises, i.e. on Poisson random measures.

Let G be some distribution on R belonging to the normal domain of attraction of the stable distribution S α (σ, ν). For λ > 0, let N λ (de, dξ) be the Poisson random measure on E × R with intensity measure n λ (de, dξ) = λm(de)G(dξ). It induces a random noise µ λ on E by the heuristic formula

µ λ (dx) = γ λ E×R ξδ e (dx)N λ (de, dξ),
where

γ λ = σ -1 λ -1 α . More formally, let F µ λ be the set of functions f : E → R such that E×R |ξf (e)| ∧ 1 n λ (dξ, de) < ∞. (8)
We consider the random noise µ λ on F µ λ defined by

µ λ [f ] = γ λ E×R ξf (e)N λ (de, dξ). ( 9
)
Our result is the following: Theorem 3.2 Suppose G belongs to the normal domain of attraction of the stable distribution S α (σ, ν). Then, for any λ > 0, L α (E, E, m) ⊆ F µ λ and as λ → ∞, µ λ converge weakly to W α on L α (E, E, m).

Remark: In the case when m is finite and normalized to m(E) = 1, we can replace the Poisson random measure N λ by the binomial random measure Ñn = σ -1 n -1 α n i=1 δ (ξ i ,e i ) where (ξ i , e i ) i≥1 is a sequence of i.i.d. random variables with distribution G(dξ)m(de). We then consider the random noise

μn [f ] = σ -1 n -1 α n i=1 ξ i f (e i ).
Similar results as those of Theorem 3.2 hold: for any n ≥ 1, L α (E, E, m) ⊆ F μn and μn weakly converge to W α on L α (E, E, m) as n → ∞. The proof is very similar to that of Theorem 3.2 and will be omitted.

Applications

We propose two applications of the above results about convergence to stable random noise. The first application based on Theorem 3.1 is an analysis of random noises arrising from linear filtering of i.i.d. random fields. Interessant fields appear in the limit that we call fractional stable random noise. The second application based on Theorem 3.2 proposes a Donsker's type theorem for stable Lvy field on the sphere or on the euclidean space.

Spatial linear filtering

In this section, we consider a random measure μh of the form

dμ h dx (x) = γh k∈Z d ξk 1 h(k+I d ) (x), (10)
where γh > 0 is a normalisation constant and the field ξ = ( ξk ) k∈Z d is not any more i.i.d. but stationary with spatial dependence (possibly long range dependence). More precisely, we consider the case when the field ξ is obtained from a random field ξ satisfying H α by linear filtering. The linear filter is given by c = (c k ) k∈Z d ∈ ℓ α and the random field ξ by ξk

= (ξ * c) k = l∈Z d c k-l ξ l .

The corresponding assumption is

Ĥα ξ = ξ * c with ξ satisfying H α and c ∈ ℓ α .

Here

ℓ α = ℓ α (Z d ) is the space of sequence u = (u k ) k∈Z d such that k∈Z d |u k | α <
∞ and * denotes the convolution product of sequences.We will see that if ξ satisfies H α and c ∈ ℓ α then ξ * c is defined almost surely.

The integrands set associated to the random measure μh is

F μh = f ∈ L 1 loc (R d ) ; R d f dμ h converges a.s. . (11)
We consider the convergence of the random measure μh to some α-stable noise. According to the behavior of c at infinity, the limit field can be either a white stable noise or a fractional stable noise (to be defined below). In the former case, spatial dependence disappear in the limit and weak dependence holds, whereas in the latter case, spatial dependence persists in the limit and strong dependence holds. For the sake of simplicity, we focus on the case 1 < α ≤ 2. As in section 3.1, W α denotes a stable white noise on R d with Lebesgue control measure and constant skewness ν.

Theorem 4.1 Let 1 < α ≤ 2 and suppose that ξ satisfies assumption Ĥα . For h > 0, consider μh the random measure given by equation [START_REF] Kokoszka | Fractional ARIMA with stable innovations[END_REF].

1. Suppose c ∈ ℓ 1 and let γh = σ -1 h (α -1 -1)d and C = ( k∈Z d c k ).
Then for any h > 0, L α ⊆ F μh and as h → 0, µ h converge weakly to CW α on L α .

