domain: spi chim

text:
Cesari Laetitia
email: laetitia.cesari@univ-lorraine.fr

Dominique Alonso

Fabrice Mutelet

Laëtitia Cesari

Experimental and Computational Investigation on the Glyceline -CO2 Working Fluid in Absorption Chiller

Introduction

Results

Deep Eutectic Solvent (DES)

Mixing of compounds at the eutectic composition Properties of DES: Low vapor pressure High thermal and chemical stability Cheaper than ionic liquids Idea: use of deep eutectic solvent as absorbent Experimental measurements of {CO 2 + glyceline} binary system were performed in order to evaluate its possible use as a working fluid in a singleeffect absorption chiller. In this work, the values of the COP easily higher than 0.7 strengthen the idea that deep eutectic solvent could be a new alternative for working mixture in absorption chiller.

Schema of the VLE apparatus. 1, thermostatic bath; 2, analytical balance; 3, vacuum pump; 4, piston; 5, temperature probe (Pt100); 6, magnetic stirrer; 7, light source; 8: calibrated pressure sensor (0 < P < 340 bar); 9, sapphire window; 10, video camera; 11, monitor.

choline chloride glycerol

Vapor-liquid equilibria of the {glyceline + CO 2 } binary system: ─ 313.

  E.-S. Abumandour, F. Mutelet, D. Alonso, Are Ionic Liquids Suitable as New Components in Working Mixtures for Absorption Heat Transformers?, Prog. Dev. Ion. Liq. (2017). [2] D.Z. Troter, M.Z. Zlatković, D.R. Đokić-Stojanović, S.S. Konstantinović, Z.B. Todorović, Citric acidbased deep eutectic solvents: Physical properties and their use as cosolvents in sulphuric acidcatalysed ethanolysis of oleic acid, Adv. Technol. 5 (2016) 53-65. [3] C.-M. Lin, R.B. Leron, A.R. Caparanga, M.-H. Li, Henry's constant of carbon dioxide-aqueous deep eutectic solvent (choline chloride/ethylene glycol, choline chloride/glycerol, choline chloride/malonic acid) systems, J. Chem. Thermodyn. 68 (2014) 216-220.

  We would like to acknowledge support from CNRS (France) and TOTAL through the PEPS project Coefficient of performance as a function of the temperature of the generator for different pressures at the generator/condenser: ─ 55 bar (Tc = 291.4 K) ─ 60 bar (Tc = 295.15 K) ─ 65 bar (Tc = 298.61 K) ─ 70 bar (Tc = 301.9 K) Others operating conditions: P low = 40 bar, Te = 278.15K, Ta = 313.15K.


text:
Fabrice Mutelet

Cesari Laetitia

Use of Ionic liquid + nanoparticles as media for fuel desulfurization

In recent years, considerable attention has been given to deep desulfurization of gasoline due to the higher restrictions concerning sulfur-compounds content level in fuels. Sulfur oxides formed in gasoil combustion process contribute to acid rains, global warming and air pollution. Those affect urban as well as industrial areas and are harmful for human health due to secondary inorganic aerosol gases formation. The European Union gradually reduced the maximum sulfur concentration in fuels to avoid high emission of gases resulting from combustion of heavy fuels. Since 2010, the total sulfur content cannot be higher than 10 ppm. Deep desulfurization of gasoline and diesel is very complex problem for petroleum industry and has to be solved to minimize the pollution from fuel oils exhaust gas.

Results of extraction of sulfur containing compounds from gasoline oil using ILs indicate that such a process could be an attractive alternative to common desulfurization by hydrotreating. Indeed, the values of thiophene selectivity (about 1000) obtained with 1-ethyl-3-methylimidazolium thiocyanate [EMIM][SCN] or 1,3-dimethylimidazolium methylphosphonate [DMIM][MP] are large but the distribution factor values still remain to be improved (0.6 -0.8 and 0.2 -1.1 respectively). The aim of this study is to investigate the possible use of these ILs in presence of nanoparticles of TiO 2 or Fe 2 O 3 as solvents for the separation problems frequently encountered in petroleum industry {aromatic sulfur compound + aliphatic hydrocarbon}.

Experimental Procedure

The liquid-liquid equilibria measurements of ternary mixtures have been performed in jacketed glass cells. The experimental set up consists of a cell with an internal volume of about 30 cm 3 kept at constant temperature of 298.15 K using a thermostated bath. The ternary mixtures, with compositions inside the immiscible region of the system, are weighted using a METTLER analytical balance. All the mixtures were vigorously mixed using a teflon coated magnetic stirrer for 2h to reach a good contact between both phases and kept overnight in order to settle down. Samples of both layers were taken from the cell using a syringe. The compositions of organic compounds such as n-heptane, thiophene were determined by gas chromatography. For determination of the ionic liquid composition, density measurements of both phases were performed at 298.15 K or 302.15 K with a vibrating tube densimeter (Anton Paar, model DMA601).

Thermodynamic study of the ternary systems {Thiophene or Pyridine / n-Heptane / IL} in the presence of nanoparticles

The size of aggregates of nanoparticules strongly depends on the anion grafted on the cation (figure 8). The interactions between nanoparticules and ILs seem to be less favorable in [EMIM] In these ternary phase diagrams, the low thiophene concentration part should be considered in order to investigate deep desulfurization of fuels using ionic liquids.

[DMIM][MP] is also highly selective for thiophene but the solute distribution ratio is twice smaller than with No structural changes in the ILs are observed

Use of {ionic liquid + nanoparticles} as media for fuel desulfurization

Fabrice

  MUTELET, Laetitia CESARI Laboratoire Réactions et Génie des Procédés (UMR CNRS 7274) , Université de Lorraine, Ecole Nationale Supérieure des Industries Chimiques, 1 rue Grandville, 54000 Nancy, France Conclusion (Liquid + liquid) equilibrium data for the two ternary systems {thiophene + n-heptane + [EMIM][SCN]} and {thiophene + n-heptane + [DMIM][MP]} in the presence of nanoparticles at T = 298.15 K were determined. All ternary systems correspond to the Type 2 category, with partial or total immiscibility of the IL with the thiophene and the hydrocarbons. The presence of nanoparticles in the ionic liquid phase increases the solute distribution ratio b. In [DMIM][MP], the interactions with the nanoparticules are more favorable than in [EMIM][SCN].

  [SCN] than in [DMIM][MP]. Nanoparticules also aggregate to form a bigger gathering. For example, with TiO 2 , the particules size distribution strongly decreases from d = 579 nm in [EMIM][SCN] to d = 243 nm in [DMIM][MP].

  [EMIM][SCN]. The capacity and the selectivity of {[EMIM][SCN] + nanoparticles} for thiophene extraction from hydrocarbon solvent increases when the mass of nanoparticles increases from 0.05 wt % to 0.1 wt %. The presence of nanoparticles in the ionic liquid phase increases the solute distribution ratio b of about 25 %.

Figure 5 .

 5 Figure 5. Solute distribution ratio b as function of solute mole fraction (thiophene) in hydrocarbon-rich phase, at 298.15 K for pure [EMIM][SCN] and its suspensions with TiO 2 and Fe 2 O 3 nanopowders.

Figure 1 .

 1 Figure 1. Effect of TiO 2 nanoparticles addition on the tie lines of ternary system {Thiophene + Heptane + [EMIM][SCN]} at 298.15 K. Nanopowder concentration in IL: 0.05 wt %:  and …….; 0.1 wt %:  and ------. Mixture without nanoparticles: .

Figure 2 .

 2 Figure 2. Effect of TiO 2 nanoparticles addition on the tie lines of ternary system {Thiophene + Heptane + [DMIM][MP]} at 298.15 K. Nanopowder concentration in IL: 0.05 wt %:  and …….; 0.1 wt %:  and ------. Mixture without nanoparticles:  .

Figure 3 .

 3 Figure 3. Effect of Fe 2 O 3 nanoparticles addition on the tie lines of ternary system {Thiophene + Heptane + [EMIM][SCN]} at 298.15 K. Nanopowder concentration in IL: 0.05 wt %:  and …….; 0.1 wt %:  and ------. Mixture without nanoparticles: .

Figure 4 .

 4 Figure 4. Effect of Fe 2 O 3 addition on the tie lines of ternary system {Thiophene + Heptane + [DMIM][MP]} at 298.15 K. Nanopowder concentration in IL: 0.05 wt %:  and …….; 0.1 wt %:  and ------. Mixture without nanoparticles:  .

Figure 8 .

 8 Figure 8. Particles size distribution for nanoparticules suspension in [DMIM][MP] and [EMIM][SCN].

Figure 6 .

 6 Figure 6. IR spectras of ILs and their suspensions with nanopowders.Figure 7. Raman spectra of [DMIM][MP] and suspensions with nanoparticles.

Figure 7 .

 7 Figure 6. IR spectras of ILs and their suspensions with nanopowders.Figure 7. Raman spectra of [DMIM][MP] and suspensions with nanoparticles.


Cesari Laetitia

Christophe Castel

published or not. The documents may come

Membrane contactors for intensified physical solvent absorption biogas purification processes: a parametric study

Introduction

Materials and Method

Main results

Conclusion and perspectives

Laëtitia Cesari, Christophe Castel et Eric Favre Laboratoire Réactions et Génie des Procédés (LRGP), UMR 7274, Nancy, France Laetitia.cesari@univ-lorraine.fr

Membrane contactors for intensified physical solvent absorption biogas purification processes: a parametric study.

Influence of solvent properties

Performance of the solvents decreases as follows :

methanol > N-methyl-2-pyrrolidone (NMP) > Propylenecarbonate (PC) > selexol > water Key factors : low CO 2 solubility in water, high viscosity of selexol.

Influence of membrane performances

Decrease of the k m value decreases the transferred molar flow.

Methanol and NMP are the most efficient solvents and the most affected.

For water, limitation is made by the liquid side instead of the membrane

Influence of flow configuration (liquid in or liquid out)

Optimum of the compacity value in liquid-out configuration. Value is dependent of the solvent used.

Always higher value of compacity is better in liquid-in configuration.

Strong dependence of the correlation used

Comparison with packed columns: Intensification possibilities

High value of intensification factors

Intensification factor independent of the inlet pressure 0.00 (u l /u g = 1, φ = 0.9, k m = 10 -3 m.s -1 ). This study intended to explore the possibilities of process intensification thanks to membrane contactors for absorption processes based on physical solvents. A systematic parametric study has been performed for CO 2 absorption in five different solvents showing a broad range of solubility and viscosity.

The results obtained here show however that solvent viscosity strongly impacts the interest and feasibility of membrane contactors compared to packed columns. Because the solubility / viscosity interplay plays a key role in the maximal intensification performances, a parametric study on operating temperature would be interesting. It is known that a lower temperature generally increases solubility but also viscosity. The intensification / energy trade-off is thus likely to strongly depend on the solvent temperature. The influence of solvent volatility, which has not been taken into account in this study, should also be considered.

It is expected that this exploratory parametric study will stimulate further researches on intensified processes with physical solvents.

Equations :

Parameters :

Table 2 :

 2 Influence of the pressure in liquid-in configuration

						Symbol	Parameter
	12.00
	10.00
	-1 )
	(mol.m -3 .s	6.00 8.00
	CO 2
	N	4.00
		2.00
		0	5	10	15	20
				E tot (kWh.Nm -3 )

Table 1 :

 1 Fixed and variable parameters in this work


domain: math math

text:
Safa Bridaa
email: bridaasafa@gmail.com

Wissem Jedidi
email: bwjedidi@ksu.edu.sa

Hristo Sendov
email: chssendov@stats.uwo.ca

GENERALIZED UNIMODALITY AND SUBORDINATORS, WITH APPLICATIONS TO STABLE LAWS AND TO THE MITTAG-LEFFLER FUNCTION

 on the distribution of the stable subordinators. Finally, we provide a new complete monotonicity property for the Mittag-Leffler function.

1. Introduction. Let CM denote the class of completely monotone functions, i.e. of those infinitely differentiable functions f : (0, ∞) → (0, ∞) satisfying (-1) n f (n) ≥ 0 for all integer n = 0, 1, 2, . . . Bernstein showed that the class CM corresponds to the Laplace transforms of measures on [0, ∞). That is f ∈ CM if, and only if,

f (λ) = L ν (λ) := [0,∞)
e -λx ν(dx), λ > 0, for some Radon measure ν on [0, ∞).

Let CF denote the class of cumulant functions, [START_REF] Basalim | Three classes of decomposable distributions[END_REF] CF := {ϕ Z (λ) = -log E[e -λZ ], s.t. λ ≥ 0, where Z ≥ 0 is a random variable}.

Let ϕ Z be the cumulant function of a non-negative random variable Z. Adapting [13, Theorem 2.1] (by looking at the Laplace transform of Z as the Mellin transform of e -Z ), we see that

(2) ϕ Z (λ) λ decreases to inf{x ≥ 0, s.t. P(Z ≤ x) > 0}, as λ increases to ∞.

