Generalized Gamma Convolutions, Dirichlet means, Thorin measures, with explicit examples

In Section 1, we present a number of classical results concerning the Generalized Gamma Convolution (:GGC) variables, their Wiener-Gamma representations, and relation with the Dirichlet processes.To a GGC variable, one may associate a unique Thorin measure. Let $G$ a positive r.v. and $\Gamma_t(G)$ (resp. $\Gamma_t(1/G))$ the Generalized Gamma Convolution with Thorin measure $t$-times the law of $G$ (resp. the law of $1/G$). In Section 2, we compare the laws of $\Gamma_t(G)$ and $\Gamma_t(1/G)$.In Section 3, we present some old and some new examples of GGC variables, among which the lengths of excursions of Bessel processes straddling an independent exponential time.


Introduction
1. This survey is concerned with the study of a rich and interesting class of infinitely divisible laws on R + called the generalized gamma convolutions (GGC), a class introduced by O. L. Thorin in 1977 see [49] and then studied thoroughly by L. Bondesson [5]; both the lectures notes by Bondesson and the book by Steutel and Van Harn [48] contain many results on this class of laws. We shall also discuss their close connections to a class of random variables known as Dirichlet means whose study was initiated by Cifarelli and Regazzini [10; 11].
We shall often make, throughout this paper, the common abuse of language which consists of talking about a random variable instead of its law; thus, we shall use slightly incorrect terms such as GGC variables, and so on . . .
In order to introduce the family of GGC variables as naturally as possible, let us consider the 3 sets of r.v.'s (or of laws): G, S, J defined as follows: a) J is the family of infinitely divisible r.v.'s (taking values in R + ) b) S is the set of self-decomposable r.v.'s, valued in R + c) G is the set of positive GGC variables The Laplace transforms of these variables satisfy: and a') If X ∈ J , then a ≥ 0 and ν(dx) is a Lévy measure, i.e.: Another definition of a GGC variable is sometimes given in the literature as the limit (in law) of sums of independent gamma variables (with different parameters).
A comparative discussion of different (but, in the end, equivalent) properties of elements of S is made in Jeanblanc, Pitman, and Yor [32]. Similarly, it is the aim of this survey to gather and to compare different properties for G. This, and more generally, the entire survey is motivated by the fact that, recently, some new elements of G were discovered; likewise, there is some recurring interest in the Gamma process see, e.g. the Festschrift Volume for Dilip Madan [55] .
2. This survey paper consists of three parts: • In the first part, the results being discussed are classical; they are about the relationships between different families of r.v.'s and/or processes, namely: GGC r.v.'s, Wiener-Gamma integrals, Dirichlet means, compound Poisson processes, mixing of Gamma variables, Poisson point processes, GGC subordinators, and so on. These results are detailed here in order to ease up the reading of this paper for probabilists coming from diverse horizons.
• In the second part, we discuss a notion of duality for the GGC r.v.'s; in particular, when one knows the density, or the Laplace transform of a GGC r.v. Γ, this notion of duality allows to compute explicitely the density, or the Laplace transform of the "dual GGC variable". We use this principle to compute the Thorin measure of a Pareto r.v. and of a power of a gamma r.v.
• The third part consists essentially in presenting the explicit computations of densities and of Laplace transforms of some particular GGC r.v.'s. In the main, these r.v.'s originate from the study of the length of excursion which straddles an independent exponential time for a recurrent Bessel process [4]. It is noteworthy that, in this third part, the notion of duality presented in the second part allows to obtain very easily explicit formulae for the density and the Laplace transform of a large number of GGC variables.
Finally, in the Appendix, we describe an interpolation principle between the gamma subordinator (γ t , t ≥ 0) and a family of GGC subordinators.
1. Classical results on GGC r.v.'s A writing convention.
1. Each time we write an equality in law between r.v.'s and that on one or the other side of this equality, several r.v.'s occur, we always assume that these r.v.'s are independent, without mentioning it systematically.
2. It will be convenient, in some instances, to speak of a r.v. instead of its law and vice versa. We hope that no confusion will ensue.

