Caroline Thévenot

Abdel Khoukh

Stephanie Reynaud

Jacques Desbrieres

Bruno Grassl
email: bruno.grassl@univ-pau.fr

Stéphanie Reynaud

Jacques Desbrières

Kinetics aspects, rheological properties and mechanoelectrical effects of hydrogels composed of polyacrylamide and polystyrene nanoparticles

Keywords: nanoparticles, hydrogel, kinetics, rheological properties, NMR spectroscopy

come

Kinetics aspects, rheological properties and mechanoelectrical effects of hydrogels composed of

polyacrylamide and polystyrene nanoparticles.

Introduction

Hydrogels are materials that swell and hold high amounts of water without losing their shape. 1 Polymers and copolymers of acrylamide (Am) are the most commonly used hydrogels and they have been widely studied. Polyacrylamide (PAM) hydrogels have found applications in water purification and irrigation, 2 in improving soil texture, in pesticide formulations to limit spray drift 3 and in medical applications. 4 They are also used in molecular biology laboratories as matrices for separating nucleic acid components during DNA sequence analysis and protein identification. 5 PAM hydrogels can absorb relatively high amounts of water furthermore, their swelling capacity is not very sensitive to pH or electrolytes. 6 For some applications, high water absorption capacity is desired because it increases the permeability and the biocompatibility of the hydrogels. However, these hydrogels present poor mechanical properties because water does not contribute to mechanical strength. 7 To increase the range of applications, two types of approaches are necessary: (i) controlling the microstructure restricting inhomogeneities and (ii) improving the mechanical strength of these materials while maintaining their water absorption capacity.

To produce hydrogels with relatively high mechanical strength combined with high water content, some procedures have been reported: (i) "topological (TP) gel". 8 The TP gels have figure-of-eight crosslinkers that can slide along the polymer chains. Reflecting this flexible crosslinker, TP gels absorb large amounts of water, and can be highly stretched without fracture. (ii) "nanocomposite (NC) gel". 9 In NC gels, polymer chains can be crosslinked by inorganic clay slabs on the scale of a several tens of nanometers, instead of organic crosslinking agents. 10,11 The NC gels are also highly stretchable, and possess other favorable physical properties such as excellent optical transparency (iii) "interpenetrating polymer network (IPN)". 12,13 IPN gels consist of two interpenetrating polymer networks: one is made of highly crosslinked rigid polymers and the other is made of loosely crosslinked flexible polymers. (iv) "snake-cage (SC) gel". In SC gels a linear hydrophobic polymer 14 or polyelectrolyte 15 (snake) with good mechanical properties is entrapped in crosslinked hydrophilic polymer (cage).

These types of gels with high mechanical strengths have made a breakthrough in finding wide applications in industrial and biomedical fields, and also raise fundamental problems in gel science. In our case, we studied particular gels, i.e. PAM hydrogels containing slightly crosslinked polystyrene (PS) nanoparticules called nanoparticles composite gels (NPC gels), which belong to the family of snake-cage (SC) gels where monodisperse, hydrophobic and non-ionic nanoparticles of PS take the place of the linear polymer. We used the particular mechanical properties to develop a stimulus responsive ("smart") material, i.e. a mechanoelectrical effect which may be used in the development of soft and wet tactile sensing devices

In this work, we studied the polymerization kinetics of these NPC gels by 1 H-NMR spectroscopy and rheological properties. The hydrophobic particles were prepared by direct (o/w) emulsion polymerization. The free radical polymerization process and the mechanical properties of these new type of snake-cage gel were compared with those of conventional hydrogels (i.e., without particles) with similar PAM content and degree of crosslinks (PAM gels). The difference in their elastic properties was used for producing an electrical potential by simple compression between two hydrolyzed NPC gels.

Experimental section

Materials. For nanoparticles synthesis, styrene (Aldrich) was purified by passing through a basic activated alumina column before use. Ammonium persulfate [APS, (NH 4 ) 2 S 2 O 8 ] and divinylbenzene were purchased from Aldrich and used as received. Two surfactants were used for the preparation of nanoparticles: Igepal CO 897 © (called NP40, hydrophilic-lypophilic balance HLB=17.8) obtained from Rhodia © and Ninol © (HLB=11.4) obtained from Stepan © . Surfactants were used as received. For NPC gels synthesis, acrylamide (Am) (purity >99 %) and N,N'-methylene-bisacrylamide (BisAm) (purity >99 %) were respectively purchased from Aldrich and Fluka. 2, 2'-Azobis (2-methylpropionamide)dihydrochloride called VAZO®56WSP from Dupont Chemicals (M w = 271.19 g.mol -1 , t 1/2 = 10 h at 56 °C) was used as initiator for hydrogel polymerization. Water was distilled three times on Pyrex-quartz glass apparatus and prefiltered with 0.22 µm Millipore filter just before use to exhibit conductivity below 0.055 µΩ.cm -1 . Reaction mixtures were prepared with the filtered water. All 1 H-NMR spectra were recorded with D 2 O as solvent (purchased from SDS, purity >99.9 %).

Crosslinked polystyrene latex particles. Latex particles were synthesized according to the procedure used by Kohut-Svelko et al. 16 The reaction vessel was purged with nitrogen to remove traces of oxygen. In a typical polymerization, reaction mixture of non ionic surfactants (NP40 62wt%, Ninol 38wt %) was dissolved in distilled water (200 mL) in a three-necked round-bottom flask under a nitrogen atmosphere. The purified monomers styrene (87 g, 0.836 mol) and divinylbenzene (3.7 wt % vs. styrene) used as the crosslinking agent, were added to the aqueous solution and emulsified under vigorous stirring. This emulsion was then heated under mechanical stirring up to 70 °C. The aqueous solution of APS (0.3 g in 5 mL of water) was then added drop by drop. The polymerization was then allowed to proceed for 24 h at 70 °C under mechanical stirring.

Nanoparticles analyses.

Nanoparticles were characterized on a dynamic light scattering apparatus DL 135-45, developed by the IFP (French Petroleum Institute). The technique works on a very thin suspension layer to avoid multiscattering phenomena and can be successfully applied to concentrated, dense, and opaque solutions (laser wavelength : 633 nm).The results were confirmed by classical DLS (Sematech) on very diluted latex solutions (solvent, water; wavelength, 514.5nm). To avoid particles aggregation, solutions were put into ultrasonic bath during one hour.. Moreover, Environmental Scanning Electron Microscope (ESEM) analysis (Figure 1), carried out using an ElectroScan instrument operating at 20 kV, showed that particles are monodisperse. Radius obtained was (200 ± 30) nm from both techniques Insert Figure 1 Gel samples. Gels were prepared by classical radical polymerization with BisAm as crosslinker and VAZO56 as initiator at a temperature reaction of 56 °C, unless otherwise stated. Reaction mixtures were prepared few hours before polymerization and thoroughly mixed for 2 h prior to use. The sample solutions were degassed with nitrogen. Hydrogels were prepared by adding monomer (Am), crosslinker (BisAm), initiator (VAZO®56WSP), in D 2 0 and water for NMR and rheological experiments, respectively. The nanoparticles composite hydrogels (NPC gels) containing hydrophobic nanoparticles were prepared by addition of PS latex into the above solution. To prepare different compositions for NPC gels, amounts of monomer, crosslinker, initiator and PS latex have been varied over a wide range in order to emphasize their own effect in the polymerization process.

1 H-NMR monitoring. NMR apparatus is a 400 MHz Bruker Advanced AM400 spectrometer. Once the reaction mixtures prepared and mixed, there were poured into NMR glass tubes (5 mm interior diameter), purged with nitrogen and sealed with paraffin film.

Around 50 measurements were taken continuously during the polymerization in order to follow the monomers units conversion as a function of time (see Figure 2). Indeed, we considered three integrations: the first one corresponds to vinylic protons of monomers (at 6.0-6.7 ppm), the second one corresponds to methylene protons of BisAm (4.95 ppm) and the third one to protons of aliphatic carbon in the polymer backbone to network gathered (at 1.7-2.7 ppm). Thus, we obtained the overall monomers conversion against time.

Insert Figure 2 Rheological measurements. Rheological measurements of the NPC gels were carried out using a Bolhin CVOR 150 rheometer equipped with a Peltier device for temperature control.

Parallel-plate geometry was used with a gap of 2 mm. The upper plate was made of stainless with a 40 mm diameter and the lower plate was a stainless Peltier surface. We noticed a strong adhesion of NPC gel in the stainless surface. Reaction mixture was prepared and mixed as described above and placed into the parallel plates at a temperature of 56 °C in order to form the hydrogel in situ. Once the polymerization was complete, dynamic frequency sweep tests were recorded in an auto-stress mode (equivalent to constant strain mode) at deformation γ 0 = 0.10 maintained constant over the a frequency range of 0.1-10 Hz at 25 °C. For all experiments, linear regime was checked and a solvent trap containing silicon oil was used to minimize evaporation. A thin film of oil was applied around the samples (once the gel state reached) when it was necessary. We compensated for the density and thermal expansions of the experimental set up by using normal force compensation.

Kinetics aspect of the copolymerization. For propagation in free radical copolymerization, it is generally considered that the reaction rate for a given comonomer depends only on which monomer bears the radical at the chain terminus [START_REF] Odian | Principles of Polymerization[END_REF] . For the molar concentrations of vinylic units in Am (A) and in BisAm (B) and by considering the equal reactivity of propagating species whatever the polymerization degree of propagated chains is, this leads to propagation equations of the form:

* * [ ] [ ][ ] [ ][ ] AA BA d A k A A k A B dt - = + (5) * * [ ] [ ][ ] [ ][ ] BB AB d B k B B k B A dt - = + (6)
where A* and B* are the propagating radicals of A and B, respectively; k AA and k BB the rate constants of homopropagation reactions, k AB and k BA the rate constants of cross-propagation reactions. The equations rate for propagating radicals A* and B*, are:

* * * * * * 2 [ ] 2 [ ] [ ][ ] [ ][ ] [ ]² - [ ] [ ] d A B B A A A A B d A F k I k B A k A B kt A kt A B dt = - + - (7)
* * * * * * 2 [ ] 2 [ ] - [ ][ ] [ ][ ] - [ ]² - [ ] [ ] d B A A B B B A B d B F k I k A B k B A kt B kt A B dt = + (8)
* * * * * * 2 2 [ ] [ ]² - [ ] [ ] [ ]² - [ ] [ ] d A A A B B B A B F k I kt A kt A B kt B kt A B = = (9)
[ ] [ ]

0.5 * * tAA BA AB tBB A A k k k B k B ⎡ ⎤ ⎛ ⎞ ⎣ ⎦ = = ⎜ ⎟ ⎡ ⎤ ⎝ ⎠ ⎣ ⎦ (10)
The instantaneous composition of copolymer was determined using the Mayo-Lewis 18 copolymer equation given below:

[ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] A B d A A r A B d B B A r B ⎛ ⎞ ⎛ ⎞ + = ⎜ ⎟ ⎜ ⎟ + ⎝ ⎠ ⎝ ⎠ ( 11
)
where the reactivity ratios r A and r B are defined as following:

AA A AB k r k = ; BB B BA k r k = (12)
The instantaneous mol fraction of [A] incorporated, F A , is given by:

[ ] ² (1 -) [ ] ² 2 (1 -) (1 -)² A A A A A A A A A B A d A r f f f F d A B r f f f r f + = = + + + (13)
where f A and f B are the mol fractions of monomers A and B, respectively. F B is obtained by interchanging the subscripts A and B. These expressions form the basis for finding composition and sequence length distributions.