As a consequence of ( 1) and ( 2), one observes that any ϕ ∈ CF is infinitely differentiable on (0, ∞), satisfies ϕ(0) = 0,

:= lim x→∞ ϕ(x) (3) d
x ∈ [0, ∞) and ϕ(λ) -dλ ≥ 0, for all λ ≥ 0.

An application of Hölder's inequality shows that cumulant functions are concave. More generally, elementary considerations show that every concave function ϕ : [0, ∞) → [0, ∞), differentiable on (0, ∞), satisfies

d := lim x→∞ ϕ(x) x = lim x→∞ ϕ (x) ∈ [0, ∞). ( 4
)
MSC2020 subject classifications: Primary 26A48, 26D07; secondary 30E20, Bernstein function, Complete monotonicity, Generalized unimodality, Mittag-Leffler function, Positive stable distribution, Subordinators. [START_REF] Basalim | Three classes of decomposable distributions[END_REF] Indeed, being concave and differentiable ϕ has non-increasing derivative and satisfies ϕ(x) ≤ ϕ(y) + ϕ (y)(x -y) for all x, y > 0. Divide both sides by y and take limit infimum as y approach infinity. Then, divide both sides by x and take limit superior as x approaches infinity. Combine the two resulting inequalities to conclude. Observe that (2) provides the following property

ϕ ∈ CF =⇒ 0 ≤ xϕ (x) ≤ ϕ(x), ∀x > 0 =⇒ lim x→0 + xϕ (x) = 0. ( 5
)
The class of Bernstein functions, usually denoted BF , consists of those functions φ : [0, ∞) → [0, ∞), satisfying φ ∈ CM.

Every Bernstein function can be represented, see the book by Schilling, Song and Vondraček [START_REF] Schilling | Bernstein Functions[END_REF]Equations (3.2), (3.3)], by [START_REF] Chaumont | Exercises in Probability: A Guided Tour from Measure Theory to Random Processes, via Conditioning[END_REF]

φ(λ) = q + dλ + (0,∞) (1 -e -λx )µ(dx) = q + λ d + ∞ 0 e -λx µ(x, ∞)dx , λ ≥ 0,
where q, d ≥ 0 are called the killing rate and the drift term respectively. The so-called Lévy measure µ satisfies the integrability condition [START_REF] Gorenflo | Mittag-Leffler Functions (Related Topics and Applications[END_REF] (0,∞)

(1 ∧ x)µ(dx) < ∞.
The classes CM and BF are closed under pointwise limits and form closed convex cones. The former is also closed under multiplication, while the latter under composition. From [START_REF] Chaumont | Exercises in Probability: A Guided Tour from Measure Theory to Random Processes, via Conditioning[END_REF], one also has the implication

φ(λ) ∈ BF =⇒ φ(λ) λ ∈ CM. (8)
For justification of these facts, see [START_REF] Schilling | Bernstein Functions[END_REF]Corollary 1.6 and Corollary 3.8]. Bernstein functions (with no killing rate) are in one-to-one correspondence with infinitely divisible non-negative random variables, see [START_REF] Schilling | Bernstein Functions[END_REF]Lemma 5.8]: If X ≥ 0 has cumulant function ϕ X , then the distribution of X is infinitely divisible if, and only if ϕ X is a Bernstein function. In this case, X is embedded into a subordinator (X t ) t≥0 . That means that (X t ) t≥0 is an increasing Lévy process starting from zero, and the celebrated Lévy-Khintchine formula holds: X d = X 1 and E[e -λX t ] = e -tϕ X (λ) , λ ≥ 0.

Let φ = q + ϕ X be the Bernstein function obtained by adding to ϕ X a killing rate q > 0. This Bernstein function is then associated by the Lévy-Khintchine formula to the killed process (X (q) t ) t≥0 defined by

X (q) t := X t if t < e q , ∞ otherwise,
where e q is an independent exponential random variable with parameter q. Bernstein and completely monotone functions are also connected by the relationship φ ∈ BF ⇐⇒ e -tφ ∈ CM, for all t > 0, as shown in [START_REF] Schilling | Bernstein Functions[END_REF]Theorem 3.7]. Other good references for these classes of functions and their properties are the books by Steutel and van Harn [START_REF] Steutel | Infinite Divisibility of Probability Distributions on the Real Line[END_REF] and Bertoin [START_REF] Bertoin | Subordinators: Examples and Applications[END_REF]. Define ε t (ϕ)(λ) := 1 -e -tϕ(λ) , ϕ ∈ CF, and observe that if ϕ(0) = 0, then [START_REF] James | Generalized Gamma Convolutions, Dirichlet means, Thorin measures, with explicit examples[END_REF] ϕ ∈ CF ⇐⇒ e -ϕ ∈ CM ⇐⇒ 1 -e -ϕ ∈ BF ⇐⇒ ε n (ϕ) ∈ BF, for all n = 1, 2, . . .

(To see the necessity in the last implication, differentiate ε n (ϕ) and use the facts that e -ϕ and e -ϕ ϕ are both completely monotone and that product of completely monotone functions is completely monotone.) Since {φ ∈ BF, s.t. φ(0) = 0} ⊂ CF, [START_REF] Jedidi | Stable processes, mixing, and distributional properties[END_REF] we also observe that φ ∈ BF ⇐⇒ t (φ -φ(0)) ∈ CF, for all t > 0 (11) ⇐⇒ ε t (φ -φ(0)) ∈ BF, for all t > 0.

Finally, observe that any Bernstein function satisfies [START_REF] Bondesson | Generalized Gamma Convolutions and Related Classes of Distributions and Densities[END_REF], [START_REF] Butzer | A Direct approach to the Mellin Transform[END_REF], and (5) too.

In [START_REF] Hansen | Reverse self-decomposability[END_REF], Hansen introduced what he called the class of reverse s-self-decomposable distributions, as follows: DEFINITION 1.1 (Hansen, Definition 3.1 [START_REF] Hansen | Reverse self-decomposability[END_REF].). Let X be real valued infinitely divisible random variable and s > 0. The distribution of X is called reverse s-self-decomposable, if its characteristic function Ψ X is such that [START_REF] Olshen | A Generalized Unimodality[END_REF] Ψ X (u) exists for u = 0, lim u→0 u Ψ X (u) = 0 and, for every c ∈ (0, 1), there exists a characteristic function Ψ c such that

(13) Ψ X (u) = Ψ c s X (u/c) Ψ c (u), u ∈ R.
Taking into account properties (3) of cumulant functions and motivated by [START_REF] Pakes | Characterization by invariance under length-biasing and random scaling[END_REF], we introduce for c > 0 and s > 0, the difference operator ω c,s defined, for functions ϕ : [0, ∞) → [0, ∞), differentiable on (0, ∞), such that the limit [START_REF] Sato | Lévy Processes and Infinitely Divisible Distributions[END_REF] d := lim x→∞ ϕ(x) x exists and is in [0, ∞), and lim x→0 + xϕ (x) = 0. Let ϕ (λ) := ϕ(λ) -ϕ(0) -dλ, [START_REF] Sendov | New representation theorems for completely monotone and bernstein functions with convexity properties on their measures[END_REF] and note that ϕ (λ) = ϕ(λ) -dλ ≥ 0, whenever ϕ ∈ CF . Define ω c,s (ϕ)(λ) := ϕ (cλ) -c s ϕ (λ), λ ≥ 0, [START_REF] Schilling | Bernstein Functions[END_REF] and note that

ω c,s (ϕ)(λ) = ω c,s (ϕ )(λ) = ϕ(cλ) -c s ϕ(λ) -(1 -c s ) ϕ(0) -(c -c s ) d λ.
Next, observe that [START_REF] Simon | On the unimodality of power transformations of positive stable densities[END_REF] ϕ ∈ CF ∪ BF =⇒ ω c,s (ϕ)(0) = 0 and lim x→∞ ω c,s (ϕ)(x) x = 0.

By analogy to Definition 1.1, we have the following one in terms of cumulant functions.

DEFINITION 1.2. Let s > 0. We say that a non-negative random variable X has a distributions in the class RSD s , and we denote X ∼ RSD s , if its cumulant function ϕ X is such that for every c ∈ (0, 1), there exists a non-negative random variable Y c such that [START_REF] Steutel | Infinite Divisibility of Probability Distributions on the Real Line[END_REF] e -ω c,s (ϕ X )(λ) = E[e -λY c ], λ ≥ 0.

In other words, X ∼ RSD s if ω c,s (ϕ X ) ∈ CF , for every c ∈ (0, 1).

REMARK 1.3. (i) Note that condition ( 18) is analogous to [START_REF] Pakes | Characterization by invariance under length-biasing and random scaling[END_REF] obtained by replacing the characteristic function of X with the normalized cumulant function ϕ X (λ) -dλ. Conditions similar to [START_REF] Olshen | A Generalized Unimodality[END_REF] are not needed in Definition 1.2, since, by (3), it is immediate that ϕ X satisfy ϕ X (λ) -d ∈ (0, ∞), for all λ > 0, and lim

x→0+ x (ϕ X (x) -d) = 0.
(ii) Theorem 2.6 below shows several important implications of [START_REF] Steutel | Infinite Divisibility of Probability Distributions on the Real Line[END_REF]. That is, if

X ∼ RSD s , then • The cumulant function ϕ X is a Bernstein function, that is X is infinitely divisible; • The cumulant function ω c,s (ϕ X ) is a Bernstein function, for every c ∈ (0, 1), that is, Y c is infinitely divisible.
We will see that the difference operators ω c,s is tightly linked to the differential operator Ω s , s > 0, defined for functions ϕ : [0, ∞) → [0, ∞), differentiable on (0, ∞) and satisfying [START_REF] Sato | Lévy Processes and Infinitely Divisible Distributions[END_REF], by ( 19)

Ω s (ϕ)(λ) := s ϕ (λ) -λϕ (λ), λ > 0, Ω s (ϕ)(0) = 0.
where ϕ is given by [START_REF] Sendov | New representation theorems for completely monotone and bernstein functions with convexity properties on their measures[END_REF]. Note that

Ω s (ϕ)(λ) = Ω s (ϕ )(λ) = s ϕ(λ) -ϕ(0) -λϕ (λ) -(s -1) d λ, Ω s (ϕ) ≡ 0 ⇐⇒ ϕ(λ) = q + d λ + cλ s , for some q, d, c ≥ 0.
Analogously to [START_REF] Simon | On the unimodality of power transformations of positive stable densities[END_REF], we have:

ϕ ∈ CF ∪ BF =⇒ lim x→∞ Ω s (ϕ)(x) x = 0.
Adopting Hansen's terminology, keeping [START_REF] Kyprianou | Fluctuations of Lévy Processes with Applications[END_REF] and Definition 1.2 in mind, we introduce the following subclasses of CF and BF . Recall the definition of φ , given in [START_REF] Sendov | New representation theorems for completely monotone and bernstein functions with convexity properties on their measures[END_REF]. DEFINITION 1.4. For any s > 0, define

(i) CF s := {ϕ ∈ CF, s.t. ω c,s (ϕ) ∈ CF, ∀ c ∈ (0, 1)}; (ii) BF s := {φ ∈ BF, s.t. Ω s (φ) ∈ BF}; (iii) BF * s := {φ ∈ BF, s.t. 1 -e -tφ ∈ BF s , ∀t > 0}.
Note that the class BF 1 has already been investigated by the first two authors in [1, Section 5]. The goal of this work is to develop a comprehensive probabilistic theory of the classes BF s and BF * s , and to complete Hansen's results and those of [START_REF] Basalim | Three classes of decomposable distributions[END_REF]. This allows us, in Section 6, to answer a conjecture stated in [START_REF] Sendov | New representation theorems for completely monotone and bernstein functions with convexity properties on their measures[END_REF]Open Problem 4.1] that is related to a problem previously raised by Simon [START_REF] Simon | On the unimodality of power transformations of positive stable densities[END_REF]. To help the reader navigate through the results in this paper, we include Figure 1, showing the dependencies between the main results.

Section 2 deals with several analytic properties of the class BF s . Theorem 2.6 below shows that

CF s = BF s ∩ {φ ∈ BF, s.t. φ(0) = 0}.
The former also shows that our extension of Hansen's classes to RSD s , s > 0, corresponds to the cumulant functions in the class BF s . A full characterization of this class is given in Theorem 2.1.