The Gamma process
It is a subordinator -i.e. a Lévy process with increasing paths and càdlàg trajectories -All through this paper, the reference process is the standard Gamma process (γ t , t ≥ 0), which is a subordinator without drift, and with Lévy measure dx x e −x (x > 0). Thus, its Lévy-Khintchine representation writes (see [2]): where formula (2) is obtained from (1) and from the elementary Frullani formula (see [34], p. 6): where, in general, ν denotes a positive measure on R + , which is σ-finite. Here, ν(dz) = δ 1 (dz) is the Dirac measure at 1, but formula (3) shall be useful in the sequel. We note that Frullani's formula (3) may be easily obtained by observing that the two sides of this formula take the value 0 as λ = 0 and have the same derivative with respect to λ. For each t > 0 fixed, γ t follows a gamma law with parameter t: This process (γ t , t ≥ 0) enjoys a large number of remarkable properties which make it a "worthy companion" of Brownian motion. In particular, Emery and Yor [18] establish a parallel between Brownian motion and its bridges on one hand, and the Gamma process and its bridges on the other hand. See also Vershik, Yor and Tsilevich [50] and Yor [55] for a survey of many remarkable properties of the gamma process.

Wiener-Gamma integrals and GGC variables
1.2.a Many times throughout this work, we shall use the properties of the integrals: where h : R + −→ R + is a Borel function such that: Under this hypothesis (6), Γ(h) is finite a.s. (see Proposition 1.1 below). Of course, since the trajectories of the process are a.s. increasing, the integral featured in (1.5) may be defined in a path-wise manner, as a usual Stieltjes integral. We call Γ(h) a Wiener-Gamma integral, in analogy with the Wiener integrals ∞ 0 f (u) dB u , f ∈ L 2 (R + , du) which constitute the Gaussian space generated by Brownian motion (B u , u ≥ 0). Thus, the family of r.v.'s Γ(h) with h such that ∞ 0 log 1 + h(u) du < ∞ constitutes the analogue for the process (γ t , t ≥ 0) of the first Wiener chaos for a Brownian motion (B u , u ≥ 0).
Let us assume for a moment that, in (1.5), the function h is constant on a finite number of intervals, i.e.: for a subdivision 0 = s 0 < s 1 < · · · < s n < ∞. Then:

351
Thus, Γ(h) is a linear combination of independent gamma r.v.'s and we obtain: where µ h is the image of Lebesgue's measure on R + by the application s → 1 h(s) · This latter formula justifies the following definition: 1.2.b Definition 1.0: Following ( [5], p. 29), we say that a positive r.v. Γ is a generalized gamma convolution (GGC) -without translation term -if there exists a positive Radon measure µ on ]0, ∞[ such that: with: |log x| µ(dx) < ∞ and The measure µ is called Thorin's measure associated with Γ. Thus, from the Lévy-Khintchine formula, a GGC r.v. is infinitely divisible. In fact, since its Lévy density l Γ (x) = 1 x ∞ 0 e −xz µ(dz) satisfies: x −→ x l(x) is decreasing, then Γ is a self decomposable r.v. see, e.g. [37] . Such a self-decomposable r.v. Γ, assumed to be non degenerate, admits a density f Γ such that f Γ (x) > 0 for every x > 0 see [45] p.404 . The study of GGC variables was initiated by O. Thorin in a series of papers see for instance [49] .

1.2.c GGC variables and Wiener-Gamma representations.
The following Proposition is classical. The reader may refer to Lijoi and Regazzini [36].
where µ h denotes the image of Lebesgue's measure on R + under the application: We note that, in (11), h may vanish on some measurable set.

Let
It is easily obtained, by approximation of Γ h by Riemann sums, using also the fact that the Lévy measure of the process (γ t , t ≥ 0) equals dx x e −x , that: after making the change of variable xh(u) = y. We observe, from (11) and (12), the equivalence of the conditions: 1. Formula (13) may be obtained in a slightly different manner: the process (γ s − γ s − := e s , s ≥ 0) of jumps of the subordinator (γ t , t ≥ 0) is a Poisson point process whose intensity measure n equals the Lévy measure of (γ t , t ≥ 0) see [2] : Thus, from the exponential formula for Poisson point processes [43], p. 476 : which agrees with the expression in (13).

2.
Let h, k : R + −→ R + two Borel functions which satisfy (6) and assume that Γ(h) = Γ(k). Relation (13) and the uniqueness of the Lévy measure in the Lévy-Khintchine representation imply: the images by h and k of Lebesgue's measure on R + are identical. Thus, choosing for k the increasing rearrangement h * , (resp.: the decreasing rearrangement h * ) of h, we obtain that there exists essentially a unique increasing function h * , (resp. a unique decreasing function h * ) such that: We recall that the function h * (resp. the function h * ) is the unique (equivalence class of) increasing (resp. decreasing) function such that for every a ≥ 0: where meas indicates Lebesgue's measure on R + . From Proposition 1.1, Γ(h) is a GGC r.v. and we shall denote by µ h the Thorin measure associated with Γ(h). In this work, we shall often consider a GGC r.v. whose associated Thorin measure has finite total mass. Thus, we shall now particularize Proposition 1.1 in this case.
1.3.a Let m > 0 and h : [0, m] → R + a Borel function such that: We call m-Wiener integral of h the r.v.: Since (γ u , 0 ≤ u ≤ m) and (γ m − γ (m−u)− , 0 ≤ u ≤ m) have the same law, we deduce from (17), after making the change of variable m − u = v: We note, in relation with (15) above, that if h is increasing, resp. decreasing, then u → h(m − u) is decreasing, resp. increasing.