[ ] [ ] [ ]

A A f A B = + ; [ ] [ ] [ ] B B f A B = + (14)
If the composition drift and initiator decomposition rate are slow, the concentrations of A* and B* can be assumed almost constant throughout the reaction, so that [A] and [B], at least at early values of conversion, will yield first-order decay kinetics of the form:

- 0 [ ] [ ] A t A A e α = ; - 0 [ ] [ ] B t B B e α = ( 15

[ ] [ ]

A A A B A k A k B α = + ; * * [ ] [ ] B B B A B k B k A α = + (17)
Otherwise, the reactivity ratios can be determined by finding the ratio of these rate constants, determined over early conversion, for separate experiments in which the relative initial concentrations [A] 0 and [B] 0 varied. Using equations 9, 10 and 12 that yields

0 0 0 0 [ ] 1 [ ] [ ] [ ] A A B B A r B A r B α α + = + (18)
Results and Discussion 1. Kinetics aspect by 1 H NMR

PAM gels at low crosslinker concentration

Insert Table 1 Figure 2 shows raw spectra for a typical 1 H-NMR monitoring experiment (run A3 in Table 1).

The signal at 4.7 ppm corresponds of the residual water in D 2 0. The signal decrease between 6.0 and 6.7 ppm and the signal increase between 1.7 and 2.7 ppm show the conversion of vinylic units A and B, whereas the signal decrease at 4.95 ppm is due chiefly to the conversion of the crosslinking agent, i.e. it corresponds to one vinylic function of BisAm.

These three signals yield the monomers conversion, according to the following equations:

p A = 1-[A]/[A] 0 ( 19
)
p B = 1-[B]/[B] 0 (20)
The first series of experiments (series A in Table 1), were prepared with different amounts of initiator while the fraction of crosslinking agent %B was kept roughly constant.

Unfortunately, the concentration of B was too small to make the signal at 4.95 ppm accurate.

Figure 3 shows only monomer conversion p A versus time for different initiator concentrations while keeping %B and %M constant.

Insert Figure 3

The curves in Figure 3 can be treated approximately as first order over a significant portion of the reaction. Nevertheless, fits of the form:

p A = 1 -exp (-α A . t) ( 21
)
are also shown in Figure 3, where α A is the first-order decay rate constant defined by equation 15. The fit is best for all the reactions presented in Table 1. The ln[A] versus time curves did not show a steepening with increasing conversion. This indicates that auto acceleration was absent or insignificant. The inset in Figure 3 The slope of the plot is hence related to the apparent rate constants k app via

dα A /d[I 2 ] 0 0.5 = k app = 1.09 × 10 -1 s -1 .mol -1/2 .L 1/2
for %B = 1 and where the approximation is made that the initiator decay rate is slow compared to the time for the complete reaction, so that [I 2 ] ≈ [I 2 ] 0 . As expected, the reaction time is short compared to the half-time life of initiator (10 h at 56 °C). Our system using azoic initiator is significantly faster than the classical one using a persulfate initiator for the free radical homopolymerization of Am: k app = 1.05 × 10 -2 s -1 .mol -1/2 .L 1/2 and 4.58 × 10 -3 s -1 .mol - 1/2 .L 1/2 at 50 and 60 °C respectively. 19 These values favor the formation of the NPC gel in reasonable time with %B = 1, a small amount of initiator and a conversion close to 100%.

The series of experiments concerning the same reaction mixture at different temperatures (cf.

Table 1, Runs B1 to B4) is plotted on the Figure 4. The inset shows a linear variation of ln(α A ) versus 1/T (T in Kelvin) according to the Arrhenius relation

α A = α A 0 . exp (-∆E / RT) ( 22
)
where ∆E is the apparent activation energy of the global process in this free radical copolymerization of acrylamide and R the perfect gas constant. The data from the inset Figure 4 yield activation energy ∆E = 67 kJ.mol -1 as the same order of classical radical polymerization. 19 Insert figure 4 1.

PAM gels at high crosslinker concentration

A series was prepared with high concentrations of crosslinker (runs C5 to C11 with %B varying from 5 to 27) to determine the influence of a great increase of the amount of vinylic unit. Indeed, if [B] is high enough, the signal at 4.95 ppm is accurate. So, kinetics study can be performed with both considering the conversion of the two monomers (p A and p B ) and determining α A and α B .

Figure 5 shows the initial decay rates, i.e. the initial slope of semi-logarithmic plot of monomer concentration ((1p A ) and (1p B )) versus time for runs C7 and C11. The monomer decay in each case follows a first-order kinetics over almost the entire range of the experiments, as shown in Figure 6. Steady state approximations and slow monomer concentration ratios variation fit well. Inset of Figure 6 shows the determination of r A and r B using the ratio of A and B rate constants from equation 18 (runs C5 to C9). The experimental values of r A and r B are 0.52 ± 0.04 and 5.2 ± 0.8, respectively. Even in the case of high drift, the initial decay rates should still yield a good approximation for the reactivity ratios. This method, using rate constants, issued from NMR data is convenient and accurate provided that the same initiator concentration is used in each experiment.

Nevertheless, from equation ( 18), it can be seen that the value of r B is dependent on low value of the ratio [A] 0 / [B] 0 where there is little variation in α A / α B . The main conclusion is the very high reactivity of BisAm compared with Am, as already reported elsewhere (r A = 0.57 and r B = 3.4 at 22 °C 20 ). As we can observe in Figure 6, in the first 1000s of the copolymerization, the major part of BisAm was consumed while the Am monomer conversion reached only 0.86. This phenomenon was enhanced by increasing [B] 0 .

Insert figure 5 and 6

Moreover, the use of the Mayo-Lewis theory allows computing the sequence length distribution. W BB denotes the probability that B* adds to B. Using the reactivity ratios, this can be expressed, at any conversion point p, during the reaction as

( ) ( ) ( )( 1) 1 B B BB B B r f p W p f p r = -+ (23)
Where conversion point p is the total conversion of monomers.

The probability of having a sequence of k monomers B in a row , followed by a monomer A, P B,k , then follows the well-known geometric distribution : 17

B n BB N p W p = - (25)
To obtain the appropriate expressions for monomer A, the subscript A is substituted for B in each variable in the above equations.

Figure 7 shows the probability that B* adds to B and the instantaneous number B sequence lengths, <N B > n vs. conversion, for experiment C6. These values were computed from eqs. 23-25, using the value of r B = 5.2.

Insert figure 7

Figure 8 shows a mechanism of the network formation, by taking into account the very strong difference between r A and r B (r A = 0.52 ± 0.04 and r B = 5.2 ± 0.8 in our system). Run C6 was chosen in order to demonstrate the occurring phenomena. Before polymerization begins, there is a small part of crosslinker molecules in the reaction mixture with high concentration of vinyl groups. Then, free radical polymerization begins with a few chains of linear PAM growing incorporating some BisAm molecules. Indeed, with [B] = 55.10 -3 mol -1 .L -1 , an average quantity of one BisAm molecule against 9 Am molecules are polymerized (F B = 0.11). Moreover, the determination of <N B > n (at p=0, <N B > n = 1.3) can be understood as an average of the incorporation in the PAM growing chain of a sequence of two BisAms at one point followed further by two individual BisAms (4/3 ≈ 1.3). In consequence (Figure 8a), the sequence of two BisAms would be the cause of the initial formation of very dense and crosslinked microdomains. However, the incorporation of only one BisAm permits the formation of simple node. Finally (Figure 8b), these phenomena, occurring during the early stages of the polymerization, lead to a network which already exhibits structural inhomogeneities.

Insert figure 8

Other factors than the reactivity ratios have been studied to explain these inhomogeneities such as the nature of the crosslinker, 21 the nature of the solvent 22 and the temperature. 23 Finally, our results confirm the commonly held view of the existence of microdomains of highly crosslinked and reticulated clusters, joined by PAM chains. 22,24,25 Thus, using NMR technique for studying polymer networks is very interesting because a mechanism of the network formation can be built thanks to the determination of some kinetics constants; in consequence, the microstructure of PAM gels can be controlled by restricting inhomogeneities. Indeed, small amount of B with %M = 7 permits limiting high crosslinked microdomains and inhomogeneities. That is why these concentrations (or less) will be used for the following kinetics study of NPC gels. After the kinetics study of PAM gels, the formation of NPC gels and their mechanical strength were studied in the aim of widest future applications.

Nanoparticle composite gel (NPC)

In this part, the influence of the introduction of latex PS nanoparticles in the reaction mixture was analyzed in terms of apparent rate constants of propagation reactions of A with a low concentration of B (Table 2).

Insert Table 2 -Insert Figure 9

For experiments D1 to D5, the incorporation of latex particles showed a decrease of α a along with an increase of the nanoparticles volume fraction, φ. Although α a is lower, the final conversion of P A for D5 and D6 experiments can reach up to 0.8 but the polymerization rate is slower and the time to have the same conversion is higher than for the PAM matrix alone (about 4000s). This result was unexpected: indeed, the nanoparticles which are just incorporated in the PAM matrix are neutral in terms of reactivity and were not supposed to influence the kinetics of the crosslinked PAM. This feature occurred without any modification of the first-order kinetic plot, as shown in Figure 9. As shown in the following section, the incorporation of latex particles increased the elastic modulus of the gel, and could explain the slower propagation rate. However, comparison of runs D5 and D6 performed with and without crosslinker exhibited the same rate constants (values of α A for D5 and D6 are respectively equal to 0.37 × 10 -3 and 0.38 × 10 -3 mol.s -1 .L -1 ) proves that gel effect does not occur in our system. Moreover, the presence of surfactants in the reaction medium could act as an inhibitor by chain transfer, since propagating radicals will be removed and very few new ones will be created. In this case, chain transfer agent actually seemed to lead to a decrease of the conversion rate. This was explained by runs D7 and D8, with no PS particles and different overall concentration of surfactants mixture used for PS nanoparticles (D7: Ninol = 0.48 wt% and NP40 = 1.18 wt%, and D8: Ninol = 0.26 wt% and NP40 = 0.59 wt %). The results were compared to those obtained for the experiment D9 containing no surfactant and showed no or very few contribution of the surfactants in the kinetic process of PAM gel formation.

The decrease in conversion rate along with the fraction of PS particles cannot be explained by gel effect or by the presence of residual surfactant. Instead, we think that this feature is rather due to a steric effect. Indeed, the latex particles are spherical objects; we can assume that they adopt a regular space organization of cubic lattice in the gel. Several types of cubic lattices could be considered, for example a simple cubic (sc), a body-centered cubic (bcc), or a facecentered cubic (fcc) structures. The lattice constant, a i , is related to the distance between the planes, d hkl , through the equations : a sc = d 100 , a bcc = 2 1/2 d 100 , and a fcc = 3 1/2 d 100 . Considering that a i 3 is the cubic lattice volume and that there are 1, 2, and 4 scattering objects per lattice for the sc, bcc, and fcc, respectively, it is then possible to write that the lattice constant a i is a function of the radius R and of the particles volume fraction φ :

( )

3 3 4 3 R a α π φ = ( 26
)
Where α is the prefactor depending of the nature of the cubic lattices, 1, 2 and 4 for sc, bcc, and fcc, respectively. In the case where R = 202 nm and φ = 0.13, the edge to egde distance d between surface particles are 239, 481 and 169 nm for simple cubic (sc), a body-centered cubic (bcc), or a face-centered cubic (fcc) structure respectively. Moreover, the radical polymerisation of Am with no crosslinker (at 56°C and with the same initiator concentration as in run A3) provided a linear PAM chains with high molecular weight (M w = 1.35 × 10 6

g.mol -1 ) and a radius of gyration of 82 nm. The corresponding diameters are of the same order of magnitude as the distance d between surface particles. In this case, the particles could be considered as a hindrance of the growing chain and slow down the propagation rate of the growing chain in the reaction medium. In fact, the rate of propagation α A decreased by a factor close to ten between run D9 containing no particle and runs D5 and D6 realized with 13% of PS particles, with and without crosslinker, respectively. It is clear that if we take into account these geometric considerations, the decrease in conversion rate with the fraction of PS particle could surely be explained by a steric effect.