FIGURE 1. Dependencies between the main results

Sections 3 and 4 give various stochastic interpretations for the class RSD s . For instance, Corollary 3.1 provides the decomposability of the distributions of the associated subordinators: if X is a non-negative infinitely divisible random variable, with Bernstein function ϕ X , and is embedded into the subordinator (X t ) t≥0 , then

(20) ϕ X ∈ BF s ⇐⇒ c X t d = X c s t + d (c -c s ) t + c Z c,s
t , for all t > 0, c ∈ (0, 1), where in the last identities, d is the drift term and Z c,s t is a non-negative random variable whose distribution is necessarily infinitely divisible. This decompositions explains the name of reverse self-decomposability and produces a self-similar temporal property which mimics the one of the classical and well-known class of self-decomposable distributions and selfsimilar processes. To explain this resemblance, we remind that a random variable X is selfdecomposable if it satisfies the identities in law

X d = cX + Z c , for all c ∈ (0, 1),
where Z c is a random variable independent of X, necessarily infinitely divisible. By [START_REF] Sato | Lévy Processes and Infinitely Divisible Distributions[END_REF]Theorem 16.1], we known that for such X, and for any γ > 0, there exist a self-similar process with independent increments (X t ) t≥0 , i.e. a process satisfying (21) (a X t ) t≥0 d = (X a γ t ) t≥0 , for any a ≥ 0, and X t n -X t n-1 , . . . , X t 2 -X t 1 are independent for any t n > • • • > t 1 ≥ 0, such that X 1 and X have the same distribution.

In Section 4, we will see that the concept of reverse self-decomposability is intimately connected to the concept of generalized unimodality which is defined as follows: DEFINITION 1.5 (Olshen and Savage [START_REF] Olshen | A Generalized Unimodality[END_REF]). Let s > 0. A real-valued random variable Z is s-star unimodal, if it is of the form

Z d = U 1/s V,
where U is uniformly distributed on (0, 1) and independent of V .

Let DF be the collection of distribution functions of non-negative random variables:

DF := {F Z (x) = P(Z ≤ x), s.t. x ≥ 0, Z ≥ 0}.
Proposition 4.1 below, asserts, among other things, that if Z is a positive random variable with distribution function F Z and cumulant function ϕ Z , then for arbitrary subordinators (η t ) t≥0 , independent of (X ( ) t ) t≥0 . We stress that Sendov and Shan [START_REF] Sendov | New representation theorems for completely monotone and bernstein functions with convexity properties on their measures[END_REF] were the first to introduce the class BF * 1 (and also BF 1 ) without making use of the identities in law (43) and (44) and the density shapes (31) and (49) below. The above discussion illustrates to what extent the classes BF s and BF * s are rich from a stochastic point of view.

1 -e -ϕ Z ∈ BF s ⇐⇒ 1 Z is s-star unimodal (22) ⇐⇒ ω c,s (F Z ) ∈ DF, ∀c ∈ (0,
Finally, in Sections 6 and 7, we draw attention to Simon's work [START_REF] Simon | On the unimodality of power transformations of positive stable densities[END_REF], who focused on positive stable distributions, namely those associated to the Bernstein functions φ α (λ) := λ α , α ∈ (0, 1). With a different approach and with techniques restricted to this special case, Simon studied the range of values s > 0, for which φ α ∈ BF s . We emphasize that, for the function φ α , we have, see (87):

φ α ∈ BF * s ⇐⇒ Ω s (1 -e -φ α ) ∈ BF.
Then, the problem becomes to find the values of s > 0 for which Ω s (1 -e -φ α ) ∈ BF . This simple looking, but in our opinion non-trivial, question is completely answered in Theorem 6.3. As a consequence, Corollary 7.1 illustrates when the usual Mittag-Leffler function E α is such that λ → 1 -r Γ(1 -α) λ E α (-λ), r > 0, is completely monotone, or at least is non-negative. Sections 8 and 9 consist of two appendices that clarify the structure of Lévy exponents and Bernstein functions.

2. The classes BF s for s > 0. We start this section with an additional account on several interesting subclasses of Bernstein and completely monotone functions, that will be needed in the sequel. The subclass of complete Bernstein functions, CBF , consists of those Bernstein functions that have associated Lévy measure of the form (26) µ(dx) = m(x) dx, x > 0, where m(x) ∈ CM.

Similarly See Section 9 for the integral representations of the functions in these subclasses. The corresponding classes of infinitely divisible distributions are respectively the famous Bondesson class and the class of generalized Gamma convolutions popularized by Bondesson [START_REF] Bondesson | Generalized Gamma Convolutions and Related Classes of Distributions and Densities[END_REF], see also [START_REF] James | Generalized Gamma Convolutions, Dirichlet means, Thorin measures, with explicit examples[END_REF], [START_REF] Steutel | Infinite Divisibility of Probability Distributions on the Real Line[END_REF], or [START_REF] Schilling | Bernstein Functions[END_REF] for more information. Sendov and Shan [START_REF] Sendov | New representation theorems for completely monotone and bernstein functions with convexity properties on their measures[END_REF] focused on another proper subclass of BF , namely those φ such that in [START_REF] Chaumont | Exercises in Probability: A Guided Tour from Measure Theory to Random Processes, via Conditioning[END_REF] the Lévy measure µ has an harmonically concave tail, that is

x → x µ(x, ∞), x > 0, is concave .
The latter is equivalent to φ ∈ BF 1 . After some analytical effort, the equivalence

φ ∈ BF 1 ⇐⇒ µ(dx) = p(x)
x 2 dx, where p(x) is a measurable, non-decreasing function, was shown in [START_REF] Basalim | Three classes of decomposable distributions[END_REF]. The class BF s , introduced in Definition 1.4, is not void and extends BF 1 . Indeed, consider the Bernstein function ϕ α (λ) = λ α , 0 ≤ α ≤ 1, associated to the positive stable distribution, cf. (75) below. We have

(28) ϕ α ∈ BF s ⇐⇒ Ω s (ϕ α )(λ) = (s -α)λ α ∈ BF ⇐⇒ s ≥ α.
The characterization of BF s , for all s > 0, is obtained by the following theorem. Recall the notation ( 14), ( 15), [START_REF] Zolotarev | One Dimensional Stable Laws[END_REF], and define (29) a s := lim λ→∞ φ (λ) λ s . (The next theorem gives conditions under which a s exists.) THEOREM 2.1. Let s > 0 and let φ : [0, ∞) → [0, ∞) be a differentiable function on (0, ∞). Then, the following assertions hold.

1)

If Ω 1 (φ) ∈ BF , then d exists and is finite, φ ≥ 0 and

lim λ→∞ Ω 1 (φ)(λ) λ = 0;
2) Assume that d exists and is finite. Assume φ ≥ 0 and Ω s (φ) ∈ BF . Then, (i) φ (λ)/λ s ∈ CM;

(ii) φ ∈ BF s (in particular φ ∈ BF ), limits [START_REF] Sato | Lévy Processes and Infinitely Divisible Distributions[END_REF], and (29) exist, and φ has the representation

(30) φ(λ) = φ(0) + d λ + a s λ s + ∞ 1 Ω s (φ)(λx) x s+1 dx, where d ≥ 0, a s ∈ [0, ∞), if s < 1, and a s = 0, if s ≥ 1.
3) The following conditions are equivalent:

(i) φ ∈ BF s ;

(ii) φ ∈ BF and the Lévy measure µ of φ is of the form

(31) µ(dx) = p s (x)
x s+1 dx, x > 0, for some non-decreasing, right-continuous, function p s : (0, ∞) → [0, ∞), such that (32)

1 0 p s (x) x s dx + ∞ 1 p s (x) x s+1 dx < ∞,
and p s (0+) = 0, if s ≥ 1;

(iii) φ has the representation

(33) φ(λ) = φ(0) + d λ + a s λ s + ∞ 1 ϕ s (λx) x s+1 dx, λ ≥ 0,
where d ≥ 0, a s ∈ [0, ∞), if s < 1, and a s = 0, if s ≥ 1, and ϕ s ∈ BF has no killing term nor drift term. 4) The coefficient a s can be expressed as a s = p s (0+)Γ(1 -s)/s, whenever s < 1. 5) BF s ⊆ BF r , for all r > s > 0.

PROOF. 1) By the definition of Ω 1 and the assumption Ω 1 (φ) ∈ BF , we see that φ is twice differentiable and and -φ (λ) = Ω 1 (φ) (λ)/λ ≥ 0. Thus, φ is concave and by (4), we conclude that, as λ approaches infinity, φ (λ) decreases to d ∈ [0, ∞) and

lim λ→∞ Ω 1 (φ)(λ) λ = lim λ→∞ φ(λ) -φ(0) λ -φ (λ) = 0. The inequality φ (λ) = Ω 1 (φ)(λ) + λ φ (λ) -d ≥ 0 follows. 2) (i): Since Ω s (φ ) = Ω s (φ) is infinitely often differentiable, then so is φ . Observe that (34) - d dλ φ (λ) λ s = 1 λ s Ω s (φ )(λ) λ , λ > 0
is a product of two completely monotone functions, hence is completely monotone, see [START_REF] Hansen | Reverse self-decomposability[END_REF].

Then, since φ is non-negative, conclude that φ (λ)/λ s ∈ CM.

2) (ii): We first discuss the conditions on a s . Since φ (λ)/λ s ∈ CM, then necessarily a s ∈ [0, ∞). On the one hand, since the limit d is assumed to exist, by (29) we see that

a 1 = 0. The assumption Ω s (φ ) ∈ BF implies that lim λ→∞ Ω s (φ )(λ) λ = s a 1 -lim λ→∞ φ (λ) = -lim λ→∞ φ (λ) ∈ [0, ∞).
On the other hand, if s > 1, then

0 = lim λ→∞ Ω s (φ )(λ) λ s = s a s -lim λ→∞ λ 1-s φ (λ) = s a s ,
hence a s = 0. Next, integrating (34), we obtain

φ(λ) = φ(0) + d λ + φ (λ) = φ(0) + d λ + λ s a s + ∞ λ Ω s (φ )(y) y s+1 dy , λ ≥ 0.
Then, making the change of variable y = λx, we arrive at (30). Observe that λ s ∈ BF if s < 1 and if s ≥ 1, then a s = 0. Since BF is a closed convex cone, see the comments above (8), formula (30) shows that φ ∈ BF . Finally, the assumption Ω s (φ) ∈ BF implies that φ ∈ BF s .

3) (i) =⇒ (ii): By the definition of the class BF s , we have that φ and Ω s (φ)(λ) are both in BF . By representation ( 6) of φ, it follows that

Ω s (φ)(λ) = λ (0,∞) e -λx s µ(x, ∞)dx -x µ(dx) ∈ BF.
Thus, necessarily

s µ(x, ∞) dx -x µ(dx) = ν s (x, ∞) dx, for some Lévy measure ν s .
We deduce that µ is absolutely continuous and could be written in the form (31) with some non-negative function p s . Further,

(35) ν s (x, ∞) = s ∞ x p s (u) u s+1 du - p s (x) x s is non-increasing, i.e. ν s (dx) = -d(ν s (x, ∞)) = dp s (x)/
x s is a positive measure, or in other words, p s (x) is a non-decreasing function. Since the integral is continuous in x and ν s (x, ∞) is right-continuous, then so is p s (x). For later use, we record that in this case,

Ω s (φ) = Ω s (φ ) is represented by (36) Ω s (φ)(λ) = (0,∞) (1 -e -λx ) dp s (x) x s , λ ≥ 0.
The integrability condition (32) on p s (x) is a reformulation of [START_REF] Gorenflo | Mittag-Leffler Functions (Related Topics and Applications[END_REF]. It is clear that since p s is non-decreasing and right continuous, then p s (0+) ∈ [0, ∞). For the last assertion, note that the integrability of p s (x)/x s on (0, 1) implies that

lim x→0+ xp s (x)/x s = lim x→0+ p s (x)/x s-1 = 0, hence p(0+) = 0, if s ≥ 1. 3) (ii) =⇒ (i): is obtained by reading the arguments in (i) =⇒ (ii) in reverse. 3) (i) =⇒ (iii): is true with ϕ s = Ω s (φ) in (30).
3) (iii) =⇒ (i): Change the variable under the integral y = λx in (33) and then differentiate with respect to λ to get Ω s (φ) = ϕ s . This implies that Ω s (φ) ∈ BF and by part 2) (ii), we conclude that φ ∈ BF s . 4) For s < 1, we have

a s = lim λ→∞ φ (λ) λ s = p s (0+) Γ(1 -s) s + lim λ→∞ 1 λ s ∞ 0 (1 -e -λx ) p s (x) -p s (0+) x s+1 dx and lim λ→∞ 1 λ s ∞ 0 (1-e -λx ) p s (x) -p s (0+) x s+1 dx = lim λ→∞ ∞ 0 (1-e -y ) p s (y/λ) -p s (0+) y s+1 dy = 0,
follows from (32), from the monotonicity of p s , and from the dominated convergence theorem. 5) Use the fact that

φ ∈ BF s ⇐⇒ φ ∈ BF and Ω s (φ) ∈ BF =⇒ φ ∈ BF and Ω r (φ) = (r -s)φ + Ω s (φ) ∈ BF ⇐⇒ φ ∈ BF r .
This concludes the proof of the theorem.