1.3.b
Let m > 0 and G be a positive r.v. such that: We say that a positive r.v. Γ is a (m, G) GGC if: Of course, from (7), a (m, G) GGC r.v. is a GGC r.v. whose Thorin measure µ equals: where P G denotes the law of G and we have: Under (21), it is clear that: We denote by Γ m (G) the (law of the) r.v. Γ defined by (20). Hence: = exp − m E log 1 + λ G 1.3.c G or 1/G ? How to choose ? We have used the notation Γ m (G) due to the relation (24). Of course, the relation (25) invites, on the contrary, to adopt the notation Γ m (1/G). However, we shall not adopt this latter notation as the notation Γ m (G) is used by L. Bondesson [5] who has contributed in an essential manner to the study of the GGC variables.
1.3.d Proposition 1.1, when the Thorin measure has a finite total mass m, admits the following translation.
More precisely: where F −1 G denotes the right continuous inverse, in the sense of composition of functions, of F G , the cumulative distribution function of G.
where U m denotes a uniform r.v. on [0, m].
1.3.e Some classical results We gather here some results which are due to L. Bondesson [5] and which we shall use in the sequel. Let m > 0 and G satisfy (19). Then, denoting f Γm(G) the density of Γ m (G): where g is a completely monotone function ( [5], p. 49). Moreover ( [5], p. 50) g admits a limit on the right of 0 and: We note that, since by hypothesis In Section 2 of this work (see Theorem 2.1) we give an explicit form of g when E |log G| < ∞.
• m may be determined from the knowledge of f Γm(G) : (see [5], p. 51). u , 0 ≤ u ≤ m) and defined as: It is well known, and it is an easy consequence of the properties of the "betagamma algebra" that this process (D (m) Indeed, if γ a and γ b are two independent gamma variables with parameter a, resp. b, the basic "beta-gamma algebra" states that: where β a,b and γ a+b are independent and are respectively beta (a, b) and gamma (a + b) distributed. In particular, γa γa+γ b is independent from γ a + γ b . Thus, for every u ≤ m: This allows to write, for h which satisfies (16): Thus, from Proposition 1.3, we may write for G which satisfies (19): with It follows that for every (m, G) GGC r.v., the r.v. Γ m (G) is a Gamma (m) mixture, i.e. it may be written as: where Z is a positive r.v. In general, a relationship of the kind: (with X and Z independent, and X and Z ′ independent) does not allow, "via simplification" to conclude that Z (law) = Z ′ . However, when X is a gamma variable, this "simplification" is licit. More precisely: The relation (35) determines the law of Z. Indeed, let Z and Z ′ two positive r.v.'s such that: Then, for every s ∈ R: Hence: E(Z is ) = E(Z ′is ) and Z (law) = Z ′ 1.4.c Remark 1.4 (We shall not use the present Remark in the sequel of this paper). We come back to the notation of point 1 of Remark 1.2 and we denote: (J , · · · ) constitute a sequence of i.i.d r.v.'s with uniform law on [0, m] which is independent from the sequence (J (m) k , k ≥ 1). Thus: where 1 G k , k ≥ 1 is the sequence of i.i.d r.v.'s with common law 1/G and is independent (as a sequence) from the r.v.'s (J (m) k , k ≥ 1). Indeed: m is uniform on [0, 1]. We deduce from (36) that: We note that: k≥1 J (m) k γm = 1. We then define the random Dirichlet measure P (1/G) 0,m (dx) by the formula: and we obtain, from (34) and (37): This relation (38) justifies the denomination, for D m (G), of a Dirichlet means. The study of the Dirichlet means can be traced back to an early work of Cifarelli and Regazzini [10] which culminates into the more recognized [11]. Additional early works on this topic include [14; 21; 26; 56]. See also Bertoin [3] for an example of (34) where he shows that a Cauchy random variable, C 1 , may be represented as Where, (M ′ s , s ∈ [0, 1[) is the right derivative of the convex minorant of a Cauchy process.