In addition, even if this spectroscopy technique is widely used for monitoring kinetics on linear polymers, our study shows that it is also easily possible in the specific case of three dimensional polymers containing additional particles or not. In our knowledge, this specific case was not yet developed in literature.

Rheological behavior

Thanks to the kinetics aspects described above, we ensured that NPC gels syntheses occur in classical ways and reach to completion. So, we wanted to study the rheological properties of these NPC gels. As previously shown, PS latex nanoparticles were considered as well-known model nanoparticles which can be easily used in our hydrogels systems. The influence of PS nanoparticles on rheological behavior of NPC gels was studied in the same experimental conditions as for kinetics study with a volume fraction of nanoparticles φ ranged from 0 to 26%. The samples composition is given in Table 3 (runs G1 to G6).

Insert Table 3 Once the polymerization was achieved, dynamic oscillatory tests in frequency were performed between 0.1 and 10 s -1 . The good reproducibility of the results of G'(w) confirmed that the polymerization has reached completion. For all hydrogels, G'(w) stayed constant with a standard deviation close to 0.5%, and G''(w) could be considered negligible against G'(w).

The stability and the relative large value of G'(w) compared to G''(w) over a range of at least three frequency decades is a classical characteristic feature of a crosslinked hydrogel. 7,26 This stability also indicated that the permanent chemical crosslinks were not destroyed by the increasing frequency at constant strain γ 0 . Moreover, γ 0 = 0.10 was within the linearity domain. Indeed, G' was measured as a function of increasing strain amplitude γ 0 . G' remained constant for low strains (γ 0 < 0.35). The linearity domain decreased as the elasticity of the hydrogel increased.

Experimental data are reported in Table 3 and Figure 10 shows the variation of G' with the volume fraction, φ of PS nanoparticles. The experimental results showed a strong reinforcement of rheological properties of NPC gels which was greater at higher nanoparticles content.

In order to evaluate in a better way our NPC gels, we used a rheological behaviour modelling of a composite of two elastic bodies without interfacial tension, proposed by Kerner 27 and revisited by Bousmina 28 . The elastic modulus of the blend G b can be related to the elastic modulus of the matrix G m and the dispersed phase G d as follows:

(

3 ) 3 ( - ) ( 2
3 ) -2 ( -)

d m d m b m d m d m G G G G G G G G G G φ φ ⎛ ⎞ + + = ⎜ ⎟ + ⎝ ⎠ (27)
We can consider that G d is close to 6.10 9 Pa 29 and the elastic modulus of the matrix G m = 2150 Pa as measured for the hydrogel without PS nanoparticles (run G1). Thus, a theoretical behavior was plotted and reported in Figure 10 along with our experimental data. It is obvious that our NPC gel systems do not follow the Kerner model i.e. that elastic modulus of the blend G b obtained are higher than the expected ones for volume fractions higher than 0.05.

This difference could be explained with the elastic modulus G m values. Indeed, when nanoparticles are incorporated into the gel, the kinetics of NPC gels is greatly dependent on the nanoparticles content (see kinetics section) and the rheological properties may be affected by the modifications of the gelation process. Such, G m drastically increased as the volume fraction of nanoparticles contained in our NPC gels. Thus, G m obtained from eq. 27 shows this phenomenon: 7476 and 19 219 Pa for G4 and G6 respectively. Otherwise, a slight adhesive interaction between PS nanoparticles and PAM matrix can explain the difference between experimental and theoretical data.

Insert Figure 10 3

. Mechanoelectrical effects of hydrolysed NPC gels

Hydrogels with high mechanical strength may have several applications and among them, we chose to evaluate the chemoelectrical properties of our systems.

It is well known that polyelectrolyte gels can contract or deform under an electrical stimulus, that is, a gel can convert the electrical energy into mechanical work. The reverse process was also previously reported, 30 it produces electrical potential from mechanical deformation. As shown in Figure 11a, a cell containing two stacked layers of hydrolysed NPC was made: the bottom layer was a gel containing 26% of nanoparticles and the top one did not contain any particles. Indeed, the bottom part which presents interesting rheological properties shows also a different behaviour under compression (the deformation of this bottom layer is lower than the one of the top layerif the same stress is applied). A pair of platinum wire electrodes, one as reference and the other one as working electrode, was inserted to measure the electrical potential. When the top gel layer was compressed (with a ∆L 1 /L deformation), the extra protons migrated to the bottom gel layer, slightly deformed (∆L 2 /L), through the interface until the Donnan equilibrium was reached. An electrical potential variation was observed during this period as shown in Figure 11b. These preliminary results showed that the tactile sensing systems were successfully made from polymer gel. Moreover, the electrical signal enhanced in proportion to the amplitude of the applied strain ∆L/L (or stress).

On the basis of this principle, a soft and wet tactile sensing device could be developed by connecting the electrodes to two NPC gel layers exhibiting different elastic modulus, as previously shown.

Insert Figure 11

On the basis of swelling and contraction of a weak polyelectrolyte gel, Katchalsky et al. 31 proposed a so-called ''muscle'' model which was referred to a chemomechanical system.

Thanks to this similarity and the common feature with the natural tissue (softness, wetness, elasticity, and some other rheological specified characteristics), the soft mechanoelectrical system obtained from a NPC gel may open new possibilities in the investigation of artificial tissue-like tactile perception for robotics or psycho sensorial material area.

Conclusion

We studied particular gels which belong to the family of snake-cage gels where monodisperse, hydrophobic and non ionic nanoparticles take the place of the linear polymer.

The polymerization kinetics of PAM hydrogel with and without polystyrene nanoparticles (NPC gels) has been performed by 1 H-NMR spectroscopy. Reactivity ratios were computed using initial kinetics in the case of PAM gels at high crosslinker concentration. Furthermore, it is possible to determine the instantaneous and cumulative sequence length distribution and average, based on the Mayo-Lewis model. The different results and their interpretation are in accordance with previous studies on inhomogeneities in this kind of hydrogel.

For the NPC gels, the decrease of conversion rate with the PS nanoparticles content could be explained by a steric effect which probably induces an increase of the elastic modulus of the matrix and is related to nanoparticles content. The results show a strong reinforcement of mechanical properties of NPC gels which is more pronounced for the higher fraction of nanoparticles. These differences of rheological properties have been used for producing an electrical potential by simple compression between two hydrolyzed NPC gel layers. This concept can be used to develop a soft and wet tactile sensitive device. The soft mechanoelectrical system made with a NPC gel may open new possibilities in the investigation of artificial tissue-like tactile perception for robotics or psycho sensorial material area, for example.

Captions of Tables and Figures

Gels compositions .

 compositions In this paper, several terms are introduced to classify the hydrogels compositions: %M refer the total concentration of monomers as a weight percentage; %B, %I refer respectively to the weight concentrations of crosslinking agent, initiator taking into account both acrylamide and crosslinker; φ represents the volume fraction of polystyrene particles to the total volume of the reaction mixture, [A] and [B] are the molar concentrations of vinylic units of acrylamide and BisAm respectively and [I 2 ] 0 is the initial molar concentration of initiator, taking into account the volume of the aqueous phase without the contribution of hydrophobic PS nanoparticles.

where 2Fk d [I 2 ]

 2 means the formation rate of active centers I* issued from the chemical decomposition of the initiator I 2 , F the efficiency parameter of initiator, k tAA [A*] 2 and k tBB [B*] 2 the rates of homotermination reactions and k tAB [A*][B*] the rate of crosstermination reactions. The quasi steady-state approximation applied to radicals assumes that the variation of [A*] and [B*] is negligible towards the other reaction rates, thus d[A*]/dt = d[B*]/dt = 0. In terms of the creation and loss of propagating radicals, the contribution of initiation and termination reactions is negligible compared to that of the cross-propagations, thus:

  and 16) where α A and α B are the apparent rate constants of propagation reactions of A and B respectively, with * *

  shows α A vs [I 2 ] 0 0.5 corresponding to equation 17 with [B*] << [A*] and plotted for runs A1 to A4. The linearity is quite good for all values.

Figure 6 .Figure 7 .Figure 8 .Figure 9 .

 6789 Figure 6. Am (p A ) and BisAm (p B ) conversion as a function of t, at %B = 9.3 and T =56°C (run C10), including the first-order fit according to eq 15 and 16. Inset: determination of r A and r B using the ratio of A and B rate constants from eq 18. The values of r A and r B are 0.52 and 5.2, respectively

Figure 10 .Figure 11 .

 1011 Figure 10. Evolution of elastic modulus as a function of the volume fraction of PS nanoparticles incorporated into the NPC gel. The line represents the Kerner's model and the points are experimental data

Figure 2 .Figure 3 .

 23 Figure 2. Spectra evolution of reaction mixture for the in situ PAM hydrogel formation in 1 H-NMR probe

Figure 4 .Figure 5 .

 45 Figure 4. Am conversion p A as a function of t, at various temperatures T, including the firstorder fit according to eq 15. Reaction designations are from Table1. Inset: first-order monomer decay rates in function of 1/T

Figure 6 .Figure 8 .

 68 Figure 6. Am (p A ) and BisAm (p B ) conversion as a function of t, at %B = 9.3 and T =56°C (run C10), including the first-order fit according to eq 15 and 16. Inset: determination of r A and r B using the ratio of A and B rate constants from eq 18. The values of r A and r B are 0.52 and 5.2, respectively

Figure 9 .

 9 Figure 9. Am conversion p A as a function of t, in the presence of PS nanoparticles at T = 56°C, including the first-order fit according to eq. 15. Reaction designations are from Table 2

Figure 10 .Figure 11 .

 1011 Figure 10. Evolution of elastic modulus as a function of the volume fraction of PS nanoparticles incorporated into the NPC gel. The line represents the Kerner's model and the points are experimental data

Table 1 .

 1 Experimental conditions and kinetics parameters for the hydrogel polymerization

Table 2 .

 2 Experimental conditions and kinetics parameters for the NPC hydrogel polymerization

Table 3 .

 3 Experimental conditions and rheological results for the NPC hydrogel polymerization

	Figure 1. ESEM images of PS latex particles
	Figure 2. Spectra evolution of reaction mixture for the in situ PAM hydrogel formation in 1 H-
	NMR probe
	Figure 3. Am conversion p A as a function of t, at T =56°C, including the first-order fit
	according to eq 15. Reaction designations are from Table 1. Inset: First-order monomer decay
	rates vs. [I 2 ] 0	1/2 for %B = 1
	Figure 4. Am conversion p

A as a function of t, at various temperatures T, including the firstorder fit according to eq 15. Reaction designations are from Table

1

. Inset: first-order monomer decay rates in function of 1/T Figure 5. Initial decay rates of monomers A and B; 1-p A (full symbols) and 1-p B (empty symbols) versus time for runs 7 and 11

Table 3 .