REMARK 2.2. Note that the set BF s is closed under pointwise limits and forms a closed convex cone. This is immediate from its definition with the help of [16, Corollary 3.9]. Theorem 6.2 in [START_REF] Schilling | Bernstein Functions[END_REF] states that the operator ∆ defined by

∆φ(λ) := λ 2 (0,∞) e -λx φ(x) dx, φ ∈ BF, is a bijection from BF onto CBF . Let BFT s , s > 0, be the class of Bernstein functions ϕ represented by (37) ϕ(λ) = q + dλ + (0,∞) λ λ + u p s (u) u s du, λ ≥ 0,
where q, d ≥ 0 and p s satisfies conditions 3(ii) of Theorem 2.1. By (104), one sees that BFT 1 = T BF . The following extension of this bijection gives an additional interest to the class BF s .

PROPOSITION 2.3. The operator ∆ is a bijection from BF s onto BFT s . Moreover, BFT s ⊆ BFT r if s < r.

PROOF. If φ ∈ BF s , then, by Theorem 2.1, there exist q, d ≥ 0 and a nondecreasing function p s such that

φ(λ) = q + dλ + (0,∞) (1 -e -λu ) p s (u) u s+1 du, λ ≥ 0,
and elementary computation shows that ∆φ(λ) is given by the right-hand side of (37). The surjectivity is obtained by reversing the calculus. The inclusion BFT s ⊆ BFT r is obvious since

p s (u) u s+1 = u r-s p s (u) u r+1 ,
and u → u r-s p s (u) remains non-decreasing.

REMARK 2.4. In [3, (9.5.1), pp. 150] and [16, Definition 8.9], a particular class of Bernstein functions was introduced. In [START_REF] Schilling | Bernstein Functions[END_REF], it is denoted by T BF s , s > 0, and it corresponds to those Bernstein functions such that the corresponding Lévy measure µ in ( 6) has a density of the form

µ(dx) = m(x) dx and x → x 2-s m(x) ∈ CM. It is not difficult to show that s ≥ 1 =⇒ BFT s ⊂ T BF s .
To check the latter, use point 3(ii) in Theorem 2.1, and the fact that p s (0+) = 0 if s ≥ 1, to obtain that any function ϕ ∈ BFT s , with s ≥ 1, is represented by (37), hence has the form

ϕ(λ) = q + dλ + ∞ 0 (1 -e -λx ) m s (x) dx,
and the function

x → x 2-s m s (x) = x 2-s ∞ 0 u 1-s p s (u) e -ux du = (0,∞) ∞ vx u 1-s e -u du dp s (v)
is completely monotone.

Point 3) in next proposition shows a nice sufficient condition for a function to be in BF s . PROPOSITION 2.5. Let s > 0 and let φ ∈ BF have drift term d. Then, the following hold.

1) If φ ∈ BF s , then φ α (λ) := φ(λ α ) ∈ BF s , for every α ∈ (0, min(1, s)].
In the case when d = 0, the implication holds for every α

∈ (0, 1]. 2) If φ ∈ BF s , s ≥ 1, then (38) ψ a,s (λ) := a s φ(λ) -φ(0) -d λ -λ φ(λ + a) -φ(λ) -a d ∈ BF, for every a > 0.
3) If φ satisfies (38) for some s > 0, then φ ∈ BF s .

PROOF. 1) It suffices to use that

d := lim x→∞ φ(x)/x ∈ [0, ∞) implies lim x→∞ φ α (x)/x = 0
and to write

Ω s (φ α )(λ) = s φ(λ α ) -φ(0) -α λ α φ (λ α ) = α s φ(λ α ) -φ(0) -λ α φ (λ α ) -(s -1) d λ α + s (1 -α) φ(λ α ) -φ(0) -d λ α + (s -α) d λ α = α Ω s (φ)(λ α ) + s (1 -α) φ (λ α ) + (s -α) d λ α ,
then, use that non-negative linear combinations and compositions of Bernstein functions are Bernstein. In the case when d = 0, the last term above disappears, leaving non-negative linear combination of Bernstein functions for every α ∈ (0, 1].

2) There is no loss of generality to assume that φ has zero killing rate and drift term, and by Theorem 2.1 the Lévy measure of φ is given by (31). For x > 0, define

(x) := 1 -e -x
x and l a,s (x) := s

∞ x p s (u) u s+1 du -(ax) p s (x) x s . ( 39
)
Observe that 0 < (x) < 1 and that

κ s (x) := s 1 -(x) + x (x) ≥ κ 1 (x) ≥ 0.
Then, using (35), it is clear that

l a,s (x) ≥ s ∞ x p s (u) u s+1 - p s (x)
x s ≥ 0.

Integrating by parts, using that (ax) is a continuous function and [5, Theorem 6.2.2], gives

∞ x (au) dp s (u) u s = -(ax) p s (x) x s - ∞ x p s (u) u s+1 (au (au) -s (au)) du = -(ax) p s (x) x s - ∞ x p s (u) u s+1 (κ s (au) -s) du = l a,s (x) - ∞ x p s (u) u s+1 κ s (au) du.
(In the first equality, use (32) to conclude that p s (x)/x s+1 approaches zero at infinity.) This expresses l a,s as the right tail of a positive measure:

l a,s (x) = [x,∞) κ s (au) p s (u) u s+1 du + (au) dp s (u) u s , implying that l a,s is non-increasing. It is also right-continuous, because p s (x) is. The fact that ν a,s (du) := κ s (au) p s (u) u s+1 du + (au) dp s (u)
u s is a Lévy measure, follows from (39). Using both representations of φ in [START_REF] Chaumont | Exercises in Probability: A Guided Tour from Measure Theory to Random Processes, via Conditioning[END_REF], one can see that

λ → ψ a,s (λ) = a ∞ 0 (1 -e -λx ) ν a,s (dx) ∈ BF.
3) Use the fact that BF is closed under taking point wise limits and notice that Ω s (φ) = lim a→0+ ψ a,s /a.

Recall the difference operators ω c,s introduced in [START_REF] Schilling | Bernstein Functions[END_REF]. It is easy to verify that we have Further ω c,s is tightly linked to the differential operator Ω s given in [START_REF] Zolotarev | One Dimensional Stable Laws[END_REF].

ω c n+1 ,s (φ)(λ) = ω c,s (φ)(c n λ) + c s ω c n ,s (φ)(λ), n = 1, 2, . . .
Indeed, if φ : [0, ∞) → [0, ∞) is differentiable on (0, ∞), then ω c,s (φ)(λ) = c s 1 c Ω s (φ)(λx)
x s+1 dx, c ∈ (0, 1), (41)

Ω s (φ) = lim c→1 - ω c,s (φ) 1 -c = lim c→1 - 1 -e -ω c,s (φ) 1 -c . (42)
Indeed, to see (41), one needs to integrate by parts the second term in the middle of [START_REF] Zolotarev | One Dimensional Stable Laws[END_REF], while formula (42) follows from L'Hôpital's rule. These observations lead to the following result. THEOREM 2.6. Let s > 0 and φ be a cumulant function.

1)

If ω c,s (φ) ∈ BF for some c ∈ (0, 1) and s ≥ 1, then φ ∈ BF .

2) The following conditions are equivalent.

(i) φ ∈ BF s ; (ii) ω c n ,s (φ) ∈ CF for some sequence c n ∈ (0, 1), such that lim n→∞ c n = 1; (iii) ω c,s (φ) ∈ BF for all c ∈ (0, 1). 3) If r > s > 0 and X ∼ RSD s , then X ∼ RSD r .
PROOF. 1) Recall that the class BF is a convex cone, closed under taking point-wise limits. If ω c,s (φ) ∈ BF , then (40) implies ω c n ,s (φ) ∈ BF for all n = 1, 2, . . .. Thus, φ(0) = 0, s ≥ 1 and ( 16) yield

lim n→∞ ω c n ,s (φ) λ c n = lim n→∞ φ(λ) -c ns φ λ c n -(c n -c ns )d λ c n = φ(λ) -1l s=1 dλ ∈ BF.
Therefore, we conclude that φ ∈ BF .

2) We use again, the fact that BF is closed under taking point-wise limits. (i) =⇒ (ii): If φ ∈ BF s , then by definition Ω s (φ) ∈ BF . By (41), we obtain that ω c,s (φ) ∈ BF for every c ∈ (0, 1). Then, [START_REF] Kyprianou | Fluctuations of Lévy Processes with Applications[END_REF] and ω c,s (φ)(0) = 0 imply that ω c,s (φ) ∈ CF for every c ∈ (0, 1). (ii) =⇒ (i): The assumption and ( 9) imply that 1 -e -ω c n ,s (φ) ∈ BF for all n. Now the second equality in (42) shows that Ω s (φ) ∈ BF and together with Theorem 2.1, part 2), we conclude that φ ∈ BF s as well. (i) ⇐⇒ (iii): The necessity is done in (i) =⇒ (ii). The sufficiency is similar to (ii) =⇒ (i), using the first equality in (42).

3) The assertion is an immediate consequence of point 5) in Theorem 2.1, the definition of the class RSD s and the above equivalence 2)(i) ⇐⇒ 2)(ii) of this theorem.

3. Time scale decomposability and Jurek-Vervaat type stochastic integral representation for subordinators associated to the class BF s . We will focus on the stochastic interpretation of the classes BF s , s > 0 under several aspects. In Theorem 2.1, we have seen that for a cumulant function φ, Ω s (φ) ∈ BF yields φ ∈ BF . Thus, using representations ( 6) and (31), we retrieve for any function φ ∈ BF s and c ∈ (0, 1), the following representation of ω c,s (φ):

ω c,s (φ)(λ) = cλ ∞ 0 e -cλx µ(x, ∞) dx -c s λ ∞ 0 e -λx µ(x, ∞) dx = λ ∞ 0 e -λx ∞ x/c p s (u) u s+1 du -c s ∞ x p s (u) u s+1 du dx = c s λ ∞ 0 e -λx ∞ x p s (u/c) -p s (u) u s+1 du dx,
where in the last equality we performed the changes of variables u → u/c. With Theorem 2.6, we deduce the following stochastic interpretation.

COROLLARY 3.1. Let s > 0. Let φ ∈ BF , φ(0) = 0, be associated to the subordinator (X t ) t≥0 . Under the notation of Theorem 2.1, the following two conditions are equivalent.

1) φ ∈ BF s ;

2) For all c ∈ (0, 1), there exists a subordinator (Z (c,s) t

) t≥0 , such that we have the identity in law

(43) c X t d = X c s t + d (c -c s ) t + Z (c,s) t ,
where (X c s t ) t≥0 and (Z (c,s) t

) t≥0 are assumed to be independent, and the Bernstein function of Z (c,s) equals ω c,s (φ).

Under the two conditions above, let (Y (s) t ) t≥0 be the subordinator with Bernstein function φ Y (λ) := φ(λ) -dλ -a s λ s , let (S (s) t ) t≥0 , s ∈ (0, 1), be a s-standard stable subordinator (c.f. (75) below), independent of (Y (s) t ) t≥0 and recall that a s = 0 for s > 1. Then, we have the identities in law:

(44) X t d = d t + (a s ) 1/s S (s) t + Y (s) t and c Y (s) t d = Y (s) c s t + Z (c,s) t ,
where the random variables Y (s) c s t and Z (c,s) t are supposed to be independent.

PROOF. The assumptions φ ∈ BF and φ(0) = 0 imply that φ is a cumulant function, cf. [START_REF] Jedidi | Stable processes, mixing, and distributional properties[END_REF]. By Theorem 2.6, we equivalently have φ ∈ BF s or ω c,s (φ) ∈ BF for all c ∈ (0, 1). Identity (43) is justified by the decomposition

φ(cλ) = c s φ(λ) + d (c -c s ) λ + ω c,s (φ).
The first identity in (44) is a consequence of representation (33), while the second one is obtained by the decomposition

φ Y (c λ) = c s φ Y (λ) + ω c,s (φ),
completing the proof.

As already noticed before (21), identities (43) and (44) resemble the proper self-similarity property. The latter suggests conducting a deeper investigation into the stochastic interpretation of the class BF s , this is the main subject of the next section.

At this stage, we can propose the following Jurek-Vervaat type stochastic integral representation associated to the class the class BF s . Recall the notations of Theorem 2.1 and of Corollary 3.1. THEOREM 3.2. Let s > 0 and let φ be Bernstein function, with φ(0) = 0, associated to a positive and infinitely divisible random variable X. Then, φ ∈ BF s , if, and only if, the law of X is defined by the stochastic integral (45)

X d = d + a 1/s s S (s) 1 + 1 0 u -1/s dZ (s) u ,
where

S (s)
1 has the standard positive stable distribution (c.f. (76) below) if 0 < s < 1, independent of the subordinator Z (s) . The term a s , the Bernstein function φ Z (s) and the Lévy measure ν s of Z (s) are given by (46)

a s = p s (0+) Γ(1 -s) s 1l s<1 , φ Z (s) = Ω s (φ)
s , and ν s (dx) = dp s (x) s x s , where p s is a non-decreasing, right-continuous function, satisfying the integrability conditions (32). PROOF. By Theorem 2.1, φ ∈ BF s is equivalent to the representation (33):

(47) φ(λ) = dλ + a s λ s + 1 s 1 0 ϕ s λ u 1/s du, λ ≥ 0.
for some Bernstein function ϕ s which, by (30), necessarily equals to Ω s (φ). Observe that representation (47) of φ fits perfectly the injective one in [START_REF] Schilling | Bernstein Functions[END_REF]Lemma 10.1], when taking the θ-function there equal to θ(u) = u -1/s , u ∈ (0, 1), and then the Bernstein function of Z (s) is Ω s (φ)/s. By (36), the Lévy measure of Z (s) is necessarily given by (46). This shows that (47) implies (45) and (46).