1.4.d Multiplication by a beta variable.
In this section we discuss what happens when D m (G) is multiplied by certain independent beta random variables. This idea, and restatements of the results (1.4.d i-iii) below, first appear in James see [28], Theorem 3.1, revised in [29] .
Indeed, γm γ m ′ is independent from γ m ′ and follows a beta law, with parameters (m, m ′ − m). Hence:

1.4.d ii)
In the same spirit as for the preceding point, we note that, if G is a positive r.v. such that E(log + G) < ∞ and if m ′ > m, then: where, on the RHS of (40), G and Y p are independent and Y p is a Bernoulli r.v. with parameter p = m m ′ : Indeed, we deduce from (24):

1.4.d iii)
We now write the relation (40) in a slightly different manner with the introduction of the r.v.'s D m (1/G) and D m 1 GYp . We have, from (40) and (35) for m ′ > m: In particular, for m < 1 and m ′ = 1: 1.4.d iv) Some elements of D (m) . Let T denote a positive r.v. which belongs to the Bondesson class B (see [5], p. 73, Th. 5.2.2) i.e. whose density f T writes: is GGC (for every m > 0) with some, possibly unknown, associated Thorin measure we denote as µ m . Assuming furthermore that E T −m < ∞, we get, from (30): But, an elementary computation, starting from (44), shows that: Thus, we have: Hence, there exists, from Proposition 1.3, a positive r.v. G m such that E log + 1 Gm < ∞ and also such that, from (33): Thus, from point 1.
We now summarize what we have just obtained: Let T denote a positive r.v. which belongs to B, such that:

2)
For every m ′ > m, β m,m ′ −m · T ∈ D (m ′ ) from (43) In particular, Proposition 1.5 may be applied in the following cases: • If T is a generalized inverse Gaussian r.v., i.e. its density is given by: (β ∈ R, c 1 , c 2 > 0) then, for every m > 0, T ∈ D (m) although T is a GGC variable with Thorin measure of infinite total mass (see [5], p. 59 . • If T is a Gamma r.v. γ θ with parameter θ then for every m > 0, γ θ ∈ D (m) .
Remark 1.5 Epifani, Guglielmi and Melilli ( [16], section 4; see also [17]), posed the natural question of which kind of probability measures are the laws of Dirichlet means. They were able to find some examples in cases where those particular random variables possessed all finite moments. One sees that Proposition 1.5, in a rather simple way, identifies a large number of possible distributions.

1.4.e
Another representation of Γ m (G). We have been interested mainly until now in the distributions of the (m, G) GGC r.v.'s. We shall now describe a realization of such a r.v. with the help of a compound Poisson process. Besides, this realization allows us to show that a (m, G) GGC solves an "affine equation". For a nice survey of these equations, see Vervaat [51].
Let m > 0 and let K be a positive r.v. We shall say that (Y t , t ≥ 0) is a (m, K) R + valued compound Poisson process if: is a sequence of i.i.d. variables, distributed as K, and with (N t , t ≥ 0) a Poisson process with parameter m, independent of the sequence (K i , i = 1, 2 · · · ). In particular, N t is a Poisson r.v. with parameter mt.
2) Γ m (G) satisfies the affine equation: Proof of Proposition 1.6. i) We first prove point 1. We consider (Y t , t ≥ 0) a (m, K) compound Poisson process. Then, approximating ∞ 0 e −t dY t by the Riemann sums we obtain: where µ is the Lévy measure of the subordinator (Y t , t ≥ 0). Since this subordinator is a (m, K) compound Poisson process, we have: Hence: ii) We now prove point 2. We have: and we observe that: Until now, we have been interested uniquely in the "individual" GGC variables. However, to each GGC r.v. Γ we may, since Γ is infinitely divisible and positive, associate a unique subordinator (Γ t , t ≥ 0) such that Γ 1 (law) = Γ. It is this subordinator which we shall now define and describe.