 3 Experimental conditions and rheological results for the NPC hydrogel polymerization

			[A]		[B]	[I 2 ] 0
	Run %M	(× 10 1 )	%B	(× 10 3 )	(× 10 3 )	φ (%)	G' (Pa)
			(mol.L -1 )		(mol.L -1 )	(mol.L -1 )
	G1	7.1	9.9	1.0	9.2	2.6	0	2 200
	G2	7.1	9.9	0.9	8.1	2.8	0.32	2 300
	G3	7.1	9.9	0.8	7.0	2.8	1.81	2 100
	G4	7.0	9.8	0.8	7.1	3.5	15.1	10 800
	G5	7.3	10.2	0.8	7.1	3.4	20.3	19 000
	G6	7.4	10.3	0.8	7.2	3.4	26.0	36 100

Acknowledgment

The authors wish to thank Nicolas Kohut-Svelko for his help in the preparation of PS latexes, Gérald Clisson and Francis Ehrenfeld for technical assistance. The Communauté d'Agglomération de Pau (France) is greatly thanked for financial support. The authors would like to acknowledge Virginie Pellerin for ESEM images, Roger C. Hiorns for help in the preparation of the manuscript and Jeanne François for initiating this work.


Claude Lefèvre

Stéphane Loisel

Stationary-excess operator and convex stochastic orders

Keywords: primary 60E15, 62P05, secondary 60E10 Insurance risks, Stochastic orders, Monotone distributions, Conjugate operator, Stochastic extrema

The present paper aims to point out how the stationary-excess operator and its iterates transform the s-convex stochastic orders and the associated moment spaces. This allows us to propose a new unified method on constructing s-convex extrema for distributions that are known to be t-monotone. Both discrete and continuous cases are investigated. Several extremal distributions under monotonicity conditions are derived. They are illustrated with some applications in insurance.

Introduction

Concepts of stochastic orderings are useful in a number of applied probability models. This is especially true in insurance and finance when different risk scenarios are possible and have to be compared. Such questions arise, for instance, in life insurance, ruin theory and portfolio analysis. A theory of stochastic orders with various applications can be found in the books by [START_REF] Goovaerts | Effective Actuarial Methods[END_REF], [START_REF] Kaas | Ordering of Actuarial Risks[END_REF], [START_REF] Ross | Stochastic Processes[END_REF], [START_REF] Müller | Comparison Methods for Stochastic Models and Risks[END_REF] and Shaked andShanthikumar (1994), (2007).

This paper is concerned with the class of s-convex stochastic orders, where s ∈ IN 0 = {1, 2, . . .}. For s = 1, this order is the classical stochastic dominance. For s = 2, it is the well-known convex order and corresponds, in actuarial sciences, to the stop-loss order with fixed mean. For an arbitrary s, the s-convex order compares the s-th right-tail distribution functions of random variables that have the same first s -1 moments. In the case of discrete distributions, this class of orders was studied in [START_REF] Lefèvre | An unusual stochastic order relation with some applications in sampling and epidemic theory[END_REF], Fishburn and Lavalle (1995), [START_REF] Lefèvre | Comparing sums of exchangeable Bernoulli random variables[END_REF], [START_REF] Denuit | Some new classes of stochastic order relations among arithmetic random variables, with applications in actuarial sciences[END_REF] and Denuit el al. (1999aDenuit el al. ( ), (1999c)), among others. The case, more traditional, of real-valued random variables was investigated by [START_REF] Rolski | Order relations in the set of distribution functions and their applications in queueing theory[END_REF], [START_REF] Levy | Stochastic dominance and expected utility: Survey and analysis[END_REF], [START_REF] Denuit | The s-convex orders among real random variables, with applications[END_REF]Denuit et al. ( ), (1999b) ) and in many other works.

A directly related question is the construction of stochastic extrema with respect to the sconvex orders. In an actuarial context, the extremal risks represent the less and more dangerous risks, and their knowledge can yield accurate lower and upper bounds on various risk quantities It is worth recalling that in a renewal process, if an interval between points is of distribution function F X , then the associated stationary-excess mapping H(F X ) gives the distribution function of the interval to the next point from an arbitrary time in equilibrium. To begin with, we are going to introduce a similar operator H that is specific to discrete distributions.

Discrete version

Let us assume that X is a discrete non-negative random variable with probability mass function P X = {P (X = j), j ∈ IN = {0, 1, . . .}} and mean E(X) > 0. Obviously, the classical stationary-excess operator may be applied here too. As explained by [START_REF] Whitt | The renewal-process stationary-excess operator[END_REF], however, it is more appropriate to work with a discrete version that is directly applicable to discrete-time renewal processes. We propose to adopt the following definition.

Definition 2.1 A discrete stationary-excess operator H maps any such random variable X to an associated discrete non-negative random variable X H whose probability mass function H(P X ) is defined by H(P X )(j) ≡ P (X H = j) = P (X ≥ j + 1) E(X) , j ∈ IN.

(2.1)

Let us notice that a slightly different operator was investigated by [START_REF] Whitt | The renewal-process stationary-excess operator[END_REF] for a discrete positive random variable X. Specifically, the associated random variable X H is of probability mass function defined by P (X H = j) = P (X ≥ j) E(X) , j ∈ IN 0 .

Contrary to that operator, H defined by (2.1) does not yield a one-to-one correspondence on the set of probability measures on IN. Indeed, it is directly checked that X H and (νX) H are equidistributed if ν is an indicator independent of X. Nevertheless, when E(X) is fixed, H gives a one-to-one correspondence since P (X = 0) = 1 -E(X) P (X H = 0), and

(2.2) P (X = j) = E(X) [P (X H = j -1) -P (X H = j)], j = 1, 2, . . . .

(2.3)

The definition (2.1) has the advantage to lead to a simple relationship between the binomial moments of X H and X. We make the convention x y = 0 if x < y. Lemma 2.2

E X H i = 1 E(X) E X i + 1 , i ∈ IN.
(2.4)

Proof. Let us consider the iterated right-tail distribution functions of X, i.e. F0 (X, j) = P (X = j) and

Fi+1 (X, j) = ∞ k=j Fi (X, k), i, j ∈ IN.
As proved in [START_REF] Lefèvre | Comparing sums of exchangeable Bernoulli random variables[END_REF], an equivalent expression is

Fi+1 (X, j) = E X -j + i i , i, j ∈ IN. (2.5)
Let us turn to the iterated right-tail distribution functions of X H . By (2.1), we have

F0 (X H , j) = P (X H = j) = F1 (X, j + 1) E(X) , j ∈ IN.
Arguing by induction, we then find

Fi+1 (X H , j) = Fi+2 (X, j + 1) E(X) , i, j ∈ IN.
(2.6) Finally, taking j = i in (2.5) and (2.6), we obtain the identity (2.4).

Continuous version

Let X be any non-negative real random variable with distribution function F X and mean E(X) > 0. The usual stationary-exces operator is defined as follows [START_REF] Cox | Renewal Theory[END_REF]). For easiness, it will be named continuous subsequently.

Definition 2.3 A continuous stationary-excess operator H maps any such random variable X to an associated non-negative random variable X H whose distribution function H(F X ) is defined by

H(F X )(x) ≡ P (X H ≤ x) = 1 E(X) x 0 [1 -F X (y)]dy, x ≥ 0.
An equivalent expression is

P (X H > x) = 1 E(X) ∞ x P (X > y)dy, x ≥ 0.
(2.7)

This definition of H guarantees a one-to-one correspondence. Indeed, denoting the density function of X H by q X H , one has

P (X > x) = E(X) q X H (x), x ≥ 0, (2.8)
and E(X) then follows as a consequence.

The moments of X H and X are linked through a simple identity, known but rederived below for comparison with Lemma 2.2.

Lemma 2.4 E(X i H ) = E(X i+1 ) (i + 1)E(X) , i ∈ IN.
(2.9)

Proof. The result is related to the i-th stop-loss transform of X, defined as

Π (i) (X, x) = E[(X -x) i + ], x ≥ 0;
(2.10) see, e.g., [START_REF] Cheng | On the nth stop-loss transform order of ruin probability[END_REF]. Indeed, considering the iterated right-tail distribution functions of X, i.e. F1 (X, x) = P (X > x) and

Fi+1 (X, x) = ∞ x Fi (X, y)dy, x ≥ 0, i = 1, 2, . . . ,
we easily see that

Fi+1 (X, x) = Π (i) (X, x) i! , x ≥ 0, i ∈ IN. (2.11)
For the transformed random variable X H , one finds by induction that

Fi+1 (X H , x) = Fi+2 (X, x) E(X) , x ≥ 0, i ∈ IN.
(2.12)

From (2.12) and (2.11), (2.10), we then deduce that

E[(X H -x) i + ] = E[(X -x) i+1 + ] (i + 1)E(X) , x ≥ 0,
which yields (2.9) when x = 0.

H-transform and convex orders

We start by giving a brief presentation of the s-convex stochastic orders and the associated moment spaces. Applying the stationary-excess operator, we then identify the image of these moment spaces and the corresponding stochastic orders. Both discrete and continuous cases will be examined.

Discrete problem

For any s ∈ IN 0 , let F s denote the set of s-convex real functions on IN, i.e.

F s = {f : ∆ s f (j) ≥ 0, j ∈ IN}, ∆ being the usual forward difference operator [∆f (j) = f (j + 1) -f (j), j ∈ IN],
and ∆ s the s-th iterate of ∆.

Definition 3.1 Consider any two probability mass functions P 1 and P 2 on IN. One says that P 1 is smaller than P 2 in the s-convex stocastic sense, written P 1 s P 2 , when

< f, P 1 >≡ E P 1 (f ) ≤ E P 2 (f ) ≡< f, P 2 > for all functions f ∈ F s . (3.1)
As the functions f (j) = j i and f (j) = -j i , j ∈ IN, belong to F s for all i = 1, . . . , s -1, the ordering P 1 s P 2 necessarily implies that P 1 and P 2 have the same first s -1 moments. In fact, if X 1 and X 2 are two random variables with probability mass functions P 1 and P 2 , then an equivalent characterization of (3.1), written as

X 1 s X 2 , is that E(X i 1 ) = E(X i
2 ), i = 1, . . . , s -1, and Fs (X 1 , j) ≤ Fs (X 2 , j), j ≥ s.

  

Now, let us consider the set of probability mass functions on IN with prescribed first s moments, where s ∈ IN 0 . For the sequence, it will be more convenient to work on the basis of the combinatorial moments. So, we introduce a set

B s+1 = P X : E X i = c i , i = 1, . . . , s , (3.2)
where c 1 , . . . , c s represent s given (admissible) constants. Clearly, the distributions in the set B s+1 are susceptible to be ordered with respect to s+1 , the (s + 1)-convex stochastic order. Let us take the discrete stationary-excess operator H defined by (2.1). We have seen before that H yields a one-to-one correspondence on the set of distributions with fixed mean. Thus, this property holds true on the previous set B s+1 . Let us now apply the operator H to B s+1 . On the image set H(B s+1 ), it is then natural to define a transformed order H as follows.

Definition 3.2 Consider any two probability mass functions

Q 1 and Q 2 in H(B s+1 ). One says Q 1 H Q 2 when H -1 (Q 1 ) ≡ P 1 s+1 P 2 ≡ H -1 (Q 2 ). (3.3)
We are in a position to identify the set H(B s+1 ) and the transformed order H . Let Q Y = {Q Y (j), j ∈ IN} denote the probability mass function of a discrete non-negative random variable Y .

Proposition 3.3 H(B s+1 ) = Q Y : E Y i = c i+1 c 1 , i = 1, . . . , s -1, and (3.4) ∆Q Y (j) ≤ 0, j ∈ IN, with c 1 Q Y (0) ≤ 1} , (3.5)
and the order H corresponds to the s-convex stochastic order s .