Suppose that c has regular variations at infinity of order

-β ∈ (-d, -α -1 d) in the sense that lim t→∞ sup |x|=1 t β c [tx] -p(x) = 0, ( 12
)
where p : R d \{0} → R is some locally integrable homogeneous function of order -β (hyp ??). Let γh = σ -1 h α -1 d-β . Then for any h > 0, L α ∩ L 1 ⊆ F μh and as h → 0, µ h converge weakly to W α,p on L α ∩ L 1 , where W α,p is the fractional stable random noise defined by

W α,p [f ] = W α [f * p] , f ∈ L 1 ∩ L α
We now give some interessant properties of the random noise W α,p .

Proposition 4.1 Let 1 < α ≤ 2 and p be as in Theorem 4.1.

1. The random noise W α,p on L α ∩ L 1 is α-stable, stationary and α -1 d -β + d self-similar.

2. In the case α = 2, W 2,p is the centered Gaussian random noise with covariance function

E (W α,p [f ]W α,p [g]) = R d ×R d f (x)K(x, y)g(y)dxdy, f, g ∈ L 1 ∩ L 2
where K : R d × R d → R is the kernel given by

K(x, y) = R d p(x -z)p(y -z)dz.
Remark: The notions of stationarity and selfsimilarity are invariance properties of the random noise under the transformation group of translations and dilatations respectively. The formal definition is given in equation ?? and ??.

Remark Self-similarity can be stated using the notion of renormalisation groups.

Let F be a linear space of functions on R d closed under dilatations, i.e. for any h > 0 and f ∈ F, f (h.) ∈ F. Let u ∈ R. For h > 0 let T (u) h be the transformation acting on a random noise W on F by (T

(u) h W )[f (.)] = h u W [f (h.)] , f ∈ F.
The group relations

T (u) 1 = Id and T (u) h • T (u) h ′ = T (u)
hh ′ holds and the group of transformation (T (u) h ) h>0 is called the normalisation group of index u. Our results have a nice interpretation in terms of normalisation group. Consider first Theorem 3.1. The scaling relation [START_REF] Kallenberg | Random Measures[END_REF] implies that the random measure µ h is equal to

T (α -1 d) h µ 1
and the Theorem states that the random noise

T (α -1 d) h µ 1 converges to W α as h → 0. Furthermore, W α is a fixed point of the renormalisation group of index α -1 d, i.e. T (α -1 d) h W α = W α , meaning that W α is (α -1 d)-selfsimilar.
In the same way, consider Theorem 4.1 part 2. Here μh = T (α -1 d-β+d) h μ1 and the Theorem states that

T (α -1 d-β+d) h μ1 converges to W α,p as h → 0. Furthermore, W α,p is a fixed point of the renormalisation group of index α -1 d -β + d, i.e. T (α -1 d-β+d) h W α,p = W α,p , meaning that W α,p is α -1 d -β + d-selfsimilar.

Convergence to Lvy Brownian motion on the sphere or on the euclidean space

The Brownian motion parametrized by a general metric space was introduced by P.Lévy. It is defined as follows: let O be a fixed point in a metric space (M, d), a Brownian motion parametrized with the metric space (M, d) and with origin O is a centered Gaussian process (B(m)) m∈M such that : -B(0) = 0 almost surely, -B(m) -B(m ′ ) has variance d(m, m ′ ). There does not always exist a Brownian motion on (M, d) for an arbitrary metric space (M, d) but constructions of such Lvy Brownian motion have been proposed in the case when M is a sphere, a euclidean space or an hyperbolic space (see [START_REF] Kubo | Brownian motion parametrized with metric space of constant curvature[END_REF]). These constructions are based on Gaussian white noise and have been extended to stable white noise in order to get stable processes with interesting properties.

First consider the case when M = S q is the unit sphere in R q+1 and d is the geodesic distance. Let O be a fixed point on the sphere S q , for example the northern pole and let W 2 be a Gaussian white noise on the sphere with control measure ds the normalized uniform measure on the sphere S q . For m ∈ S q , let H m be the hemisphere centered at m defined by

H m = {m ′ ∈ S q | d(m, m ′ ) ≤ π 2 }.
Then the process (B(m)) m∈S q defined by

B(m) = √ πW 2 (H O ∆H m ) , m ∈ S q
is a Brownian motion on (S q , d). Here ∆ denotes the symmetric difference. More generally, the Lvy stable motion on the sphere S q is defined by the same formula with the Gaussian white noise W 2 replaced by a symmetric α-stable random measure W α with control measure ds.

When the metric space (M, d) is the euclidean space R q , a Brownian motion is constructed in the following way (Lvy-Chenstov construction). Let W 2 be a white Gaussian noise on S q × R ⋆ + with control measure ds × dr. A pair (s, r) represents the hyperplane in R q with equation < x, s >= r and S q × R ⋆ + is thought as the set of hyperplanes that do not contain the origin. For m ∈ R q define V m as the set of hyperplanes that separate O and m, i.e.