For the opposite direction, note that conditions (10.13) and (10.14) in [START_REF] Schilling | Bernstein Functions[END_REF]Proposition 10.4] are satisfied since the Bernstein function Ω s (φ)/s has no drift nor killing terms. We now check that ν s satisfies the integrability condition (10.15) in [START_REF] Schilling | Bernstein Functions[END_REF]Proposition 10.4]. After taking ϑ(y) := -(θ -1 ) = s/y s+1 , when y > 1, and ϑ(y) := 0 otherwise, the integrability condition is translated as follows:

I s := ∞ 0 1/x 0 y ϑ(y) dy ν s (x, ∞)dx = 1 0 1/x 1 y ϑ(y) dy ν s (x, ∞)dx = 1 0 ∞ x 1/x 1 y ϑ(y) dy ν s (du) dx = ∞ 0 1∧u 0 1/x 1 y ϑ(y) dy dx ν s (du) = ∞ 0 η s (u) dp s (u) u s < ∞, (48)
where

η s (u) := 1∧u 0 1/x 1 1 y s dy dx = ∞ 1 (1/y)∧(1∧u) 0 1 y s dx dy = ∞ 1 (uy) ∧ 1 y s+1 dy. Now, observe that J s := ∞ 0 (x ∧ 1)x -(s+1) dx < ∞ if s < 1,
and recall that p s (0+) = 0 if s ≥ 1. Then, using Tonelli-Fubini's theorem and a change of variable, express the terms in (32) as

1 0 p s (x) x s dx + ∞ 1 p s (x) x s+1 dx = ∞ 0 x ∧ 1 x s+1 p s (0+) + (0,x] dp s (u) dx = p s (0+) J s 1l s<1 + (0,∞] ∞ u x ∧ 1 x s+1 dx dp s (u) = p s (0+) J s 1l s<1 + (0,∞] η s (u) dp s (u) u s .
Thus, conditions (32) and ( 48 PROPOSITION 4.1. Let s > 0. Let the random variable U be uniformly distributed on (0, 1) and let G be exponentially distributed with parameter 1. Let Z be a positive random variable with cumulant function ϕ and distribution function F . Then, the following conditions are equivalent.

1) 1 -e -ϕ ∈ BF s ;

2) Z has a probability density function of the form

(49) f (x) = p s (x)
x s+1 , x > 0, where p s (x) is non-decreasing and right-continuous;

3) For all c ∈ (0, 1),

(50) F c (x) := F (x/c) -c s F (x) 1 -c s , x ≥ 0,
is the distribution function of some positive (continuous) random variable Z c . 4) 1/Z is s-star unimodal in the sense of Definition 1.5, i.e., there exists a positive random variable V s such that

(51) Z d = V s U 1/s d = e G/s V s ,
where U and G are independent of V s ;

5) For any positive random variable Y independent of Z, the quotient Y /Z s has a nonincreasing probability density function; 6) For any positive random variable

Y independent of Z, λ → λ E[e -λY /Z s ] ∈ BF ; 7) λ → λ E e -λ/Z s ∈ BF ; 8) λ → λ E Z s /(λ + Z s ) is a Thorin Bernstein function in the sense of (27).
PROOF. We start with this observation: since 1 -e -ϕ(0) = 0 and lim λ→∞ (1 -e -ϕ(λ) )/λ = 0, then (1 -e -ϕ(λ) ) = 1 -e -ϕ(λ) . Thus, by part 2)(ii) of Theorem 2.1, we have

Ω s (1 -e -ϕ ) ∈ BF ⇐⇒ 1 -e -ϕ ∈ BF s =⇒ 1 -e -ϕ ∈ BF =⇒ 1 -e -ϕ ∈ CF. (52)
1) ⇐⇒ 2): This is an application of the equivalence between 3i) and 3ii) in Theorem 2.1, applied to 1 -e -ϕ with µ(dx) = dF (x), after noticing that

1 -e -ϕ(λ) = E[1 -e -λZ ] = ∞ 0 (1 -e -λx ) dF (x).
1) ⇐⇒ 3): Recall the operator ω c,s is defined in [START_REF] Schilling | Bernstein Functions[END_REF]. The implication is obtained by writing

ω c,s (1 -e -ϕ )(λ) 1 -c s = E[1 -e -cλZ ] -c s E[1 -e -λZ ] 1 -c s = λ 1 -c s ∞ 0 e -λx P(cZ > x) -c s P(Z > x) dx = λ ∞ 0 e -λx 1 -F c (x) dx (53) = ∞ 0 e -x 1 -F c (x/λ) dx. (54)
By definition, F c ≥ 0, since F is non-decreasing. Now, using (52), and Theorem 2.6, we get equivalences

1 -e -ϕ ∈ BF s ⇐⇒ ω c,s (1 -e -ϕ ) ∈ BF, ∀c ∈ (0, 1)
⇐⇒ F c ≤ 1 and is non-decreasing, ∀c ∈ (0, 1), while the second equivalence follows from representations ( 6) and (53). Since Z is a positive random variable, we have F c (0) = F (0) = 0. Then, we use the dominated convergence theorem together with (54) and the fact that ω c,s (1 -e -ϕ )(0) = 0, to obtain that lim x→∞ F c (x) = 1. All this shows that F c is the distribution function of a positive random variable for all c ∈ (0, 1).

2) =⇒ 4): Observe that if a positive random variable V is independent of U, then the probability density function of the product U V is expressed by

(55) f UV (x) = E f U (x/V ) V = (0,∞) 1 v 1l { x v ∈(0,1)} P(V ∈ dv) v = (x,∞) P(V ∈ dv) v .
By assumption, the probability density function of Z has the form (49), thus the one of Z -s is expressed by

(56) f Z -s (x) = f x -1 s s x 1+ 1 s = p s x -1 s s , x > 0.
Then, f Z -s is a non-increasing function of the form (55), i.e. Z d = U -1/s V s for some positive random variable V s independent of U. The last identity in (51) is trivial.

4) =⇒ 5): This is simply seen by Y /Z s d = U (Y /V s s ) and referring to formula (55). 5) =⇒ 6): If q s is the non-increasing probability density function of Y /Z s , then

λ E[e -λY /Z s ] = λ ∞ 0 e -λx q s (x)dx,
complies with the second representation of a Bernstein function in [START_REF] Chaumont | Exercises in Probability: A Guided Tour from Measure Theory to Random Processes, via Conditioning[END_REF]. 6) =⇒ 7): Take Y = 1. 7) =⇒ 2): Compare with the second representation in [START_REF] Chaumont | Exercises in Probability: A Guided Tour from Measure Theory to Random Processes, via Conditioning[END_REF] to conclude that the Bernstein function λ → λ E[e -λ/Z s ] needs to have killing rate and drift term zero. Then, one sees that Z -s has a non-increasing, right-continuous probability density function and we conclude by using the first equality in (56).

2) =⇒ 8): Since Z -s has a probability density function of the form (49) we have

E λZ s λ + Z s = ∞ 0 λx s λ + x s p s (x) x s+1 dx,
then making the change of variable x = y 1/s , we obtain that

E λZ s λ + Z s = 1 s ∞ 0 λ λ + y p s (y 1 s ) y dy,
meets the form (104) in the Appendix, of a Thorin Bernstein function. 8) =⇒ 2): By the uniqueness of representation (104), we see that the Thorin Bernstein function λ → E[λZ s /(λ + Z s )] has no killing nor drift terms (q = d = 0) and that the probability density function of Z s has the form f Z s (z) = σ (0, z] /z 2 , z > 0, for some positive measure σ. Since

f Z s (z) = 1 s z 1 s -1 f z 1 s
, then, with the change of variable x := z 1/s , we obtain

f (x) = s x s-1 f Z s (x s ) = s σ (0, x s ] x s+1 , x > 0.
This gives us (49) with p s (x) := s σ (0, x s ] . REMARK 4.2.

(i) Observe that the equivalent conditions in Proposition 4.1 do not hold if Z has degenerate distribution, that is if it has cumulant function of the form d λ, for d > 0. Indeed, for s > 0, we have

Ω s (1 -e -dλ )(λ) = s -e -dλ (s + d λ) / ∈ BF,
as can be verified by taking successive derivatives. Thus, 1 -e -d λ ∈ BF * s . (ii) The integrability of f in (49) on (0, 1) implies that

lim x→0 + x f (x) = lim x→0 + p s (x)/x s = 0.
This is an improvement on the similar observation that one can make from the first integral in (32).

EXAMPLE. If Z

d = e G/s , then its probability density function is f (x) = s/x s+1 , x > 1, and we have

1 -e -ϕ(λ) = E[1 -e -λZ ] = 1 -s λ s ∞ λ e -u u s+1 du.

Thus, we obtain

Ω s 1 -e -ϕ (λ) = s (1 -e -λ ), λ ≥ 0.
Let ω c,s , ω c,s , and Ω s stand for the following modifications of the operators ω c,s and Ω s :

ω c,s (h)(λ) := ω c,s (h)(λ) 1 -c s , for c ∈ (0, 1) Ω s (h)(x) := s h(x) + x h (x).
The introduction of the operator ω c,s is justified by the forms (53). The next result completes 1) The density of Z, satisfies

(57) f (x) = s E[V s s 1l (V s ≤x) ] x s+1
, for Lebesgue almost every x > 0;

2) The function f is almost everywhere differentiable, and we have the representation

(58) Ω s (1 -e -ϕ )(λ) = s ∞ 0 (1 -e -λx ) f s (x) dx, λ ≥ 0,
where

f s (x) := 1 s Ω s+1 (f )(x) = p s (x)
s x s is a probability density function;

3) The functions ω c,s (1 -e -ϕ ) and f c , for c ∈ (0, 1), are represented by and Z s is a positive random variable (independent of W c,s ) with probability density function f s .

ω c,s (1 -e -ϕ )(λ) = ∞ 0 (1 -e -λx )f c (x)dx, λ ≥ 0, (59)
f c (x) = c s 1 -c s 1 c Ω s+1 (f )(x/u) u s+2 du, x > 0. ( 60

PROOF. 1)

The relationship is obtained by the change of variable x := u -1/s V s and then exchanging the order of integration in:

E[1 -e -λZ ] = E[1 -e -λU -1/s V s ] = E 1 0 1 -e -λu -1/s V s du = ∞ 0 (1 -e -λx ) s E V s s 1l (V s ≤x) x s+1 dx.
2) By (49), since p s is non-decreasing, the function f is almost everywhere differentiable on (0, ∞). The claim that f s (x) is a probability density function on [0, ∞) can be checked directly by integration by parts, utilizing (ii) in Remark 4.2. Using the fact that Ω s (1e -ϕ )(0) = 0 and an integration by parts, representation (58) follows from

Ω s (1 -e -ϕ )(λ) = Ω s u → E[1 -e -uZ ] (λ) = Ω s u → ∞ 0 (1 -e -ux ) f (x) dx (λ) = ∞ 0 Ω s (u → 1 -e -ux ) f (x) dx = ∞ 0 s (1 -e -λx ) -λxe -λx f (x) dx = ∞ 0 s (1 -e -λx ) f (x) -e -λx xf (x) + f (x) dx = s - ∞ 0 e -λx (s + 1) f (x) + x f (x) dx = s - ∞ 0 e -λx Ω s+1 (f )(x) dx = s ∞ 0 (1 -e -λx ) f s (x) dx.
Finally, use (49) to observe that

Ω s+1 (f )(x) = (s + 1)f (x) + xf (x) (63) = (s + 1) p s (x) x s+1 + x p s (x) x s+1 -(s + 1) p s (x) x s+2 = p s (x) x s .
3) Using ( 6) and (53), we obtain

ω c,s (1 -e -ϕ )(λ) = λ ∞ 0 e -λx 1 -F c (x) dx = ∞ 0 (1 -e -λx )f c (x)dx.
In order to derive (60), we differentiate (50) and use (49), to find

f c (x) = c s (1 -c s )x s+1 [p s (x/c) -p s (x)] = c s (1 -c s )x s+1 x/c x p s (y) dy = c s (1 -c s )x s 1 c p s (x/u) u 2 du,
where we made the change of variable y = x/u. Finally, expressing p s (x) from ( 63) and substituting it in the latter integral, we conclude

f c (x) = c s (1 -c s )x s 1 c (x/u) s Ω s+1 (f )(x/u) u 2 du = c s (1 -c s ) 1 c Ω s+1 (f )(x/u) u s+2 du.
4) Noting that F (x/c) = P(cZ ≤ x) and differentiating (50), we obtain

f cZ (x) = c s f (x) + (1 -c s ) f c (x), x > 0,
which immediately proves identity (61). 5) Expressing (60) in the form

f c (x) = s c s 1 -c s 1 c f s (x/u) u du u s+1 ,
and using the formula for the density of the product of two independent random variables, gives that the probability density function of W c,s is s c s (1 -c s ) -1 u -s-1 , c < u < 1. It is identified in (62) by the expression

P(c e G/s ≤ u | c e G/s < 1) = P(G ≤ s log(c/u)) P(G ≤ s log c) = 1 -(c/u) s 1 -c s , c < u < 1,
after differentiating with respect to u.