Proof. From (2.4) and (3.2), we find that the s -1 first binomial moments of Y are provided by the formula (3.4). The remaining restrictions stipulated in (3.5) are straightforward from the relations (2.3) and (2.2).

Let us now identify H . By virtue of (3.1) and

(3.3), Q 1 H Q 2 when < f, H -1 (Q 1 ) > ≤ < f, H -1 (Q 2 ) > for all f ∈ F s+1 . (3.6) Due to (2.2), (2.3), for any Q = Q Y in H(B s+1 ), the expansion of < f, H -1 (Q) > yields < f, H -1 (Q) > = f (0) [1 -c 1 P (Y = 0)] + c 1 ∞ j=1 f (j) [P (Y = j) -P (Y = j -1)] = f (0) + c 1 ∞ j=1 P (Y = j) [f (j + 1) -f (j)]. Therefore, < f, H -1 (Q) > = < (δ 0 + c 1 ∆)(f ), Q >, (3.7)
where δ 0 is the operator such that δ 0 (f ) = f (0). In fact, the equivalence

< f, H -1 (Q) > = < [(H -1 ) * ](f ), Q > (3.8)
holds in general if (H -1 ) * is the conjugate operator of H -1 ; thus, in our case, (H -1 ) * = δ 0 + c 1 ∆.

(3.9)

Substituting (3.7) in (3.6), we have Q 1 H Q 2 when < ∆(f ), Q 1 > ≤ < ∆(f ), Q 2 > for all f ∈ F s+1 .
As ∆ s+1 = ∆ s (∆), that condition means, by (3.1), Q 1 s Q 2 , hence the orders H and s are quite identical. ♦ Note that the image set H(B s+1 ) involves s -1 moment relations, the nonincreasingness and an additional technical constraint. Reciprocally, returning to B s+1 through the operator H -1 has the effect of removing the nonincreasingness condition. To put it speaking, if for instance, all the combinatorial moments are equal to 1, Proposition 3.3 yields the following simple result.

Corollary 3.4 The s-convex extremal distributions in the set

Q Y : E Y i = 1, i = 1, . . . , s -1, and ∆Q Y (j) ≤ 0, j ∈ IN
correspond to the H-transform of the s + 1-convex extremal ones in the set

P X : E X i = 1, i = 1, . . . , s .

Continuous problem

A similar approach is applicable to random variables valued in IR + . Given any s ∈ IN 0 , the set F s of s-convex real functions is defined here as

F s = {f : D s f (x) ≥ 0, x ≥ 0},
D being the usual derivative operator. For two probability distributions P 1 and P 2 on IR + , P 1 s P 2 when the condition (3.1) is again satisfied. Here too, an equivalent characterization is provided by the two conditions stated just after (3.1).

We still work on the set of distributions on IR + with prescribed first s moments, i.e.

B s+1 = P X : E(X i ) = µ i , i = 1, . . . , s . (3.10)
Let us apply to B s+1 the continuous stationary-excess operator H defined by (2.7). This yields a one-to-one correspondence with the image set H(B s+1 ). A transformed order H may be defined on H(B s+1 ) exactly as by (3.3). Let Q Y denote the probability distribution of a continuous nonnegative random variable Y with density function {q Y (y), y ≥ 0}. The continuous analogue of Proposition 3.3 is then the following.

Proposition 3.5

H(B s+1 ) = Q Y : E(Y i ) = µ i+1 (i + 1)µ 1 , i = 1, . . . , s -1, and (3.11) Dq Y (y) ≤ 0, y ≥ 0, with µ 1 q Y (0) ≤ 1} , (3.12)
and the order H corresponds to the continuous s-convex stochastic order s .

Proof. The set H(B s+1 ) is straightforward from (2.8) and (2.9). Let us now identify the order

H . As with (3.7), consider any Q = Q Y in H(B s+1
), with density function q = q Y . From (2.8) and after integration, we get (3.13) in place of (3.9). Thus, H corresponds here also to the order s .

< f, H -1 (Q) > = -µ 1 ∞ 0 f (y) Dq(y) dy = f (0) + µ 1 ∞ 0 Df (y) dQ(y), i.e., the conjugate oprator of H -1 is (H -1 ) * = δ 0 + µ 1 D,

Extension to H t -transforms

At this point, a natural step is to apply successively the operator H and then investigate the effects on the s-convex orders and the associated moment spaces. So, given t ∈ IN 0 , let H t denote the t-th iterate of H; its construction is simple, although not totally immediate.

Discrete problem

Given s ∈ IN 0 , we start by introducing, similarly to (3.2), a set B s+t of all probability mass functions P X on IN with prescribed first s+t-1 combinatorial moments (given by c 1 , . . . , c s+t-1 ). This set may be ordered with respect to the (s + t) convex order. Now, at step 1, consider the set

D 1 = D (c 1 ) 1
of all probability mass functions on IN with fixed mean c 1 . By construction, the operator

H ≡ H 1 = H (c 1 ) 1 is a one-to-one mapping from D 1 into a new set H 1 (D 1 ) ≡ D 2 . By (3.4), the mean in D 2 is equal to c 2 /c 1 ; put D 2 = D (c 2 /c 1 ) 2 . At step 2, the operator H 2 = H (c 2 /c 1 ) 2 maps D 2 into H 2 (D 2 ) ≡ D 3 , and again by (3.4), D 3 = D (c 3 /c 2 ) 3 . Finally, at step t, H t = H (ct/c t-1 ) t maps D t into H t (D t ) ≡ D t+1 . The operator H t is then the composition (denoted •) of the successive operators H 1 , . . . , H t , i.e. H t = H t • . . . • H 2 • H 1 : D 1 → H 1 D 2 → H 2 . . . → H t-1 D t → Ht D t+1 . (4.1)
Let us apply H t to the set to any random variable X in B s+t . Note that the moments c 1 , . . . , c t are well fixed within B s+t . From (2.1) and (4.1), we see that X is mapped to an associated discrete non-negative random variable X H t defined as follows.

Definition 4.1 X H t has a probability mass function H t (P X ) given by

H t (P X )(j) ≡ P (X H t = j) = Ft (X, j + t) c t , j ∈ IN. (4.2)
By (2.5), an equivalent expression is

P (X H t = j) = E X-j-1 t-1 c t , j ∈ IN.
We underline that each H k being one-to-one from D k to D k+1 , H t provides a one-to-one correspondence on B s+t . Furthermore, let us introduce, as with (3.3), an order H t on H t (B s+t ).

Definition 4.2 Two probability mass functions

Q 1 , Q 2 ∈ H t (B s+t ) satisfy Q 1 H t Q 2 when (H t ) -1 (Q 1 ) s+t (H t ) -1 (Q 2 ). (4.3)
We are now able to generalize Proposition 3.3 to the transform by H t . The previous notation Q Y will be adopted here too, and it is convenient to put c 0 = 1.

Proposition 4.3 H t (B s+t ) = Q Y : E Y i = c i+t c t , i = 1, . . . , s -1, and (4.4) (-1) k ∆ k Q Y (j) ≥ 0, j ∈ IN, k = 1, . . . , t, with (4.5) (-1) t-k c t Q Y (0) ≤ t-1 l=k-1 (-1) l-k+1 c l , k = 1, . . . , t , (4.6)
and the order H t corresponds to the s-convex stochastic order s .

Proof. The key tool is the conjugate operator of (H t ) -1 , built from the conjugate operators of (H 1 ) -1 , . . . , (H t ) -1 . By arguing as with (3.9), we see that

[(H k ) -1 ] * = δ 0 + (c k /c k-1 )∆, k = 1, . . . , t.
From (4.1), we then obtain

[(H t ) -1 ] * = [(H 1 ) -1 ] * • [(H 2 ) -1 ] * • . . . • [(H t ) -1 ] * = [c 1 ∆ + δ 0 ] • [(c 2 /c 1 )∆ + δ 0 ] • . . . • [(c t /c t-1 )∆ + δ 0 ].
Note that, for t = 2 for example,

[a∆ + δ 0 ] • [b∆ + δ 0 ] = ab∆ 2 + δ 0 b∆ + δ 0 ,
as ∆δ 0 = 0 and δ 2 0 = δ 0 . So, we get

[(H t ) -1 ] * = t k=0 δ k 0 t l=k+1 [(c l /c l-1 )∆] = c t ∆ t + δ 0 t k=1 (c t /c k )∆ t-k . (4.7)
Applying (4.7) to any function f on IN thus gives

[(H t ) -1 ] * (f ) = c t ∆ t f + t k=1 (c t /c k ) (∆ t-k f )(0). (4.8)
Let us first identify the order H t . By (4.3) and (3.8),

Q 1 H t Q 2 when < [(H -1 ) * ](f ), Q 1 > ≤ < [(H -1 ) * ](f ), Q 2 > for all f ∈ F s+t .
From (4.8), this means that

< ∆ t f, Q 1 > ≤ < ∆ t f, Q 2 > for all f ∈ F s+t ,
which shows the equivalence with the order s . We now determine the set H t (B s+t ). Let us choose the function f (j) = j i+t , j ∈ IN, for any fixed i, t ∈ IN. Since

∆ t j i + t = j i , j ∈ IN, we have by (4.8) that [(H t ) -1 ] * j i + t = c t j i , j ∈ IN. (4.9)
By (3.8), we then obtain from (4.9) the relations

E X i + t = c t E X H t i , i ∈ IN,
which generalize (2.4). The binomial moments in (4.4) then follow. Moreover, from (4.1) and using (3.5), we find that a probability mass function Q Y necessarily satisfies the restrictions (4.5). This can also be seen from (4.2) as for k = 1, . . . , t -1,

∆ k E X -j -1 t -1 = (-1) k E X -j -1 -k t -1 -k , j ∈ IN.
So, Q Y is a t-monotone function in the sense given in the Introduction. Finally, the t conditions (4.6) guarantee that the t successive transforms of P X are true probability mass functions. Indeed, writing

X H k = H k (X H k-1 ), k ≥ 1, with X H 0 = X, one gets (see (2.2)) P (X H k-1 = 0) = 1 -(c k /c k-1 )P (X H k = 0), that is c k-1 P (X H k-1 = 0) = c k-1 -c k P (X H k = 0), k = 1, . . . , t. (4.10)
Note that for each k, P (X H k-1 = 0) ≤ 1 but the non-negativity is not guaranteed so far. By iterating (4.10) t -k times, we then obtain

c k-1 P (X H k-1 = 0) = t-1 l=k-1 (-1) l-k+1 c l + (-1) t-k+1 c t P (X H t = 0), k = 1, . . . , t.
Thus, the constraints P (X H k-1 = 0) ≥ 0 provide the restrictions (4.6).

Observe that the image set H t (B s+t ) involves s -1 moment relations, the t-monotonicity and t other technical conditions. Going back to B s+t through the operator H -t would allow us to remove the monotonicity condition.

Continuous problem

For s, t ∈ IN 0 , consider now the set B s+t of distributions on IR + with prescribed first s + t -1 moments (given by µ 1 , . . . , µ s+t-1 ).

We construct the operator H t , the t-th iterate of the operator H defined by (2.7), by following a similar argument as for (4.1). Let D 1 = D (µ 1 ) 1 the set of all continuous distributions on IR + with fixed mean µ 1 . Then, H t is defined as

H t = H t •. . .•H 2 •H 1 where, for each k = 1, . . . , t, H k is an operator mapping the set D k into the set H k (D k ) = D k+1 . From (2.9), one easily checks the mean in D k+1 is equal to µ k /kµ k-1 . So, H k = H (µ k /kµ k-1 ) k
and it provides a one-to-one correspondence.