V m = {(s, r) ∈ S q × R ⋆ + ; 0 < r << s, m >}.

Then the process (B(m)) m∈R q defined by

B(m) = W 2 (V m ) , m ∈ R q
is a Brownian motion on (R q , d). Replacing the Gaussian white noise W 2 by the symmetric α-stable random measure W α with control measure dsdr, we obtain the symmetric α-stable Lvy-Chenstov random field, which is a α -1 -self-similar stationary increments process. We apply our results on convergence of Poisson random measures to stable noises to obtain a Donsker's theorem for stable Lvy motion on the sphere S q or on R q .

Theorem 4.2 Suppose G is in the normal domain of attraction of the distribution S α (σ, 0) for some σ > 0. The following convergence hold in the sense of finite dimensional distributions as λ → ∞.

1. Let N λ (ds, dξ) be a Poisson random measure on S q ×R with intensity λdsG(dξ).

The random process B λ on S q defined by

B λ (m) = √ πσ -1 λ -1/α S q ×R ξ1 H O ∆Hm (s)N λ (ds, dξ) , m ∈ S q
weakly converges to the symmetric α-stable Lvy motion on the sphere S q .

2. Let N λ (ds, dr, dξ) be a Poisson random measure on S q × R ⋆ + × R with intensity λdsdrG(dξ). The random process B λ on R q defined by

B λ (m) = σ -1 λ -1/α S q ×R ⋆ + ×R
ξ1 Vm (s, r)N λ (ds, dr, dξ), m ∈ R q weakly converges to the symmetric α-stable Lvy motion on the euclidean space R q .

Remark : The random fields B λ can be easily simulated, which is not clear for the limit field B. For example in the case of Lvy motion on the sphere S q , simulating the Poisson integral can be made as follows. Draw T according to a Poisson distribution with mean λ, conditionally to T = t, draw (s i , ξ i ) 1≤i≤t identically distributed with distribution dsG(dξ) and let

B λ (m) = √ πσ -1 λ -1/α t i=1 ξ i 1 H O ∆Hm (s i ).
Note that the quantity x ∈ H m is easily calculated since x ∈ H m if and only if < x, m >≥ 0. In the case of Lvy motion on the sphere R q , the method is just the same once we observe that simulating B λ (m), m ∈ D in a bounded domain D requires the knowledge of the Poisson random measure on a bounded domain of S q × R ⋆ + and hence only of a finite number of random points.

Proofs

Preliminaries on stable distributions

We recall some known facts about normal domains of attraction of stable distribution (see [START_REF] Feller | An Introduction to Probability Theory and its Applications[END_REF]). Let ξ belong to the normal domain of attraction of the stable distribution S α (σ, ν). Then, the following estimate holds for its characteristic function as θ → 0

λ(θ) = E e iθξ = λ(θ) + o(|θ| α ). ( 13
)
where λ is given by [START_REF] Cambanis | Two classes of self-similar stable processes with stationary increments[END_REF]. Furthermore, in the case 0 < α < 2, the tail estimate (4) implies that there exists C > 0 such that for any s > 0

Var[ξ1 {|ξ|≤s} ] ≤ Cs 2-α and E[|ξ|1 {|ξ|≤s} ] ≤ Cs 1-α . (14)

Proof of Proposition 3.1

The random measure defined by ( 5) is linked with the discretization of the space R d by hZ d , h > 0. Introduce the operator

ψ h : L 1 loc → R (Z d ) f → h -d h(k+I d ) f (u)du k∈Z d . Define the random signed measure ν on Z d by ν = k∈Z d ξ k δ k . ( 15
)
It defines a random noise on the integrands set F ν defined by

F ν = f : Z d → R; sum k∈Z d f k ξ k converges a.s.
where once again convergence means semi-convergence of the integral. We have the formal relation

µ h [f ] = γ h h d ν[ψ h f ]
from which we deduce that f ∈ F µ h if and only if ψ h f ∈ F ν . But we will see below that F ν = ℓ α (see Lemma 5.1) and that ψ h (L α ) ⊂ ℓ α if 1 ≤ α ≤ 2 and ψ h (D α ) ⊂ ℓ α if 0 < α < 1 (see Lemma 5.2 in the Appendix). This proves Proposition 3.1.

Lemma 5.1 Suppose α ∈ (0, 2] and ξ satisfies H α . Then, F ν = ℓ α .