REMARK 4.4. Note that (59) can be re-written as

ω c,s (1 -e -ϕ )(λ) = 1 -e -ϕ c (λ) ,
where ϕ c is the cumulant function of the non-negative random variable Z c with probability density function f c .

The class BF *

s and generalized unimodality. Let s > 0. Let φ be a Bernstein function with killing term q = φ(0) and drift term in d. The Bernstein function φ -q corresponds to a non-killed subordinator (X t ) t≥0 , whereas φ (λ) = φ(λ) -dλ -q is the Bernstein function of subordinator (X t -d t) t≥0 . Observing that the function

ε t (φ)(λ) = 1 -e -tφ(λ) = 1 -e -tq + e -tq (0,∞) (1 -e -λx )P(X t ∈ dx)
is also a Bernstein function for every t > 0, it is natural to introduce the class (64)

BF * s := {φ ∈ BF, s.t. Ω s ε t (φ ) ∈ BF, ∀t > 0},
as was done in Definition 1.4. Since lim x→∞ (1 -e -tφ (x) )/x = 0, we have (ε t (φ )) = ε t (φ ) ≥ 0, and Theorem 2.1 guarantees that Definition (64) is equivalent to

BF * s = {φ ∈ BF, s.t. ε t (φ ) ∈ BF s , ∀t > 0}.
Using the fact that the class BF is closed under pointwise limits and the fact that

lim t→0 + Ω s (ε t (φ )) t = Ω s (φ ),
we see that BF * s ⊂ BF s . The inclusion could be strict. Indeed, if φ(λ) = λ 0.9 , then Ω 1 (φ)(λ) = 0.1 λ 0.9 ∈ BF, Ω 1 (1 -e -φ )(λ) = 1 -e -λ 0.9 (1 + 0.9 λ 0.9 ) / ∈ BF.

Therefore, we have

φ ∈ BF 1 \ BF * 1 .
The fact that 1 -e -λ 0.9 (1 + 0.9 λ 0.9 ) / ∈ BF is checked by the sign change of its second derivative. By definition of the classes BF * s , and by the relation

Ω r (ε t (φ )) = Ω s (ε t (φ )) + (r -s) ε t (φ ),
we have that BF * s ⊂ BF * r ⇐⇒ 0 < s < r. We now present a simple characterization of the functions in BF * s . Due to the representation ( 65)

Ω s ε t (φ ) (λ) = s ε t (φ )(λ) -λε t (φ ) (λ) = s -e -tφ (λ) s + tλφ (λ) ,
we see that if φ ∈ BF * s , then both functions Ω s ε t (φ ) and s -Ω s ε t (φ ) are nonnegative, and by [START_REF] James | Generalized Gamma Convolutions, Dirichlet means, Thorin measures, with explicit examples[END_REF], we obtain that

(66) λ → ϕ s,t (λ) := s -Ω s ε t (φ ) (λ) = e -tφ (λ) s + tλφ (λ) ∈ CM, ∀t > 0.
Conversely, assume (66) holds and φ ∈ BF , the latter implies that φ ∈ BF . Since lim λ→0+ λφ (λ) = 0, see for example (2.11) in [START_REF] Sendov | New representation theorems for completely monotone and bernstein functions with convexity properties on their measures[END_REF], the non-increasing function ϕ s,t is bounded by s, thus Ω s ε t (φ )) = s -ϕ s,t ∈ BF , we have the equivalences (67) φ ∈ BF * s ⇐⇒ φ ∈ BF and λ → s -Ω s ε t (φ (λ)) = e -tφ s + tλφ (λ) ∈ CM, ∀t > 0.

REMARK 5.1. We now show that BF * s is not void. Consider the Bernstein function ϕ(λ) := √ λ -log(1 + √ λ) ∈ BF and let φ be such that

φ(0) = lim x→∞ φ(x) x = 0, φ ∈ BF s , and λ → (λφ (λ)) 2 ∈ BF.
For instance, one can take φ(λ) = λ α , α ∈ (0, 1/2], s ≥ α. We show that φ ∈ BF * s . Indeed, observing that

λ → log s + tφ(λ) -log s + tλφ (λ) = t s Ω s (φ) + ϕ (t λ φ (λ)/s) 2 ∈ BF, ∀t > 0, we conclude that λ → e -(tφ(λ)-log(s+tλφ (λ))) = e -tφ(λ) s + tλφ (λ) ∈ CM, ∀t > 0.
As a consequence of Proposition 4.1, we connect the class BF * s to generalized unimodality, a relation that reads on the level of the subordinators. COROLLARY 5.2. Let s > 0 and let φ be a Bernstein function, such that φ is not identically equal to 0. Let (X ( ) t ) t≥0 be the subordinator corresponding to φ . Then, the following conditions are equivalent.

1) φ ∈ BF *

s ; 2) ψ(φ ) ∈ BF s , for all ψ ∈ BF ; 3) for every t > 0, the random variable Z := X ( ) t satisfies any of the equivalent conditions of Proposition 4.1.

PROOF. Take ϕ = tφ in Proposition 4.1, for t > 0, and deduce the equivalence between parts 1) and 3). To see the equivalence of parts 1) and 2), we use the representation

ψ(φ (λ)) = q ψ + d ψ φ (λ) + (0,∞) ε t (φ )(λ) µ ψ (dt).
The facts that BF s is a closed convex cone and that the integral is a limit of a sequence of Bernstein functions, ensure that we can use the linearity of the operator Ω s to exchange it with the integral, see [START_REF] Schilling | Bernstein Functions[END_REF]Corollary 3.9]. REMARK 5.3. Let φ be Bernstein function associated to the subordinator (X t ) t≥0 .

(i) With the observation

lim t→0 + Ω s (ε t (φ)) t = Ω s (φ) + (s -1) dλ,
we deduce that, if s ≤ 1 and Ω s ε t (φ) ∈ BF , for all t > 0, then φ ∈ BF s .

(ii) One can notice that for s > 0: Ω s ψ(φ) ∈ BF, for all ψ ∈ BF =⇒ Ω s ε t (φ) ∈ BF, for all t > 0;

for s ∈ (0, 1]: Ω s ψ(φ) ∈ BF, for all ψ ∈ BF ⇐= Ω s ε t (φ) ∈ BF, for all t > 0.

Indeed, for the first implication take ψ(λ) = 1 -e -tλ , while the second uses the same argument as in the proof of Corollary 5.2 and part (i) of this remark. (iii) The harmonic and potential harmonic measures are given, in the vague sense, by

U (dx) = ∞ 0 P(X t ∈ dx) dt and H(dx) = ∞ 0 P(X t ∈ dx) dt t ,
respectively. They are linked to φ via

(68) 1 φ(λ) = [0,∞) e -λ x U (dx) and φ (λ) φ(λ) = [0,∞) e -λ x x H(dx), λ > 0.
Thus, due to the shape of the distribution of P(X ( ) t ∈ dx) provided by Corollary 5.2 in conjunction with (49), it is immediate that if φ ∈ BF * s , then the harmonic and potential harmonic measure, associated to φ , have the form (31), i.e.

U (dx) = u(x) x s+1 dx and H (dx) = h(x)
x s+1 dx, where both u and h are non-decreasing functions. Writing,

ψ (λ) := s φ (λ) + λ φ (λ) φ 2 (λ) = s φ (λ) -λ 1 φ (λ) = s ∞ 0 e -λx u(x) x s+1 dx -λ ∞ 0 e -λx u(x) x s dx = λ ∞ 0 e -λx v(x) dx = (0,∞) e -λx dv(x), λ > 0,
where v is the non-decreasing function

v(x) := s x 0 u(y) y s+1 dy - u(x) x s ,
we retrieve that ψ is a completely monotone function. Further, by an evident change of variable, we may use the second representation (68) for φ and affirm that

λ → φ (λ) λ s-1 φ (λ) = (0,∞) e -x h(x/λ)
x s dx is non-increasing.

6.

A solution to a conjecture by Sendov, Shan and its relationship to a result by Simon. In this section, we give an answer to a conjecture stated by Sendov and Shan, [START_REF] Sendov | New representation theorems for completely monotone and bernstein functions with convexity properties on their measures[END_REF]Open Problem 4.1], that has a strong connection with stable laws. The authors proved that if

λ α 0 ∈ BF * 1 , then λ α ∈ BF * 1 , for all α ≤ α 0 . Thus, it is natural to ask (69)
what is the largest value of α 0 ∈ (0, 1), such that λ α ∈ BF * 1 , for all α ≤ α 0 ? It is shown in [START_REF] Sendov | New representation theorems for completely monotone and bernstein functions with convexity properties on their measures[END_REF] that (69) holds for α 0 ≤ 2/3 and was conjectured that α 0 = 1/ √ 2 = 0.70710678118. In this section we will find the exact optimal value, which turns out to be larger. We do this by resolving the more general problem of (70) finding all pairs (s, α) ∈ (0, ∞) × (0, 1), such that λ α ∈ BF * s ?

According to (65), this is equivalent to finding all pairs (s, α) ∈ (0, ∞) × (0, 1), such that s -e -tλ α (s + tαλ α ) ∈ BF, for all t > 0.

Since a function φ(λ) is a Bernstein function, if and only if φ(tλ) ∈ BF for all t > 0, the problem simplifies to the equivalent one of finding all pairs (s, α) ∈ (0, ∞) × (0, 1), such that s -e -λ α (s + αλ α ) ∈ BF. (71) By (67), the problem is also equivalent to finding all pairs (s, α) ∈ (0, ∞) × (0, 1), such that e -λ α (s + αλ α ) ∈ CM.

This problem has been previously raised by Simon [START_REF] Simon | On the unimodality of power transformations of positive stable densities[END_REF]. He showed that there exists an increasing function R :

[0, 1] → [0, ∞], such that (73) e -λ α (s + αλ α ) ∈ CM ⇐⇒ α ≤ 1 2 or s ≥ R(α),
where the function R satisfies

R(α) = α if α ∈ [0, 1/2] and 1 4(1 -α) ≤ R(α) ≤ α sin 2 (πα) if α ∈ [1/2, 1]. (74)
In particular, taking s = 1, it is clear that (73), hence (69), holds true whenever α ∈ [0, 1/2] or whenever α ∈ [1/2, 1] and 1 ≥ α/ sin 2 (πα). Due to the inclusion BF * s ⊂ BF s and due to (28), we see that a pair (s, α) does not satisfy (70) if s < α.

The importance of problems ( 69) and ( 70) is highlighted by the fact that φ(λ) = λ α , for 0 < α < 1, is the Thorin Bernstein function associated to the stable subordinator (S (α) t ) t≥0 through the following representations:

λ α = ∞ 0 (1 -e -λx ) c α x α+1 dx, c α := α Γ(1 -α) = Γ(α + 1) sin(πα) π , (75)
e -tλ α = E e -λS (α) t ) = ∞ 0 e -λs f t,α (s) ds, t > 0, (76)
where f t,α is the probability density function of the positive stable random variable S (α) t . The so-called scaling property for stable processes could be noticed from (76) and gives S (α) t d = t 1/α S α , where we denote from now on, S α := S (α) 1 .