By definition of H t and using (2.7), we see that any random variable X in B s+t is mapped to an associated non-negative random variable X H t whose survival function is given by

P (X H t > x) = t! Ft+1 (X, x) µ t , x ≥ 0. (4.11)
By (2.10) and (2.11), it can be rewitten as

P (X H t > x) = E[(X -x) t + ] µ t , x ≥ 0,
which generalizes (2.7). Here too, H t is one-to-one on B s+t . A transformed order H t is then defined on H t (B s+t ) by stipulating again the condition (4.3). We adopt the same notation Q Y and q Y (y) as before.

Proposition 4.4

H t (B s+t ) = Q Y : E(Y i ) = µ i+t i+t t
µ t , i = 1, . . . , s -1, and (4.12)

(-1) k D (k) q Y (y) ≥ 0, y ≥ 0, k = 1, . . . , t, with (4.13) (-1) t-k D (t-k) q Y (0) ≤ t! µ k-1 (k -1)! µ t , k = 1, . . . , t , (4.14)
and the order H t corresponds to the s-convex stochastic order s .

Proof. Working with conjugate operators, one gets (see (3.13)) that

[(H k ) -1 ] * = δ 0 + (µ k /kµ k-1 )D, k = 1, . . . , t.
Therefore,

[(H t ) -1 ] * = [µ 1 D + δ 0 ] • [(µ 2 /2µ 1 )D + δ 0 ] • . . . • [(µ t /kµ t-1 )D + δ 0 ] = µ t t! D t + δ 0 t k=1 k! µ t t! µ k D t-k . (4.15)
First, consider the function f (x) = x i+t , x ∈ IR + , for fixed i, t IN. Applying (4.15) to this function and using (3.8), we find that

E(X i+t ) = i + t t µ t E(X i H t ), i ∈ IN,
hence (4.12). Now, as for (2.8), one has

P (X H k-1 > x) = (µ k /kµ k-1 ) q H k (x), k = 1, . . . , t, (4.16)
which gives by differentiation

µ k-1 q H k-1 (x) = (-1/k) D{µ k q H k (x)}, k = 1, . . . , t.
(4.17)

By iterating (4.12) t -k more times, we get

µ k-1 q H k-1 (x) = µ t k . . . t (-1) t-k+1 D (t-k+1) q H t (x), k = 1, . . . , t. (4.18)
Thus, q H k-1 ≥ 0 yields the conditions (4.13). Moreover, (4.16) implies that

µ k q H k (0) ≤ kµ k-1 , k = 1, . . . , t;
inserting (4.18) with x = 0 and k instead of k -1 then leads to the constraints (4.14).

Convex extrema for monotone distributions

A further interest of Propositions 4.3 and 4.4 is to point out that the s-convex extrema fo tmonotone distributions may be constructed from the simple s + t-convex extrema (i.e. without monotonicity restriction).

Hereafter, we are going to derive the explicit expression of a few extrema for monotone distibutions that are defined on a bounded non-negative support. In the discrete case, the random variables are valued in a set {0, . . . , n}, n ∈ IN 0 ; in the continuous case, the random variables are valued on an interval [0, b], b > 0. We first obtain the s = 1, 2, 3-convex extrema when t = 1 (nonincreasing distributions). These were found earlier by Denuit et al. (1999b) in the discrete case and [START_REF] Denuit | The s-convex orders among real random variables, with applications[END_REF] in the continuous case. The method of proof used there, however, is quite different as it relies on a Khinchine representation for unimodal distributions. Then, we examine the s = 1, 2-convex extrema when t = 2 (convex nonincreasing distributions). These are seen to be the same as with t = 1 except for the minimum in the discrete case.

Let us introduce the class B s,t (n) of all such discrete random variables Y that have prescribed first s -1 moments ν i = E(Y i ), i = 1, . . . , s -1, and possess a t-monotone probability mass function Q Y , i.e. for k = 1, . . . , t, (-1

) k ∆ k Q Y (j) ≥ 0, j = 0, . . . , n -k. The s-convex extrema in this set are denoted by Y (s,t) min (n) and Y (s,t) max (n). Similarly, B s,t (b)
is the class of all continuous random variables Y with prescribed first s -1 moments ν i and a t-monotone density function q Y , i.e. for k = 1, . . . , t, (-1

) k D (k) q Y (y) ≥ 0, y ∈ [0, b]. The s-convex extrema in the set are written Y (s,t) min (b) and Y (s,t) max (b).
For the discrete case, we observe that by the definition (2.1) of H, a random variable Y valued on {0, . . . , n} is obtained as the transform of a random variable X that is valued on {0, . . . , n + 1}. This remark will be used in the sequel.

Let us recall that the usual s-convex extrema, i.e. when t = 0, inside B s (n) or B s (b), are explicitly known for s = 1, 2, 3, 4. They are recalled in the Appendix and denoted by X

Corollary 5.1 Inside B 1,1 (n) [B 1,1 (b)], Y (1,1) min (n) [Y (1,1) min (b)] = 0 a.s.,
(5.1)

Y (1,1) max (n) [Y (1,1) max (b)] = d U(n) [U(b)].
(5.2)

Proof. For the discrete case, consider the set H[B 2 (n + 1)] defined by (3.4), (3.5), with n + 1 substituted for n (see the remark above). We want to keep only the restriction of nonincreasing distributions (i.e. ∆Q Y (j) ≤ 0, j = 0, . . . , n -1). Note that as s = 1, there is no moment specification. Moreover, the condition c 1 Q Y (0) ≤ 1 will be satisfied by choosing c 1 = 1, for instance. Now, by Proposition 3.3, the associated 1-convex extrema are provided by the Htransform (2.1) of the 2-convex extrema in B 2 (n + 1). Using (7.1) and (7.2) where ξ = 0, we then get

P [Y (1,1) min (n) = j] = P [X (2)
min (n + 1) ≥ j + 1] = δ j,0 , j = 0, . . . , n, where δ j,0 = 1 (resp. 0) if j = 0 (> 0), and

P [Y (1,1) max (n) = j] = P [X (2)
max (n + 1) ≥ j + 1] = 1/(n + 1), j = 0, . . . , n, hence the extrema indicated above. For the continuous case, H[B 2 (b)] defined by (3.11), (3.12) contains now an undesired condition µ 1 q Y (0) ≤ 1. It can be satisfied, however, by choosing µ 1 → 0. Then, applying the continuous operator H to the 2-convex minimum gives

P [Y (1,1) min (b) ≤ x] = 1 µ 1 x 0 [1 -F X (2) min (b) (y)]dy = x µ 1 , 0 ≤ x ≤ µ 1 , i.e. Y (1,1) min (b) is uniform on [0, µ 1 ].
As µ 1 tends to 0, this minimum degenerates to the single point 0. Using the 2-convex maximum (7.3), we have

P [Y (1,1) max (b) ≤ x] = 1 µ 1 x 0 [1 -F X (2) max (b) (y)]dy = x b , 0 ≤ x ≤ b,
(independently of µ 1 ), i.e. Y

(1,1) max (b) is uniform on [0, b] as announced. 2-convex extrema for nonincreasing distributions. This time, we work with s = 2 and t = 1. Denote by

ν 1 = E(Y ) the fixed mean in B 2,1 (n) (and H[B 3 (n + 1)]) or B 2,1 (b) (and H[B 3 (b)]).
Corollary 5.2 Inside B 2,1 (n), putting ξ for the integer in [0, n -1] such that ξ < 2ν 1 ≤ ξ + 1,

Y

(2,1) min (n) = 0, . . . , ξ with equal probabilities 2( ξ + 1 -ν 1 )/( ξ + 1)( ξ + 2), ξ + 1 with probability (2ν 1 -ξ)/( ξ + 2), (5.3)

Y (2,1) max (n) = 0
with probability 1 -2ν 1 /(n + 1), 1, . . . , n with equal probabilities 2ν 1 /n(n + 1).

(5.4)

Inside B 2,1 (b), putting I 0 for a distribution degenerated in 0, Y (2,1) min (b) = d U(2ν 1 ), (5.5) Y (2,1) max (b) = d (1 -2ν 1 /b) I 0 + (2ν 1 /b) U(b).
(5.6)

Proof. As indicated in the Appendix, the unsconstrained 3-convex extrema are given in terms of the moments µ i = E(X i ), i = 1, 2. Let us start with the discrete case. By (3.4), ν 1 = c 2 /c 1 which implies that µ 2 = µ 1 (2ν 1 + 1). For X

(3) min (n + 1) given by (7.4), we have

ξ 1 -1 < 2ν 1 ≤ ξ 1 , i.e. ξ < 2ν 1 ≤ ξ + 1 by putting ξ = ξ 1 -1. Moreover, X (3) min (n + 1) =    0
with probability p 1 = 1 -p 2 -p 3 , ξ + 1 with probability p 2 = µ 1 ( ξ -2ν 1 + 1)/( ξ + 1), ξ + 2 with probability p 3 = µ 1 (2ν 1 -ξ)/( ξ + 2).

For X

(3) max (n + 1) given by (7.5), we have ξ 2 < (n -2ν)/[(n + 1)/µ 1 -1] ≤ ξ 2 + 1, so that ξ 2 = 0 as µ 1 → 0. Then,

X (3) max (n + 1) =    0 with probability p 1 = 1 -p 2 -p 3 , 1
with probability p 2 = µ 1 (n -2ν 1 )/n, n + 1 with probability p 3 = µ 1 2ν 1 /n(n + 1).

Applying the discrete operator H then leads to the extrema (5.3) and (5.4).

Let us now discuss the continuous case. By (3.11), ν 1 = µ 2 /2µ 1 . We may let µ 1 → 0 and choose µ 2 = 2µ 1 ν 1 accordingly. For the minimum, we get from (7.6) that

P [Y (2,1) min (b) ≤ x] = (µ 1 /µ 2 )x, 0 ≤ x ≤ µ 2 /µ 1 ,
i.e. the result (5.5). For the maximum, (7.7) gives as µ 1 → 0

X (3) max (b) = µ 1 (1 -2ν 1 /b) with probability 1 -2ν 1 µ 1 /b 2 , b with probability 2ν 1 µ 1 /b 2 , so that P [Y (2,1) max (b) ≤ x] = 1 -2ν 1 /b + 2ν 1 x/b 2 , 0 ≤ x ≤ b,
i.e. the maximum (5.6).

3-convex extrema for nonincreasing distributions. We have now s = 3 and t = 1. Let ν 1 = E(Y ) and ν 2 = E(Y 2 ) be the first two moments in B 3,1 (n) (and H[B 4 (n + 1)]) or B 3,1 (b) (and H[B 4 (b)]).