Proof of Lemma 5.1 This is a direct application of Kolmogorov's three series Theorem (see [START_REF] Feller | An Introduction to Probability Theory and its Applications[END_REF]). The random series k∈Z d f k ξ k with independent summands converges almost surely if and only if for any s > 0, the following three numeric series converge:

k∈Z d P [|f k ξ k | > s] < ∞, k∈Z d Var f k ξ k 1 {|f k ξ k |≤s} < ∞,
and

k∈Z d E f k ξ k 1 {|f k ξ k |≤s} converges.
Note that the convergence of the first series for small s implies that f k → 0 as k → ∞, so we do suppose f k → 0 as k → ∞. In the case α = 2, this implies

Var f k ξ k 1 {|f k ξ k |≤s} ∼ k→∞ f 2 k E[ξ 2 k ]
and hence the second series converges if and only if f ∈ ℓ 2 . Suppose now f ∈ ℓ 2 . Cebycev's inequality implies

P [|f k ξ k | > s] ≤ f 2 k E[ξ 2 k ]s -2
so that the first series converges. Applying Borel-Cantelli's Lemma, this in turn entails that the event lim sup{|f k ξ k | > s} has probability 0. Hence, the set {k; |f k ξ k | > s} is almost surely finite, so that in the third series almost all terms vanish and the third series converges.

In the case 0 < α < 2, we deduce from equation ( 4) that

P [|f k ξ k | > s] ∼ (p + q)|f k | α s -α (16)
and hence the first serie converges if and only if f ∈ ℓ α . Suppose now that f ∈ ℓ α . Equation ( 14) implies the second and third series converge.

Proof of Theorem 3.1 and of Proposition 3.2

Proof of Theorem 3.1 Using Proposition 2.1, it is enough to prove the convergence of one dimensional distribution: for f ∈ F α ,

µ h [f ]⇒W α [f ] as h → 0.
We will prove in fact a stronger result that will be useful in the sequel and we consider diagonal convergence. Recall that

F α = L α if 1 ≤ α ≤ 2 and F α = D α if 0 < α < 1. If α ∈ [1, 2],
F α is a Banach space when endowed with the norm ||.|| L α and the notion of convergence f h → f in L α is clear. In the case 0 < α < 1, we need the following definition of convergence in D α .

Definition 5.1 We say that (f h ) h>0 converge to f in D α if the following two conditions hold:

• for any compact K ⊂ R d , f n 1 K converges to f 1 K in L 1 ,
• there is some

η > α -1 d such that f h (x) = o(|x| -η ) and f (x) = o(|x| -η ) as x → ∞ uniformly in h > 0.
Theorem 3.1 is then a direct consequence of the following Proposition.

Proposition 5.1 Suppose ξ satisfies H α and let (f h ) h>0 converge to f in F α . Then, the following diagonal weak convergence holds:

µ h [f h ]⇒W α [f ] as h → 0.
We prove convergence of the characteristic functions. Let θ ∈ R. From the definition of stable white noise,

E [exp(iθW α [f ])] = exp -σ α f |θ| α (1 -iν f ε(θ) tan(πα/2)) (17)
with σ f and ν f given by (3). On the other hand, we can rewrite

µ h [f h ] = σ -1 h α -1 d k∈Z d ξ k ( ψh f h )(hk), (18)
where ψh : L 1 loc → L 1 loc is the linear functional defined by

( ψh f )(x) = h -d h([x] h +I d ) f (u)du, (19) and [
x] h = h[h -1 x].
Note that ψh f is the approximation of f when the space R d is discretized by hZ d and the function f is replaced by its mean value on each cell of the form h(k + I d ), k ∈ Z d . With these notations, the characteristic function of

µ h [f h ] is given by E [exp(iθµ h [f h ])] = k∈Z d λ σ -1 h α -1 d ( ψh f h )(hk)θ . (20)
Here we have used equation [START_REF] Samorodnitsky | Stable Non-Gaussian Random Processes: Stochastic Models with Infinite Variance[END_REF], the independence of the ξ k 's and the almost sure convergence of the random serie [START_REF] Samorodnitsky | Stable Non-Gaussian Random Processes: Stochastic Models with Infinite Variance[END_REF] implying the convergence of the above infinite product.