Notice that the probability density function f α := f 1,α of S α is explicit only for the value α = 1/2 and

f 1/2 (x) = e -1/4x 2 √ πx 3/2 , x > 0,
corresponds to the inverse-gaussian distribution, whereas in general, it is only evaluated by the series expansion given by [19, formula (2.4.8), p.90]:

(77)

f α (x) = 1 π ∞ n=1 (-1) n-1 Γ(nα + 1) Γ(n + 1) sin(πn α) x -(nα+1) .
The Mellin transform of S α is given by ( 78)

E[S -λ α ] = Γ(1 + λ α ) Γ(1 + λ) = exp β α ψ(1)λ + ∞ 0 (e -λx -1 + λx) e -αx (1 -e -βx ) x(1 -e -x )(1 -e -αx ) dx ,
where the Digamma function is defined by ψ(u) = Γ (u)/Γ(u), u > 0. The stable random variable S α is also linked to the Mittag-Leffler function via

(79) E α (x) := k≥0 x k Γ(kα + 1) = E e x/(S α ) α , x ∈ C.
An important and known fact is that when we take two independent and identically distributed random variables S α , S α , then T α := S α /S α has the explicit probability density function, see [6, (4.23.3)]:

f T α (x) = sin(πα) π x α-1 x 2α + 2 cos(πα)x α + 1 (80) = 1 Γ(α)Γ(1 -α) x α-1 x α + cos(πα) 2 + sin 2 (πα) , x > 0.
Hence, using (76), one has the representation

E α (-λ α ) = E e -λ α /(S α ) α = E E e -λS α /S α |S α = E e -λS α /S α (81) = ∞ 0 e -λx f T α (x)dx, λ ≥ 0.
In Section 7 below we will show that there is much more consistent links between stable distributions and the Mittag-Leffler functions. If one takes G exponentially distributed, with scale parameter 1, and independent from S α , then we have the well-known, see [6, (4

.21.2)], identity (82)
G d = G S α α .
In order to state the main result of this section, we need the following technical lemma. Together with e α,s , they satisfy α ≤ A(s) ⇐⇒ s ≥ R(α) ⇐⇒ e α,s ≥ 0;

2) We have A(s) ≤ min(s, 1) and A(s) = s if s ≤ 1/2.

PROOF. 1)

The function A is nondecreasing since s → a s is such. Since a 0 (x) = 1 -e -x is increasing and a 0 (x)/x is decreasing, then the product function

x → log(1 + a 0 (x)) x = log(1 + a 0 (x)) a 0 (x)
a 0 (x) x decreases to its limit at infinity which is 0 = A(0). Clearly, A(∞) = 1 because a ∞ (x) = e x -1 and then log(1 + a ∞ (x))/x = 1. We deduce that A(s) ∈ [0, 1], for all s ≥ 0.

The continuity of A is justified as follows: suppose that the function A is not continuous at s, where s is a fixed finite nonnegative number. We have that either A(s-) < A(s) or A(s+) > A(s). a) Assume A(s-) < A(s) and choose any α ∈ A(s-), A(s) . Since α < A(s), we have e αx < 1 + a s (x) for all x > 0. Substituting the expression for a s (x) and solving the inequality for e -sx we arrive at the equivalent e α,s (x) > 0 for all x > 0. In fact, these arguments show that α ≤ A(s) if, and only if e α,s (x) ≥ 0 for all x > 0. Next, since α > A(s -1/n), for every n = 1, 2, . . ., one may build a sequence x n > 0, such that e α,s-1/n (x n ) < 0. Putting the two together, we have e α,s (x n ) ≥ 0 and e α,s-1/n (x n ) < 0.

By continuity of the function s → e α,s (x) for all x > 0, we deduce that there exists a sequence s n ∈ (s -1/n, s] such that e α,s n (x n ) = 0. It is clear that lim n→∞ s n = s. We are going to show now that as n → ∞, x n converges to ∞. If that is not the case, then, taking a subsequence, if necessary, we may assume that x n converges to a finite limit x * which satisfies

e α,s (x * ) = 1 e αx * -1 - 1 e x * -1 - 1 e sx * = 0 =⇒ e sx * = (e x * -1)(e αx * -1) e x * -e αx * .
Substituting e sx * into a s (x * ), we obtain

log(1 + a s (x * )) = log(e αx * ) = αx * =⇒ A(s) ≤ α.
This leads to the contradiction α < A(s) ≤ α, confirming that lim n→∞ x n = ∞. Finally, let y n := e -x n and from e α,s n (x n ) = 0, we obtain

1 y -α n -1 - 1 y n -1 -1 -y s n n = 0 =⇒ s n = α + 1 log(y n ) log 1 -y 1-α n (1 -y n )(1 -y α n )
.

Using lim n→∞ y n = 0 and α ≤ 1 (recall that α < A(s)), one can see that the right-hand side approaches α, while the left-hand side approaches s, hence s = α. But s is fixed, while we can choose α freely in the interval (A(s-), A(s)), a contradiction. b) Assume A(s) < A(s+) and choose any α ∈ (A(s), A(s+)). Then for any > 0, α < A(s + ) implies that e α,s+ (x) ≥ 0 for all x > 0. In other words, for any fixed x > 0, we have e α,s+ (x) ≥ 0 for all > 0. Taking the limit as goes to 0 gives e α,s (x) ≥ 0 for all x > 0. In other words, α ≤ A(s), a contradiction.

By the representation (85) of e α,s , it is also readily seen that e α,s (x) ≥ 0, ∀x > 0 ⇐⇒ s ≥ R(α) and e α,s (x) > 0, ∀x > 0 ⇐⇒ s > R(α).

Thus, we have

α ≤ A(s) ⇐⇒ s ≥ R(α) and α < A(s) ⇐⇒ s > R(α).
Hence, R is the inverse of A. The continuity of R as well as the strict monotonicity of A follow.

2) If α ≤ 1/2, then r α ≤ 0, implying that R(α) = α, which yields A(s) = s. If α > 1/2, then r α has sign changes, implying that R(α) ≥ α, which yields A(s) ≤ s.

We now straightforwardly retrieve the following preliminary observation. PROPOSITION 6.2. The function λ α belongs to BF * α , if and only if, α ≤ 1/2. In this case, λ α ∈ BF * s for all s ≥ α.

PROOF. By the equivalence between (70) and (71), we know that λ α ∈ BF * α , if and only if, 1 -e -λ α (1 + λ α ) ∈ BF . Since e -λ α (1 + λ α ) ≤ 1, the latter is equivalent to e -λ α (1 + λ α ) ∈ CM. By (73), the latter holds, if and only if, α ≤ 1/2 or α ≥ R(α). Using (74), we see that if α > 1/2 then α < 1/(4(1 -α)) ≤ R(α). Thus, e -λ α (1 + λ α ) ∈ CM, if and only if, α ≤ 1/2. The last claim is obtained by part 5) of Theorem 2.1.

We can now improve Proposition 6.2 by solving Sendov and Shans's problem [START_REF] Sendov | New representation theorems for completely monotone and bernstein functions with convexity properties on their measures[END_REF] stated in (69) and expliciting the homeomorphism R in Simon's result [START_REF] Simon | On the unimodality of power transformations of positive stable densities[END_REF] described in (73). THEOREM 6.3. Let U denote a random variable with uniform distribution on (0, 1) and let A, R be given by (83) and (84) respectively. Then, following statements are equivalent.

1) There exists a real-valued random variable W s , independent of U, such that the following factorisation in law holds:

(86) 1
S α d = U 1/s e W s ;
2) λ α ∈ BF * s ; 3) α ≤ A(s) or equivalently s ≥ R(α); 4) The probability density function of the random variable 1/S s α ,

f 1/S s α (x) := 1 sπ ∞ n=1 (-1) n-1 Γ(nα + 1) Γ(n + 1) sin(πn α) x αn s -1 , x > 0,
is non-increasing.

In particular, the random variable W s is infinitely divisible. Finally, λ α ∈ BF * 1 if, and only if, α ≤ A(1) ≈ 0.717461058844. PROOF. 1) ⇐⇒ 2): In (71), we have already noticed that (87)

λ α ∈ BF * s ⇐⇒ Ω s (x → 1 -e -x α ) ∈ BF,
and by (76), λ α is the cumulant function of the positive random variable S α . Thus, by the equivalence between 1) and 4) in Proposition 4.1, there is a positive random variable V s , independent of U, such that S α d = V s /U 1/s . Thus, 1/S α d = U 1/s /V s and the result follows by letting W s := -log(V s ).

1) ⇐⇒ 3): Using the Mellin transform (78) and the Frullani representation of the logarithm,

log 1 + λ s = ∞ 0 (1 -e -λx ) e -sx x dx, λ ≥ 0, s > 0, we get E[U λ/s ] = 1 + λ s -1 and then, E[S -λ α ] E[U λ/s ] = exp ∞ 0 (1 -e -λx ) e -sx x dx + β α ψ(1)λ + ∞ 0 (e -λx -1 + λx) e -αx (1 -e -βx ) x(1 -e -x )(1 -e -αx ) dx = exp γ α,s λ + ∞ 0 (e -λx -1 + λx) e α,s (x) x dx , λ ≥ 0, (88)
where γ α,s = ( β α ψ(1) + 1 s ) and e α,s is given by (85). Observe that the expression in the last exponent is of the form (99), given in Appendix 8. Suppose that e α,s takes strictly negative values. Decompose e α,s , in a standard way, into the difference of two non-negative functions e α,s = e + -e -. Then, the measures µ + (dx) := e + (x) dx/x and µ -(dx) := e -(x) dx/x are Lévy measures, that is, they satisfy the integrability condition [START_REF] Gorenflo | Mittag-Leffler Functions (Related Topics and Applications[END_REF], because they are both dominated by the Lévy measure

e -αx (1 -e -βx ) (1 -e -x )(1 -e -αx )
+ e -sx dx x .

Moreover, for the same reason the measures µ + (dx) and µ -(dx) satisfy the integrability conditions (100). Thus, by the discussion at the end of Appendix 8 and (102), there exist two independent random variables X + and X -, with infinitely divisible distributions of the spectrally negative type, associated to the Lévy measures are µ + and µ -, and such that we have

E[S -λ α ] E[U λ/s ] = E[e λ(X + -X -) ], λ ≥ 0.
Choosing X + and X -to be independent of S α and U, the last identity becomes

E e X - S α λ = E[(U 1/s e X + ) λ ], λ ≥ 0,
giving an identity in law of the form

e X - S α d = U 1/s e X + .
Last identity can be simplified to the form (86), if, and only if, e α,s is non-negative. By Lemma 6.1, the non-negativity condition is equivalent to 3). Finally, thanks to representations (99) and (101), we see that the exponent in the right-hand side of (88) corresponds to the (bilateral) Laplace transform of a random variable W s , whose distribution is necessarily infinitely divisible of a spectrally negative Lévy type. The latter is equivalent to the factorization in law (86).

2) ⇐⇒ 4): Since λ α is the cumulant function of S α , then (87) and the equivalence between parts 1 and 2 in Proposition 4.1, show that λ α ∈ BF * s if, and only if, the probability density function, f α (x), of S α has the form f α (x) = p s (x)/x s+1 for some non-decreasing and rightcontinuous p s (x). That is, λ α ∈ BF * s is equivalent to

x s+1 f α (x) = 1 π ∞ n=1 (-1) n-1 Γ(nα + 1) Γ(n + 1) sin(πn α) x s-nα is non-decreasing. It is simple to verify that f 1/S s α (x) = s -1 x -1-1/s f α (x -1/s ) = s -1 p s (x -1/s ), showing that λ α ∈ BF *
s is equivalent to f 1/S s α (x) being a non-increasing function. Infinite divisibility of W s is shown at the end of the proof of 1) ⇐⇒ 3). The last assertion is evident and the evaluation of A(1), using formula (83), was done by Maple. REMARK 6.4. By (77), for every α < 1, we have

f α (x) ∼ c α x α+1 , as x → ∞,
where c α is defined by (75). Since lim x→0+ x α+1 f α (x) = 0 (see [START_REF] Jedidi | Stable processes, mixing, and distributional properties[END_REF], for instance), the function x α+1 f α (x) has a continuous extension to [0, ∞). For α ≤ 1/2, we know that λ α ∈ BF * α . The last inclusion is equivalent to x α+1 f α (x)/c α being non-decreasing. Since it is positive and converging to one as x → ∞, we discover the remarkable fact that for α ≤ 1/2, x α+1 f α (x)/c α is a cumulative distribution function. By (87), (57), and (76), we deduce that

x α+1 c α f α (x) = Γ(1 -α) E V α α 1l (V α ≤x) , x ≥ 0, (89)
where, comparing (51) and (86) we see that V α = e -W α for an infinitely divisible random variable W s . In particular, letting x approach infinity in (89), we see that

E [V α α ] = 1/Γ(1 - α).
7. The Mittag-Leffler function and the class BF * s . Proposition 6.2 and Theorem 6.3 illustrate the extent to which the case α > 1/2 is more intricate than the case α ≤ 1/2. The explicitness of the probability density function f T α , given in (80), will be helpful for problem (69). Recall (87), that λ α ∈ BF * s precisely when 1 -e -λ α ∈ BF s . Since, λ α is the cumulant function of the positive random variable S α , we can use the equivalence between parts 1) and 5) in Proposition 4.1. More precisely, applying part 5) in Proposition 4.1 with Y = S α , that is independent from, but identically distributed with S α , we conclude that the probability density function f T s α is non-increasing. Thus, after an elementary calculation using (80), we see that

f T s α (x s/α ) = sin(πα) s π x (α-s)/α x 2 + 2 cos(πα)x + 1 is non-increasing in x ∈ (0, ∞).
The derivative of -f T s α (x s/α ) has the same sign as (s + α)x 2 + 2s cos(πα)x + s -α.