Corollary 5.3 Inside B 3,1 (n), put ξ1 and ξ2 for the integers in [0, n -1] such that

ξ1 < (3ν 2 -ν 1 )/2ν 1 ≤ ξ1 + 1, and ξ2 < (2nν 1 -3ν 2 + ν 1 )/(n -2ν 1 ) ≤ ξ2 + 1. Then, Y (3,1) min (n) =      0
with probability q 3 ξ1 +2 + q 2 ξ1 +1 + q 1 , 1, . . . , ξ1 with equal probabilities q 3 ξ1 +2 + q 2 ξ1 +1 , ξ1 + 1 with probability q 3 ξ1 +2 ,

(5.7)

where q 1 + q 2 + q 3 = 1 and

q 2 = -3ν 2 + 3ν 1 + 2 ξ1 ν 1 ξ1 , q 3 = 3ν 2 -ν 1 -2 ξ1 ν 1 ξ1 + 1 , while Y (3,1) max (n) =      0, . . . , ξ2 with equal probabilities χ 3 n+1 + χ 2 ξ2 +2 + χ 1 ξ2 +1 , ξ2 + 1
with probability χ 3 n+1 + χ 2 ξ2 +2 , ξ2 + 2, . . . , n with equal probabilities χ 3 n+1 ,

(5.8)

where χ 1 + χ 2 + χ 3 = 1 and

χ 1 = ( ξ2 + 1)(n -2ν 1 ) + 3ν 2 -ν 1 -2nν 1 n -ξ2 , χ 2 = 2ν 1 ( ξ2 + n) -3ν 2 + ν 1 -n ξ2 n -1 -ξ2 . Inside B 3,1 (b), Y (3,1) min (b) = d 3ν 2 -4ν 2 1 3ν 2 I 0 + 4ν 2 1 3ν 2 U 3ν 2 2ν 1 ,
(5.9)

Y (3,1) max (b) = d 3ν 2 -4ν 2 1 b 2 -4bν 1 + 3ν 2 U(b) + b 2 -4bν 1 + 4ν 2 1 b 2 -4bν 1 + 3ν 2 U 2bν 1 -3ν 2 b -2ν 1 .
(5.10)

Proof. In the discrete case, we have, by (3.4), ν 1 = c 2 /c 1 , which yields µ 2 = µ 1 (2ν 1 + 1), and (ν 2 -ν 1 )/2 = c 3 /c 2 , which implies that µ 3 = µ 1 (3ν 2 + 3ν 1 + 1). Applying (2.1) to X

(4) min (n + 1) given by (7.8) yields

Y (3,1) min (n) =        0, . . . , θ -1
with equal probabilities 1/µ 1 , θ with probability (p 2 + p 3 + p 4 )/µ 1 , θ + 1, . . . , η -1 with equal probabilities (p 3 + p 4 )/µ 1 , η with probability p 4 /µ 1 .

Let us choose µ 1 → 0. To avoid absurdity, we must have θ = 0. Moreover, we then find, after some simplifications, that

   p 2 /µ 1 = (3ν 2 + ν 1 -4ν 1 η + η(η -1))/η(η + 1), p 3 /µ 1 = (-3ν 2 + ν 1 + 2ν 1 η)/η(η -1), p 4 /µ 1 = (3ν 2 + ν 1 -2ν 1 η)/η(η -1).
The condition p 4 > 0 implies that η < 1 + (3ν 2 -ν 1 )/2ν 1 , and p 3 ≥ 0 yields η ≥ (3ν 2 -ν 1 )/2ν 1 . Putting η = 1 + ξ1 , we then obtain for ξ1 the value announced above. Note that the condition p 2 ≥ 0 is always verified. Indeed, this condition means that 3ν 2 -3ν 1 -4 ξ1 ν 1 + ξ1 ( ξ1 + 1) ≥ 0. It can be rewritten as ν2 -(2 ξ1 + 1)ν 1 + ξ1 ( ξ1 + 1) ≥ 0 where ν1 = 2ν 1 and ν2 = 3ν 2 -ν 1 . This is verified if ν1 and ν2 are the first two moments of some random variable on {0, . . . , n}, since i 2 -(2 ξ1 + 1)i + ξ1 ( ξ1 + 1) ≥ 0 for all 0 ≤ i ≤ n. In fact, the random variable Z in Denuit et al. (1999c) has precisely its first two moments equal to these ν1 and ν2 ; see formula (5.7) in that paper. So, we finally get the above result (5.7).

Consider X

(4) max (n + 1) given by (7.9). First, we observe that (µ 2 n -µ 3 )/(µ 1 n -µ 2 ) = 1 + (2nν 1 -3ν 2 + ν 1 )/(n -2ν 1 ); thus, ζ = ξ2 + 1 where ξ2 is defined as above. Applying (2.1), we then get that

Y (3,1) max (n) =    0, . . . , ξ2
with equal probabilities (p 2 + p 3 + p 4 )/µ 1 , ξ2 + 1 with probability (p 3 + p 4 )/µ 1 , ξ2 + 2, . . . , n with equal probabilities p 4 /µ 1 .

Finally, one can check that p 2 /µ 1 = χ 1 /( ξ2 + 1), p 3 /µ 1 = χ 2 /( ξ2 + 2) and p 4 /µ 1 = χ 3 /(n + 1), hence the result (5.8).

In the continuous case, (3.11) gives ν 1 = µ 2 /2µ 1 and ν 2 = µ 3 /3µ 1 . So, we choose µ 2 = 2µ 1 ν 1 and µ 3 = 3µ 1 ν 2 , with µ 1 → 0. For the mininimum, we see from (7.10) that

X (4) min (b) = r -→ 0 with probability (r + -µ 1 )/(r + -r -) → 1, r + → 3ν 2 /ν 1 with probability (µ 1 -r -)/(r + -r -) → 0. Moreover, one can check that r -/µ 1 → (3ν 2 -4ν 2 1 )/3ν 2 , implying also that (µ 1 -r -)/(r + -r -)µ 1 → (4ν 2 1 /3ν 2 )(2ν 1 /3ν 2 )
. All this then leads to the result (5.9) for Y (3,1) min (b). For the maximum, we have by (7.11) that

X (4) max (b) =      0 with probability p 1 = 1 -p 2 -p 3 , 2bν 1 -3ν 2 b-2ν 1 with probability p 2 = µ 1 b-2ν 1 2bν 1 -3ν 2 (b-2ν 1 ) 2 b 2 -4bν 1 +3ν 2 , b with probability p 3 = µ 1 1 b 3ν 2 -4ν 2 1 b 2 -4bν 1 +3ν 2 .
The result (5.10) for Y

(3,1) max (b) then easily follows. 1-convex extrema for nonincreasing convex distributions. Here, s = 1 and t = 2. We observe that the 1-convex extrema obtained before for nonincreasing distributions are convex. As a consequence, they are extremal too for nonincreasing convex distributions.

2-convex extrema for nonincreasing convex distributions. This time, s = 2 and t = 2. The 2-convex extrema for nonincreasing distributions are again convex, except the minimum in the discrete case. Thus, apart this case which is discussed below, they remain extremal for nonincreasing convex distributions. Let ν 1 = E(Y ) be the fixed mean in the set B 2,2 (n) (and H 2 [B 4 (n + 1)]).

Corollary 5.4 For the minimum inside B 2,2 (n), putting η for the integer in [2, n]

such that η < 2 + 3ν 1 ≤ η + 1, Y (2,2) min (n) = j with probability (η -j -1)π 1 + (η -j)π 2 , j = 0, . . . , η -1, (5.11)
where

π 1 = 2(η -1 -3ν 1 ) η(η -1)
, and π 2 = 2(2 + 3ν 1 -η) η(η + 1) .

Proof. In (4.6), the two constraints (for

k = 1, 2) yield the condition 0 ≤ c 1 -c 2 Q(0) ≤ 1.
From now, we assume that µ 1 → 0. To satisfy the previous condition, one may choose, for instance, E(X 2 ) = 2µ 1 . Moreover, by (4.4),

ν 1 = c 3 /c 2 , i.e. 3(1 + ν 1 ) = E(X 3 -X)/E(X 2 -X).
Given our choice of E(X 2 ), one then gets E(X 3 ) = (4 + 3ν 1 )µ 1 . Finally, as µ 1 → 0, we use the approximations µ 2 E(X 2 ) and µ 3 E(X 3 ). Consider the 4-convex minimum given by (7.8). Under the first three moments above, one sees that θ = 0 and the minimum then becomes

X (4) min (n + 1) =        0 with probability p 1 = 1 -(η 2 -η + 3ν 1 + 1)µ 1 /η(η + 1), 1
with probability p 2 = (η 2 -3η + 2 + 3ν 1 )µ 1 /η(η -1), η with probability p 3 = (η -1 -3ν 1 )µ 1 /η(η -1), η + 1 with probability p 4 = (2 + 3ν 1 -η)µ 1 /η(η + 1). Furthermore, p 1 ≥ 0 (as µ 1 → 0), p 3 ≥ 0 means 1 + 3ν 1 ≤ η, and p 4 > 0 requires η < 2 + 3ν 1 , which implies p 2 > 0 (since η ≥ 2). All this leads to the announced value η.

By (4.2), the operator H 2 provides the distribution P (Y = j) = [P (X = j + 2) + 2P (X = j + 3) + . . . + (n -j)P (X = n + 1)]/c 2 , j = 0, . . . , n. Applied to X (4) min (n + 1), it gives, as µ 1 → 0,

P [Y (2,2) min (n) = η -1] = p 4 /c 2 , P [Y (2,2) min (n) = η -2] = (p 3 + 2p 4 )/c 2 , P [Y (2,2) min (n) = η -3] = (2p 3 + 3p 4 )/c 2 and in general, P [Y (2,2) min (n) = j] = (η -j -1) p 3 /c 2 + (η -j) p 4 /c 2 , j = 0, . . . , η -1.
By construction, c 2 = µ 1 /2, so that p 3 /c 2 = π 1 and p 4 /c 2 = π 2 as stated in (5.11).

Some numerical illustrations

We present hereafter three applications, somewhat non-standard, in insurance and biostatistics. More traditional illustrations in ruin theory and life insurance (as in [START_REF] Denuit | Some new classes of stochastic order relations among arithmetic random variables, with applications in actuarial sciences[END_REF] and [START_REF] Denuit | The s-convex orders among real random variables, with applications[END_REF]Denuit et al. ( ), (1999b))) could be considered too.

(i) Solvency Capital Requirements for large claims. The new European regulation, Solvency II, to be in force in October 2012, imposes insurance companies to hold enough capital to deal with unfavorable events. The Directive was voted in April 2009 but many implementation measures are still being discussed currently. Some stakeholders recommend the creation of a so-called large loss module in order to cover claim amounts that are larger than attritional ones (but not of catastrophic type). First, for each risk, some buffer capital called Solvency Capital Requirement (SCR) has to be held. Then, at the company or group level, some aggregation formula takes diversification into account and defines the Basic Solvency Capital Requirement that the company must satisfy as

i j ρ ij SCR i SCR j ,
where SCR i represents the Solvency Capital Requirement for the business line i and ρ ij is a correlation-type parameter (without precise statistical meaning) between the lines i and j.

For each risk, the Solvency Capital Requirement is generally defined as the difference between a certain Value-at-Risk of the random loss and the expected loss. In most cases, a simplification is used and the SCR is defined by SCR = qσ where σ is the standard error of the random loss and q > 0 is called a quantile factor. So, q = 3 is usually chosen for claim amounts with a moderate tail distribution (as it is in many Solvency II modules). For heavy-tailed risks, a more relevant value is q = 5 or 6; hereafter, we will take q = 6. A collective risk model is often adopted to describe the occurrence of large claims. Thus, the number of claims is a counting random variable N , and the large claim amounts are independent identically distributed random variables (distributed as W , say), independently of N . By the variance decomposition formula, the variance of the large loss aggregated claim amount is given by

σ 2 = Var(W )E(N ) + [E(W )] 2 Var(N ).
Various specialized softwares or internal models provide an estimation of the distribution of W and the mean E(N ). As only limited information is available, N is usually assumed to have a Poisson distribution. For example, with respect to some French data, one could use a Poisson distribution with parameter λ(C 27 ) = 0.37 for the business line C 27 (Drought and earthquake) and λ(C 35 ) = 0.69 for the business line C 35 (Construction -damages to building). The Poisson assumption is partially satisfying as, in practice, N is usually over-dispersed (Var(N ) is significantly larger than E(N )) and only a bounded number of large claims are susceptible to occur (practitioners consider that observing more than a certain number n claims corresponds to a catastrophe). So, for the two previous business lines, one could set, for instance, n(C 27 ) = 10 and n(C 35 ) = 20.