First step: we begin to show that as h → 0,

E [exp(iθµ h [f h ])] = k∈Z d λ σ -1 h α -1 d ( ψh f h )(hk)θ + o(1). (21)
To see this, we estimate the difference and use the following inequality : let (z i ) i∈I and (z ′ i ) i∈I two families of complex numbers in D such that the products i∈I z i and i∈I z ′ i are convergent, then

i∈I z ′ i - i∈I z i ≤ i∈I z ′ i -z i .
Using equation (20), this yields

E [exp(iθµ h [f h ])] - k∈Z λ σ -1 h α -1 d ( ψh f h )(hk)θ ≤ k∈Z λ (u h (k)) -λ (u h (k)) (22) with u h (k) = σ -1 h α -1 d ( ψh f h )(hk)θ.
Equation ( 13) implies that the function g defined by g(0) = 0 and

g(v) = |v| -α λ(v) -λ(v) , v = 0,
is continuous and bounded so that for any

k ∈ Z d , λ (u h (k)) -λ (u h (k)) = g(u h (k))|u h (k)| α .
In order to obtain an uniform estimation, define the function g : [0, +∞)

→ [0, +∞) by g(u) = sup |v|≤u |g(v)|.
Note that g is continuous, bounded and vanishes at 0, and that for any

k ∈ Z d such that |u h (k)| ≤ ε, λ (u h (k)) -λ (u h (k)) ≤ g(ε)|u h (k)| α . (23)
Let ε > 0. Equations (20), ( 22) and (23) together yield

E [exp(iθµ h [f h ])] - k∈Z λ σ -1 h α -1 d ( ψh f h )(hk)θ ≤ g(ε) k∈Z d |u h (k)| α 1 {|u h (k)|≤ε} + 2 k∈Z d 1 {|u h (k)|>ε} .
Here we have used equation (23) to bound | λ(u h (k))-λ(u h (k))| from above whenever |u h (k)| ≤ ε, and the trivial upper bound 2 otherwise. Now we remark that

k∈Z d |u h (k)| α 1 {|u h (k)|≤ε} ≤ k∈Z d |u h (k)| α = σ -α |θ| α || ψh f h || α L α is bounded since from Corollary 5.1 | ψh f h | α → |f | α in L 1 whenever f h → f in F α .
By the continuity of g in 0, g(ε) is small for ε small enough. Furthermore,

k∈Z d 1 {|u h (k)|>ε} = h -d R d 1 {| ψh f h (x)|>εσ|θ| -1 h -α -1 d } dx ≤ ε -α σ -α |θ| α ||( ψh f h )1 {| ψh f h |>εσ|θ| -1 h -α -1 d } || α L α
and this quantity goes to 0 as h → 0 because Corollary 5.

1 implies that | ψh f h | α → |f | α in L 1 whenever f h → f in F α and hence the familiy | ψh f h | α , h > 0 is uniformly in- tegrable.
These estimates imply equation ( 21).

Second step: we prove that as h → 0,

k∈Z d λ σ -1 h α -1 d ( ψh f h )(hk)θ = exp -σ α f |θ| α (1 -iν f ε(θ) tan(πα/2)) + o (1).
(24) To see this, we take here advantage of the exponential form of λ and write the l.h.s. of (24) as exp

  -|θ| α h d k∈Z d |( ψh f h )(hk)| α + iν tan(πα/2)θ <α> h d k∈Z d (( ψh f h )(hk)) <α>   = exp -|θ| α R d |( ψh f h )(x)| α dx + iν tan(πα/2)θ <α> R d |( ψh f h )(x)| <α> dx (25)
where u <α> = ε(u)|u| α denotes the signed power function.

Corollary 5.1 implies that

R d |( ψh f h )(x)| α dx = R d |f (x)| α dx + o(1)
and

R d |( ψh f h )(x)| <α> dx = R d |f (x)| <α> dx + o(1).
This together with equation (25) implies equation (24).

Conclusion:

The convergence of the characteristic functions of µ h [f h ] given by (20) to the characteristic function of W [f ] given by ( 17) is a direct consequence of equations ( 21) and ( 24). This proves the weak convergence

µ h [f h ] ⇒ W [f ]. Proposition
if f ∈ F α ∩ C(R d ), then f h → f in F α .

Proof of Proposition 3.2

The proof is rather straightforward once we observe that μh

[f M ]-W α [f ] = W α [φ h f M - f ] has the same distribution as ||φ h f M -f || Lα X α where X α has distribution S α (1, ν).
Hence,

E [|μ h [f M ] -W α [f ]| p ] = || ψh f M -f || p Lα E[|X α | p ], and C α,p = E[|X α | p ].