We deduce that λ α ∈ BF * s =⇒ (s + α)x 2 + 2s cos(πα)x + s -α ≥ 0, ∀x > 0 ⇐⇒ α ≤ s sin(πα), and in particular λ α ∈ BF * 1 =⇒ α < sin(πα) ⇐⇒ α ≤ α 1 := 0.736484448242. The inequality A(1) < α 1 comforts the last assertion of Theorem 6.3. As a consequence of Theorem 6.3, we can state the following result which improves Lemma 2.3 in [START_REF] Simon | On the unimodality of power transformations of positive stable densities[END_REF]. COROLLARY 7.1. For r > 0 and α ∈ (0, 1), let

(90) η α,r (x) := 1 -r Γ(1 -α) x E α (-x) and φ α,r (λ) := λ α -log(1 + rλ α ), x, λ ≥ 0.
Recall that E α stands for the Mittag-Leffler function given in (79).

1) The following assertions are equivalent.

(

i) α ≤ 1/2; (ii) λ α ∈ BF * α ; (iii) x E α (-x) ∈ BF ; (iv) η α,1 ≥ 0; (v) η α,r ∈ CM, for all r ≤ 1; (vi) φ α,r ∈ T BF, for all r ≤ 1. 2) More generally, below we have (i) ⇒ (ii) ⇔ (iii) ⇒ (iv). (i) r ≤ sin 2 (πα); (ii) η α,r ≥ 0; (iii) φ α,r ∈ BF ; (iv) λ α ∈ BF * α/r .
REMARK 7.2. Let α 2 ≈ 0.688483504697 be the zero of the function x → sin 2 (πx) -x in (1/2, 1). For α ∈ (1/2, α 2 ), we have sin 2 (πα) ≥ α. So, taking r = sin 2 (πα) and applying of point 2) of last corollary, we obtain λ α ∈ BF * α/r ⊂ BF * 1 . Of course, this is not optimal compared to what we obtained in Theorem 6.

1 λ 1 - 1 1 + r λ α = 1 Γ(β) ∞ 0 e -λx x α dx - 1 λ 1 -E e -λ α r G .
Then, taking S α to be an independent copy of S α , both independent from the exponentially distributed r.v. G, and using (76) followed by the identity in law for positive stable distributions (82), we obtain

1 -E e -λ α r G = 1 -E e -λS α (r G) 1/α = 1 -E e -λG r 1/α (S α /S α )

= λ ∞ 0 e -λx P G > x S α r 1/α S α dx.

Replacing in c α,r , we arrive at the simplified expression Finally, integrating (92) between 0 and λ, and using (93) and (94), we obtain the following representation for φ α,r , valid for all r > 0, λ ≥ 0; 2) The implications (ii) ⇔ (iii) ⇒ (iv) are given in (96). For the implication (i) ⇒ (ii), if α ≤ 1/2, then by Proposition 6.2, we have λ α ∈ BF * s and we are done by part 1). Thus, it is enough to check the claim for 1/2 < α < 1, where it holds that α ≤ α 2 ⇐⇒ α ≤ sin 2 (πα). Using representation (80) and (81), then performing the change of variable v = xu/r 1/α , we arrive at η α,r (x α /r) = 1 -Γ(β)

x α E α (-x α /r) = 1 -Γ(β) x α ∞ 0 e -xu/r 1/α f T α (u)du = 1 -r ∞ 0 g α (r 1/α v/x) e -v v α-1 Γ(α) dv = 1 -rE[g α (r 1/α G α /x)],
where g α (w) := 1 w α + cos(πα) 2 + sin 2 (πα) ,

and G α has the Gamma-distribution with shape parameter α and rate parameter 1. Observing that g α ≤ 1 sin 2 (πα) =⇒ η α,r (x α /r) ≥ 1 -r sin 2 (πα)

, for all x > 0, completes the proof. x 2 ∧ 1 ν(dx) < ∞, or (0,∞)

x 2 ∧ x ν(dx) < ∞, if the integral in (99) is finite when we take h(x) = x. The Lévy-Laplace exponents have the following stochastic interpretation: there is a bijection between the class of (non-killed) of the spectrally negative Lévy processes, i.e. processes Z = (Z t ) t≥0 , Z 0 = 0, with stationary and independent increments and non-positive jumps satisfying E[Z 1 ] ≥ 0, and the class of Lévy-Laplace exponents Ψ, with a = Ψ(0) = 0, via the Lévy-Khintchine formula:

(101) E[e λZ t ] = e tΨ(λ) , for t ≥ 0, λ ≥ 0, where Ψ is represented by (99). In fact, the distributions of Z t , t > 0, are entirely determined by the infinitely divisible random variable Z 1 . In (99), it is customary to label, as in ( 6), the quantity b the drift term and c the Brownian coefficient and killing the process Z, amounts to adding the killing rate a in Ψ, cf. the beginning of [START_REF] Kyprianou | Fluctuations of Lévy Processes with Applications[END_REF]Section 8.1]. Observe that splitting µ into the sum of two Lévy measures µ = µ + µ amounts to split Ψ into the sum of two Lévy-Laplace exponent Ψ = Ψ + Ψ . Equivalently, the random variable X := Z 1 splits into the sum of two independent random variables with infinitely divisible distributions:

(102) X = X + X ,

where the Lévy measures associated to X and X are µ , µ , respectively. for some measure σ on (0, ∞), such that (0,∞) σ(0, u] u(1 + u) du < ∞.

See [START_REF] Schilling | Bernstein Functions[END_REF]Theorem 8.2].

  and by an induction, that (40) ω c,s (φ) ∈ BF =⇒ ω c n ,s (φ) ∈ BF, for all n = 1, 2, . . .

4 .

 4 ) are equivalent and condition(10.15) in[START_REF] Schilling | Bernstein Functions[END_REF] Proposition 10.4] holds. Proposition and[START_REF] Schilling | Bernstein Functions[END_REF] Lemma 10.1] imply that λ → s -1 1 0 Ω s (φ)(λu -1/s )du is a Bernstein function that is the cumulant function of 1 0 u -1/s dZ (s)u . Representation (47) follows. The classes BF s and generalized unimodality. Here, we provide another justification for the introduction of the class BF s and also a stochastic interpretation for Hansen's result [8, Lemma 2.3, case n = 1].

PROPOSITION 4 . 3 .

 43 Proposition 4.1 by giving more information on the probability density function f c (respectively f ) of the continuous random variable Z c (respectively Z) in Proposition 4.1. Under the notation and the conditions of Proposition 4.1, we have the following assertions.

) 4 ) 5 )=

 45 We have the identities in law (61) c Z d = B c,s Z + (1 -B c,s )Z c , for all c ∈ (0, 1), where B c,s has the Bernoulli distribution with parameter c s and the random variables B c,s , Z, Z c are assumed to be independent. Furthermore, Z c d -→ Z, as c → 1 -. We have the identities in law (62) Z c d = Z s W c,s , where W c,s d Law c e G/s | c e G/s < 1 , for all c ∈ (0, 1),

LEMMA 6 . 1 . 1 )

 611 For α ∈ (0, 1), β = 1 -α, s ≥ 0 and x > 0, let a s (x) := e x -1 1 + e -sx (e x -1) and A(s) := inf x>0 log (1 + a s (x)) x ,(83)r α (x) := log (1 -e -αx )(1 -e -x ) (1 -e -βx ) and R(α) := α + max x>0 r α (x) x , (84) e α,s (x) := e -αx (1 -e -βx ) (1 -e -x )(1 -e -αx )-e -sx . Function A is an increasing homeomorphism from [0, ∞] onto [0, 1], with inverse R.

3 .

 3 PROOF. Recall β = 1 -α. With the help of the asymptotic[7, (3.4.15)], observe first that (91)η α,r = 1 -r + r η α,1 and lim x→∞ η α,1 (x) = lim x→∞ 1 -Γ(β) x E α (-x) = 0.Thus, the inequality η α,r ≥ 0 fails for r > 1. Second, since(92) φ α,r (λ) = d dλ λ α -log(1 + r λ α ) = α c α,r(λ), where c α,r (λ) := 1 λ β -r λ α-1 1 + r λ α , and thanks to (75), we obtain the representation c α,r (λ) = 1 λ β -

e= 1 -

 1 -λx ρ α,r (x)dx, where, by the Mittag-Leffler function representation (79),ρ α,r (x) := 1 Γ(β)x α -P G > x S α r 1/α S α = 1 Γ(β)x α -E e -x S α /(r 1/α S α ) = 1 Γ(β)x α -E e -x α /(r S α α ) Γ(β) x α E α (-x α /r) Γ(β)x α = η α,r (x α /r) Γ(β)x α , x > 0. (94)

0 ( 1 ( 1

 011 φ α,r (λ) = α λ 0 c α,r (u)du = ∞ -e -λx )ρ α,r (x) -e -λx ) η α,r (x α /r) x α+1 dx. (95) At this point, thanks to the equivalence between (70) and (72), we have found that for fixed r ∈ (0, 1], (96)η α,r ≥ 0 ⇐⇒ φ α,r ∈ BF =⇒ α r e -φ α,r (λ) = e -λ α α r + αλ α ∈ CM ⇐⇒ λ α ∈ BF * α/r ,and, by (95), that (97)x → η α,r (x α )/x α ∈ CM ⇐⇒ φ α,r has the form (103) of a Thorin Bernstein function.Further, due to (91), we haveλE α (-λ) ∈ BF ⇐⇒ η α,1 (λ) = 1 -Γ(β) λ E α (-λ) ∈ CM ⇐⇒ η α,r (λ) = 1 -r + r η α,1 (λ) ∈ CM, for all r ≤ 1. (98)1) By Proposition 6.2, part 7) of Proposition 4.1, (91), (96), (97) and (98), we obtain the equivalences between (i), . . . , (vi), via the following schemeα ≤ 1 2 ⇐⇒ λ α ∈ BF * α ⇐⇒ λ α ∈ BF * α/r , ∀r ≤ 1 ⇐= η α,1 ≥ 0 ⇐⇒ =⇒ λE[e -λ/S α α ] = λE α (-λ) ∈ BF ⇐⇒ η α,r ∈ CM, ∀r ≤ 1 =⇒ η α,r (x α )/x α ∈ CM, ∀r ≤ 1 ⇐⇒ φ α,r ∈ T BF, ∀r ≤ 1 =⇒ λ α ∈ BF * α/r , ∀r ≤ 1.

8. Appendix 1 :

 1 Spectrally negative Lévy processes and Lévy-Laplace exponents. A Lévy-Laplace exponent Ψ is a function represented by (99) Ψ(λ) = a + bλ + cλ 2 + (0,∞) e -λx -1 + λh(x) µ(dx), λ ≥ 0, where a, c ≥ 0, b ∈ R, and h is a truncation function, i.e. any bounded function such lim x→0 + (h(x) -x)/x exists, and the Lévy measure ν satisfies

9. Appendix 2 :

 2 Complete Bernstein functions (CBF ) and Thorin Bernstein functions (T BF ). The set CBF of complete Bernstein functions[16, p. 69] corresponds to those functions φ ∈ BF represented byφ(λ) = q + d λ + (0,∞) λ λ + u ∆(du) = q + d λ + (0,∞) (1 -e -λx ) L ∆ (x) dx, λ ≥ 0,for some constants q, d ≥ 0, where ∆(du)/u, u > 0, is necessarily a Lévy measure, and L ∆ stands for the Laplace transform of ∆. The class of Stieltjes functions consists of those of the form ), λ > 0, where d, q ≥ 0 are constants and (0,∞) (1 + u) -1 ∆(du) < ∞. The set T BF is the subclass of CBF formed by those functions φ of the form φ(λ) = q + d λ + (0,∞)

  There, we also show that if X is a non-negative infinitely divisible random variable, with Bernstein function ϕ X , with drift term d, and X is embedded into the subordinator (X t ) t≥0 , then ϕ X ∈ BF * s if, and only if, the r.v. X

	(23)	1)
		⇐⇒ Z has a probability density function	p s (x) x s+1 , x > 0,
	(24)	where p s (x) is non-decreasing and right-continuous.
	In Section 5 we show that BF * s is a strict subclass of BF s . Corollary 5.2 shows that BF * s
	corresponds to the subset of stabilizers of BF s with respect to composition, namely
	(25)	BF *
	( ) t or (24) for all t > 0. The interest of the characterization (25) of BF * := X t -d t satisfies any of conditions (22), (23) s is that φ ∈ BF * s , if,
	and only if, the decomposition (20), applied to the subordinator (X	( ) t ) t≥0 is preserved by
	subordination:
	φ ∈ BF * s ⇐⇒ the subordinated process, (X η t ) t≥0 , also decomposes as in (20), ( )

s := {φ ∈ BF s , s.t. ψ • φ ∈ BF s for all ψ ∈ BF}.

Acknowledgment: We are grateful to anonymous referees for their valuable comments and suggestions that improved the presentation and the results of this paper. The work of the second author was supported by the "Researchers Supporting Project number (RSP2023R162), King Saud University, Riyadh, Saudi Arabia".

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