A property of the Poisson distribution (with parameter λ) is that its probability mass function is nonincreasing convex if λ ≤ 2 -√ 2 0.5858, and nonincreasing but not convex if 2 -√ 2 < λ ≤ 1. Note that for our illustration, λ(C 27 ) = 0.37 < 2 -√ 2 < λ(C 35 ) = 0.69 < 1. In fact, this shape for a counting distribution is quite frequent for standard (non-catastrophic) risks. Instead of a Poisson distribution, one could prefer to use a distribution with the same mean that is nonincreasing (convex) and has a bounded support {0, . . . , n}. The choice of such a distribution, however, is not an easy task. It is thus interesting to dispose of upper and lower bounds for the Solvency Capital Requirement.

The bounds for SCR = qσ are simply given by SCR(N

(2,2) min ) = q Var(W)E(N) + [E(W)] 2 Var(N (2,2) min ), and SCR(N (2,2) max ) = q Var(W)E(N) + [E(W)] 2 Var(N (2,2) max
). These provide a prudent version of the SCR and allow us to situate any particular choice for the distribution of N (within the class of interest). Let us fix the mean ν 1 = E(N ) = λ. From (7.1), (7.2), one gets (writing

N (2,0) ≡ N (2) ) Var(N (2,0) min ) = ν 1 -ξ(1 -2ν 1 + ξ) -ν 2 1 , Var(N (2,0) max ) = nν 1 -ν 2 1 .
From (5.3), (5.4),

Var(N

(2,1)

min ) = [-ξ3 + ξ2 (-3 + 4ν 1 ) + ξ(-2 + 11ν 1 -3ν 2 1 ) + 6ν 1 (1 -ν 1 )]/3( ξ + 2), Var(N (2,1) max ) = ν 1 (2n + 1)/3 -ν 2 1 ,
and from (5.11) with the remark before Corollary 5.4,

Var(N (2,2) min ) = (η -1)(6ν 1 + 2 -η)/6 -ν 2 1 , Var(N (2,2) max ) = Var(N (2,1) max ),
(as 1 + 2 2 + . . . + n 2 = n(n + 1)(2n + 1)/6 and 1 + 2 3 + . . . + n 3 = n 2 (n + 1) 2 /4).

For n(C 27 ) = 10, λ(C 27 ) = 0.37, one has ξ = 0, ξ = 0, η = 3 and Var(N We present some numerical results in Tables 1 and2 below. Table 1 shows that the Poisson case is much closer to the lower bounds than the upper bounds. This means that the Poisson assumption may lead to serious underestimations if one uses it by default. Besides, the upper bound is significantly improved when the distribution is known to be nonincreasing. The indicates that the corresponding Poisson distribution is not convex; so, the bound given there is questionable if the Poisson distribution shape is taken for granted. From Table 2, one observes that, as expected, the choice of the maximal value n influences considerably the value of the variance of N ; its effect on the value of the SCR is, however, less important by comparison. Note that the lower bound does not depend on n. Until age 105, they slightly increase at a rate of around 0.015 per year; they tend to become stable after age 105. There is still some uncertainty about the pattern of mortality rates at high ages. Concerning, however, the residual lifetime floor at age 100, denoted by T 100 , its distribution is observed to have a nonincreasing convex form whatever the pattern of mortality rates. This is shown in Figure 1 (resp. Figure 2) where the annual mortality rates increase of 0.015 -Hypothesis H1-(resp. remain constant -Hypothesis H2-); the life expectancy is then 1.68 (resp. 1.86) years. Note that at a lower age of 95, the mortality rate is of 0.20 with an increase of 0.03 per year up to age 105 say, after which the rate is taken constant -Hypothesis H3-; then, the residual lifetime floor T 95 has a nonincreasing distribution which is no longer convex (see Figure 3). Provided the residual lifetime floor has a nonincreasing (convex) distribution, its variance can be bounded by using again the previous extrema. Considering T 100 , we choose n = 15 as the probability to live more than 115 years is extremely small (positive of course). Results are presented in Table 3 for both Hypotheses 1 and 2. We observe that the lower bounds are not very far from the empirical case, especially when information on the shape of the distribution is provided.

Line

(iii) Number of asthma exacerbations. Asthma exacerbation is another term for an asthma attack in which the bronchial tubes through which air flows to the lungs suddenly tighten and become constricted. This makes it extremely difficult to breathe, resulting in an asthma exacerbation. There are many things that can trigger an asthma exacerbation and a 4) for the number N of asthma exacerbations. This distribution is nonincreasing but is easily seen to be non-convex.

There is almost no overdispersion as E(N ) 1.1832 and Var(N ) 1.1852.

Given the observed mean of N , we computed the probability mass functions of the extrema for arbitrary or nonincreasing distributions valued on {0, . . . , n = 5} (using (7.1),(7.2) and (5.3),(5.4), respectively). The results are presented in Table 5. Moreover, bounds for the variance of N are provided in Table 6: for n = 5, the bounds are rather narrow and for n = 10, the upper bound becomes much higher (as expected).

  extrema for nonincreasing distributions. This is the problem of extrema when s = 1 and t = 1. Let us write U(n) [U(b)] for a uniform distribution on {0, . . . , n}([0, b]).

  min ). For n(C 35 ) = 20, λ(C 35 ) = 0.69, then ξ = 0, ξ = 1, η = 4 and Var(N (2,1) min ) > Var(N (2,0) min ). Concerning the associated claim amounts, let us choose, for instance, E[W (C 27 )] = 1000, Var[W (C 27 )] = 2500 2 and E[W (C 35 )] = 2000, Var[W (C 35 )] = 7000 2 (in thousands of euros).

Figure 1 :

 1 Figure 1: Probability mass function of T 100 under H1.

Figure 2 :

 2 Figure 2: Probability mass function of T 100 under H2.

Figure 3 :

 3 Figure 3: Probability mass function of T 95 under H3.

Table 1 :

 1 Bounds on Var(N ) and SCR for C 27 and C 35 when s = 2 and t = 0, 1, 2.

			C 27	Line C 35
	λ	λ(C 27 ) = 0.37 λ(C 35 ) = 0.69
	n	n(C 27 ) = 10	n(C 35 ) = 20
	Var(N min ) (2,0)		0.2331	0.2139
	Var(N min ) (2,1)		0.2331	0.4672
	Var(N min ) (2,2)		0.2698	0.5939
	Var(N ) (Poisson case)	0.37	0.69
	Var(N max ) (2,2)		2.4531	8.9539
	Var(N max ) (2,1)		2.4531	8.9539
	Var(N max ) (2,0)		3.5631	13.3239
	SCR(N (2,0) min )		9573.0	35326.5
	SCR(N (2,1) min )		9573.0	35839.0
	SCR(N (2,2) min )		9641.7	36092.7
	SCR (Poisson case)		9827.0	36283.9
	SCR(N (2,2) max )		13098.2	50065.2
	SCR(N (2,1) max )		13098.2	50065.2
	SCR(N (2,0) max )		14543.8	55998.2
	n(C 27 ) Var(N max ) SCR(N (2,2) (2,2) max )
	5	1.220	11276.6
	10	2.453	13098.2
	20	4.920	16135.7
	30	7.386	18685.9
	40	9.853	20927.5

Table 2 :

 2 Bounds on Var(N ) and SCR for C 27 in function of n when s = t = 2. Yearly mortality rates at high ages (larger than 100) are difficult to estimate. Nowadays, mortality rates at age 100 in France are close to 0.36.

(ii) Residual lifetime at high ages.

Table 3 :

 3 Bounds on Var(T 100 ) when s = 2 and t = 0, 1, 2. quick treatment is necessary to prevent a medical emergency. A study involving 202 patients during one year has given the following empirical distribution (Table

	Hypothesis	H1	H2
	E(T 100 )	1.68 1.86
	Var(T 100,min ) (2,0)	0.22 0.22
	Var(T 100,min ) (2,1)	1.58 1.83
	Var(T 100,min ) (2,2)	2.26 2.69
	Var(T 100 ) (empirical case) 3.77 5.27
	Var(T 100,max ) (2,2)	14.54 15.73
	Var(T 100,max ) (2,1)	14.54 15.73
	Var(T 100,max ) (2,0)	22.38 24.39

Table 4 :

 4 Empirical distribution for N .

	Values j of N Frequencies Proportions
	0		67		0.33168
	1		60		0.29703
	2		53		0.26238
	3		16		0.07921
	4		5		0.02475
	5		1		0.00495
	j N min (2,0)	N max (2,0)	N min (2,1)	(2,1) max N
	0	0	0.7634 0.3028 0.60560
	1 0.8168	0	0.3028 0.07888
	2 0.1832	0	0.3028 0.07888
	3	0	0	0.0916 0.07888
	4	0	0	0	0.07888
	5	0	0.2360	0	0.07888

Table 5 :

 5 Probability mass functions of the extrema when s = 2 and t = 0, 1, with n = 5.

		n = 5 n = 10
	Var(N min ) (2,0)	0.1496 0.1496
	Var(N min ) (2,1)	0.6942 0.6942
	Var(N ) (empirical case) 1.1852 1.1852
	Var(N max ) (2,1)	2.9384 6.8823
	Var(N max ) (2,0)	5.9259 11.8519

Table 6 :

 6 Bounds on Var(N ) for s = 2 and t = 0, 1, with n = 5 or 10.

Acknowledgement

We are grateful to S. Utev for a number of helpful discussions and advice, and to M. Mesfioui for useful comments on a preliminary draft. We also thank the Referee for many remarks and suggestions. This work was conducted while C. Lefèvre was visiting the I.S.F.A., Université de Lyon. It was partly funded by an ANR research project (ANR-08-BLAN-0314-01).

Appendix: convex extrema

The s-convex extrema are known under explicit form when s = 1, 2, 3, 4 (see the references given in the Introduction). They are recalled below, for discrete random variables valued in a set {0, . . . , n}, n ∈ IN 0 , and for real-valued random variables with support in an interval [0, b], b > 0. We denote by B s (n) and B s (b) the classes of all such discrete and real-valued random variables X that have prescribed first s -1 moments µ

min (b) = 0] a.s., and X

(1)

with probability ξ + 1 -µ 1 , ξ + 1 with probability µ 1 -ξ, (7.1)

min (b) = µ 1 a.s., and

with probability

4-convex extrema. Inside B 4 (n), let θ and η be integers in [0, n -1] with θ + 1 < η such that α(θ, η), α(η, θ), -α(θ -1, η) and -α(η -1, θ) ≥ 0, where α(θ, η)

with probability p 3 = α(η, θ)/(η -θ)(η -θ -1), η + 1 with probability p 4 = -α(η -1, θ)/(η -θ)(η -θ + 1), (7.8)

.

(7.9)

r -with probability (r + -µ 1 )/(r + -r -), r + with probability (µ 1 -r -)/(r + -r -), (7.10)

with probability p 2 =

(bµ 1 -µ 2 ) 3

(bµ 2 -µ 3 )(b 2 µ 1 -2bµ 2 +µ 3 ) , b with probability p 3 = µ 1 µ 3 -µ 2 2 b(b 2 µ 1 -2bµ 2 +µ 3 ) .

(7.11)