Proof of Theorem 3.2

We first prove that for any λ > 0, the integrand set

F µ λ contains L α (E, E, m). Let f : E → R be such that E |f | α dm < ∞.
We have to prove that equation ( 8) is satisfied and compute

E×R |ξf (e)| ∧ 1 n λ (dξ, de) = λ E |f (e)|E[|ξ|1 |f (e)ξ|≤1 ] + P[|f (e)ξ| > 1] m(de) ≤ λ E C|f (e)| α m(de) < ∞
where the last inequality is a consequence of equations ( 16) and ( 14). This proves the required inclusion.

We now prove the convergence of the random measures to the stable random noise. Following Proposition 2.1, we compute the characteristic function of

N λ [f ] for f ∈ L α (E, E, m): E [exp(iθN λ [f ])] = exp E×R Ψ(θγ λ ξf (e))λm(de)G(dξ) ,
where Ψ(u) = e iu -1. Performing integration with respect to G, the integral in the right hand side rewrites E Ψ G (θγ λ ξf (e))) λm(de)G(dξ), with Ψ G (u) = R Ψ(uξ)G(dξ). We now prove the following asymptotic result:

lim λ→∞ E Ψ G (θγ λ f (e)) λm(de) = - E |θf (e)| α (1 -νε(θf (e)) tan(πα/2)).
This is a consequence of Lebesgue's convergence Theorem. Indeed γ λ → 0 as λ → ∞ and the asymptotic behaviour of Ψ G at 0 given by equation ( 13

Proof of Theorem 4.1

In view of Lemma 5.1, the conditions c ∈ ℓ α and ξ satisfies H α ensure that ξ = ξ * c is well defined. This convolution relation implies that the random measures µ h and μh defined by equations ( 5) and [START_REF] Kokoszka | Fractional ARIMA with stable innovations[END_REF] respectively are linked by a simple relation. This is the object of the following Lemma. In order to unify the two cases considered in Theorem 4.1, we denote the integrands set by Fα with Fα = L α in the first case and Fα = L α ∩ L 1 in the second case. We define č ∈ R Z d by čk = c -k , k ∈ Z d .

Lemme Under the assumptions of Theorem 4.1 (both cases), Fα ⊆ F μh and the relation μh

[f ] = µ h [ ψc h f ] holds for any f ∈ Fα with ψc h f = γh γ h φ h ((ψ h f ) * č), (26)
where

ψ h : L 1 loc → R Z d f → h -d h(k+I d ) f (u)du k∈Z d , (27)
and

φ h : R Z d → L 1 loc u = (u k ) k∈Z d → f : x → u [h -1 x] . (28)
Proof:

We have formally,

μξ h [f ] = γh k∈Z d (ψ h f ) k ξk = γh (k,l)∈Z d ×Z d (ψ h f ) k c k-l ξ l = γh l∈Z d ((ψ h f ) * č) l ξ l = γh γ h µ h [φ h ((ψ h f ) * č)].
Furthermore, this formal computations are valid because the assumptions of Theorem 4.1 entail (ψ h f ) * č ∈ ℓ α and hence ψc h f ∈ L α ⊆ F µ . Recall indeed that Hlder inequality implies that u * v ∈ ℓ α as soon as u ∈ ℓ 1 and v ∈ ℓ α . Here in the first case (resp. in the second case) c ∈ ℓ 1 and ψ h f ∈ ℓ α (resp. c ∈ ℓ α and ψ h f ∈ ℓ 1 ). Proof of Theorem 4.1: Theorem 4.1 is a direct consequence of the above Lemma, of Theorem 3.1 (or rather of Proposition 5.1 about diagonal convergence) and of the convergence of ψc h f in L α as h → 0. Lemma 5.5 in the appendix states indeed that ψc h f → Cf converges to Cf (resp. to f * p) in the first case (resp. second case).

Appendix: some results on space functions

The following Lemma gathers some useful properties of the linear operators ψh , ψ h and φ h defined by equations ( 19), ( 27) and (28) respectively. It is standard material for α ≥ 1 but some extra care is needed when 0 < α < 1.

Lemma 5.2 1. The relations ψh = φ h • ψ h , ψ h • φ h = Id R Z d and ψh • ψh = ψh hold. 2. φ h (ℓ α ) ⊂ L α and for any u ∈ ℓ α , ||φ h u|| L α = h d/α ||u|| ℓ α . 3. If α ≥ 1, ψ h (L α ) = ℓ α and ψh (L α ) ⊂ L α and ψh induces a linear projection on L α . Furthermore, if f h → f in L α as h → 0, then ψh (f h ) → f in L α . 4. If 0 < α < 1, ψ h (D α ) ⊂ ℓ α , and ψh (D α ) ⊂ (D α ). Furthermore, if f h → f strongly in D α as h → 0, then ψh (f h ) → f strongly in D α .
Remarks: Extra care is needed in the case 0 < α < 1 because ψh (L 1 loc ∩ L α ) is not included in L α as the following example shows: for γ > 0 and δ ∈ R consider the

function f ∈ L 1 loc defined by f (x) = k δ if x ∈ [k, k + k -γ
] for k ≥ 1, and f (x) = 0 otherwise. Then it is easily seen that f ∈ L α if and only if αδ -γ < -1, and that ψ1 f ∈ L α if and only if α(δ -γ) < -1. Comparing these to conditions, we can even find f ∈ L α ∩ L 1 such that ψ1 f / ∈ L α . That's why we need some stronger condition on f in the case 0 < α < 1 and we introduce the space of rapidly decaying functions D η .

Proof of Lemma 5.2:

1. This is routine verification using the definitions.

Let

u ∈ ℓ α . Then, bbsR d |φ h u(x)| α dx = bbsR d |u [h -1 x] | α dx = h d k∈Z d |u k | α , and this implies φ h u ∈ L α and ||φ h u|| L α = h d/α ||u|| ℓ α . 3. Let α ≥ 1 and f ∈ L α . Using Jensen inequality with the convex function y → |y| α , ||ψ h f || α ℓ α = h -d k∈Z d h -d h(k+I d ) f (u)du α ≤ k∈Z d h -d h(k+I d ) |f (u)| α du = h -d ||f || α L α . This implies that ψ h f ∈ ℓ α and ||ψ h f || ℓ α ≤ h -d/α ||f || L α .
Together with point 1. and 2., we easily prove that ψh induces a linear projection on L α . We now prove that ψh converge weakly as a linear operator to the idendity. First consider the case when f is continuous with compact support. Then from the continuity of f , for every x ∈ Rd ( ψh f )(x) → f (x) as h → 0. The assumption that f is compactly supported implies that Lebesgue's convergence theorem applies and ψ h f → f in L α . The space of continuous compactly supported function is dense in L α and the ψ h are (uniformly) bounded linear operator. Hence the convergence ψ h f → f holds for every f ∈ L α . Diagonal convergence is a consequence of the equicontinuity of the ψ h which all have linear norm less than 1.

4. Let f ∈ D α . From the definition of D α , there is some integer M > 0 and C > 0 and some η > α

-1 d such that if |x| ∞ > M then |f (x)| ≤ C|x| -η ∞ . As a consequence, if h ≤ 1 and |x| ∞ ≥ M + h, we have h([h -1 x] + I d ) ⊆ B(O, M ) c and | ψh f (x)| α = h -d h([h -1 x]+I d ) f (u)du α ≤ h (1-α)d (|x| ∞ -h) -αη . This implies R d | ψh f (x)| α dx < ∞ since αη > d and hence ψh f ∈ L α . Now let f h → f strongly in D α .
From the above discussion and the definition of strong convergence in D α , we see that a similar inequality holds uniformly in h ≤ 1: there is some M > 0, C > 0 and η > α -1 d such that if |x| ∞ > M then |f (x)| ≤ C|x| -η ∞ and also for any h ≤ 1 |f h (x)| ≤ C|x| -η ∞ . As a consequence, for any ǫ > 0, there is some M > 0 such that for any h ≤

1 |x|∞>M | ψh f h (x)| α dx ≤ ǫ (29)
On the other hand, for fixed M > 0, we prove that | ψh f h | α 1 B(0,M ) is uniformly integrable. Let K = B(0, M + 1). Then f h 1 K → f 1 K in L 1 , and point 3. implies that ψh (f h 1 K ) → f 1 K in L 1 . Since the restrictions of ψh (f h 1 K ) and ψh f h to B(0, M ) are equal, this implies (ψ h f h )1 B(0,M ) converge to f 1 B(0,M ) in We consider the convergence of ( ψh f h (x)) <α> to f <α> in L 1 , the other case being treated in the same way. First of all, as above, for any ǫ > 0 there is some M > 0 such that for any h ≤ 1 |x|∞>M ( ψh f h (x)) <α> -f (x) <α> dx ≤ ǫ.

For fixed M > 0, the application L 1 (B(0, M )) → R, f → B(0,M ) f (x) <α> dx is continuous. Hence the convergence (ψ h f h )1 B(0,M ) → f 1 B(0,M ) in L 1 implies lim n→∞ B(0,M ) (ψ h f h )(x) <α> dx = B(0,M ) (ψ h f h )(x) <α> dx.

Together with equation (30), this yields the result.

We now consider the convergence of ψc h f defined by (26). Lemme Under the assumptions of Theorem 4.1 1. In the first case, for any f ∈ L α , ψc h f → Cf in L α as h → 0. 2. In the second case, for any f ∈ L 1 ∩ L α , ψc h f → f * p in L α as h → 0. Proof of Lemma 5.5:

1. Consider first the case when c ∈ ℓ 1 . We easily see that

ψc h f = k∈Z d c k ψh f (• + hk).
For each fixed k, the sequence of function ψh f (• + hk) converge to f in L α (use the fact that according to Lemma 5.2 point 3, ψh f → f in L α and that the translation operator f → f (. + hk) is continuous and converge to identity as h → 0). Then for any fixed M , At last, we prove that ψc h f → f * p in L α . We have indeed

ψc h f -( ψh f ) * ph α L α ≤ R d R d | ψh f ([x] h -y) -ψh f (x -y)||p h (y)|dy α dx ≤ ||p h 1 B || α L 1 sup 0≤u≤h || ψh f (. + u) -ψh f || α L α + ||p h 1 B c || α L α sup 0≤u≤h || ψh f (. + u) -ψh f || α L 1
and these quantities vanish as h → 0.

5 . 1 and

 51 Theorem 3.1 are proved. Proof of Corollary 3.1 Note that formally μh [f ] = µ h [f h ], where f h is the function defined by f h (x) = f ([x] h ). The result follows from Proposition 5.1 and from the fact that

  ) yields Ψ G (θγ λ f (e)) = -|θf (e)| α (1 -νε(θf (e)) tan(πα/2)) + o(1).Furthermore, there is some C > 0 such that |Ψ G (θ)| ≤ C|θ| α for some C > 0 and this implies the domination conditionλΨ G (θγ λ f (e)) ≤ C|θ| α |f (e)| αwhere the r.h.s. is integrable with respect to m(de) and does not depend on λ. Hence we have proved thatlim λ→∞ E [exp(iθN λ [f ])] = exp -E |θf (e)| α(1 -νε(θf (e)) tan(πα/2))m(de) , and this is precisely the characteristic function of W α [f ].

L 1 .Corollary 5 . 1

 151 This in turn implies that | ψhf h | α 1 B(0,M ) is bounded in L p for p = α -1 > 1,and hence uniformly integrable. This together with equation (29) implies the uniform integrability of the sequence| ψn f n | α . Let α ∈ (0, 2]. Suppose (f h ) h>0 converge to f in F α as h → 0. Then | ψh (f h )| α → |f | α in L 1 as h → 0 and also ( ψh (f h )) <α> → f <α> in L 1Proof of Corollary 5.1:

|k|≤M c k

  ψh f (• + hk) → ( |k|≤M c k )fin L α as h → 0. Furthermore, since c ∈ ℓ 1 , the remainder|| |k|>M c k ψh f (• + hk)|| L α ≤   |k|>M |c k |   ||f || L αis small when M is large. Thanks to these estimates, we prove that ψch f → Cf in L α . 2. Consider the second case when c is such that lim t→∞ sup |x|=1 t β c [tx] -p(x) = 0,and note that ψc h f can be rewritten asψc h f (x) = R d ψh f ([x] h -y)p h (y)dy = (( ψh f ) * ph )([x] h ) (31) with ph (y) = h -d γh γ h c -[h -1 y] = h -β č[h -1 y] .Using equation 12, we see that for any y = 0, ph (y) → p(y) as h → 0. More precisely, equation 12 entails that ph 1B → ph 1 B in L 1 and ph 1 B c → ph 1 B cin L α , where B denotes the unit ball B(0, 1) in R d and B c its complementary set. Using Lemma 5.2 and the assumption f ∈ L 1 ∩ L α , we have also ψh f → f in L 1 and in L α . The bilinear application L 1 × L α → L α , (g 1 , g 2 ) → g 1 * g 2 is continuous and||g 1 * g 2 || L α ≤ ||g 1 || L 1 ||g 2 || L α . This entails ( ψh f ) * (p h 1 B ) → f * (p1 B )in L α and ( ψh f ) * (p h 1 B c ) → f * (p1 B c ), and finally ( ψh f ) * ph → f * p in L α .

Acknowledgements

I am grateful to Professor Serge Cohen for fruitful discussions at the origin of the limit theorems for the Lvy Brownian motion on the sphere and related processes.