1.5.b
Let G denote a positive r.v. which satisfies (19). Then, there exists a subordinator Γ t (G), t ≥ 0 which is characterized by: In particular, for every t > 0, Γ t (G) is a (t, G) GGC r.v. Thus, there exists, following (33), a family of r.v.'s D t (G), t ≥ 0, whose laws are characterized by: and, for every t > 0, from (34) and Proposition 1.3: We note that the relations (54) and (55) are only true for fixed t, for any t > 0, but do not hold as equalities in law between processes. On the other hand, since: E(e −λγt ) = 1 (1+λ) t , we deduce from (53) that: (19).
The family of laws of the r.v.'s D t (G), t ≥ 0 solves the equation: for every t, s ≥ 0: 2) The family of laws of the r.v.'s D t (G), t ≥ 0 solves the equation: for every s, t ≥ 0: Point 2 of this Proposition 1.7 is due to Hjort and Ongaro see [27]), which can be seen as a consequence of Ethier and Griffiths [19, Lemma 1]. We note the following remarkable feature of points 1 and 2 of Proposition 1.7: the affine equations (where the unknowns are the laws of the (X t , t ≥ 0): both admit infinitely many solutions: X u = D u (G), u ≥ 0 for every r.v. G which satisfy (1.19).
Proof of Proposition 1.7 Point 1: follows from: since Γ t (G), t ≥ 0 is a subordinator and since, from the definition of D a (G): We now show (58). From the beta-gamma algebra, we have: hence, plugging (63) in (62), we obtain: which, using point 1.4.b, implies relation (58). We now prove 3 i). We have: and we deduce from (53), that: We now prove 3 ii). The a.s. convergence (hence, the convergence in law) of Indeed, we may write: and we shall prove that: Hence the result.
We now prove point 3 iii).
1, then, combining this result with the classical In fact, we shall proceed in the other direction. We shall show further (see point Remark 1.7 Lijoi and Reggazini [36] have shown that the support of the law of D t (G) is the closure of the convex hull of the support of the law of 1/G. In particular: In Section 3 of this survey, we shall verify this assertion on inspection of numerous examples.

Some examples of GGC subordinators
Let Γ denote a GGC variable with associated Thorin measure µ, and let (Γ t , t ≥ 0) denote the subordinator such that Γ 1 (law) = Γ. We have: Such a subordinator is called a GGC subordinator.
Here are now some examples of such subordinators. They are lifted from a paper by H. Matsumoto, L. Nguyen and M. Yor [38] on one hand, and from the study of hyperbolic subordinators made in Pitman and Yor [41] on the other hand. The reader may refer to these papers for further information. In the study of these examples, we shall denote subordinators with curly letters, especially to avoid some possible confusion with the modified Bessel functions, which are traditionally written with ordinary capital letters.
1.6.a The hyperbolic subordinators see [41] for a probabilistic description of these subordinators.
i) The subordinator (C t , t ≥ 0) is characterized by: Its Lévy density l C equals: Hence, its associated Thorin measure, (i.e.: the Thorin measure of C 1 ) equals: It has infinite total mass.
ii) The subordinator (S t , t ≥ 0) is characterized by: Its Lévy density equals: Hence, its Thorin measure equals: It has infinite total mass. We note that the subordinator (T t , t ≥ 0) which is characterized by: satisfies: and that its Lévy density equals: However, this subordinator (T t t ≥ 0) is not GGC, as its would be 'Thorin measure' µ T is a signed measure: t , t ≥ 0) We denote by I ν and K ν the modified Bessel functions with index ν see [34], p. 108 .

Remark
We now end up this Introduction by indicating how the study of the GGC subordinators may be embedded in a more general one.
A subordinator (N t , t ≥ 0) is said to belong to the Thorin class T (χ) (R + ), with χ > 0 see [Grig] , if its Lévy measure admits a density l N of the form: where k is a completely monotonic function, i.e. it may be represented as: for a positive Radon measure µ carried by R * + . Thus, the subordinators which we study in this work, i.e.: the GGC subordinators, belong to the class T (1) (R + ). The class T (2) (R + ) has been studied by Goldie [24], Steutel [47] and Bondesson [6]. The r.v's which belong to this class are the generalized convolutions of mixtures of exponential laws. Note that We also note that [BNMS] present extensions of these notions to R d .
In the same manner as condition (9) is necessary and sufficient for a measure µ to be the Thorin measure associated to a positive r.v., B. Grigelionis [25] obtains a necessary and sufficient analytical condition so that a measure µ defines, via (96) and (97), a subordinator (N t , t ≥ 0) which belongs to the class T (χ) (R + ).

1.9
We now detail the contents of the sequel of this paper: • In Section 2, we present a duality result which connects on one hand the r.v.'s Γ t (G) and D t (G) to the r.v.'s Γ t (1/G) and Γ t (G) on the other hand.
• In Section 3, we study in depth the examples of subordinators Γ t (G α ) (0 ≤ α ≤ 1) for which we know how to compute explicitly their Laplace transforms, i.e.: their Lévy exponents, as well as their densities at any time, and their Wiener-Gamma representations.

A duality principle
Throughout this section, G denotes a positive r.v. such that E |log G| < ∞. Thus, we have: Consequently, the subordinators Γ t (G), t ≥ 0 and Γ t (1/G), t ≥ 0 are well defined. We denote by ψ G (resp. ψ 1/G ) the characteristic exponent (i.e.: the Bernstein function or Lévy exponent) of the subordinator Γ t (G), and the same formula for ψ 1/G obtained when replacing G by 1/G. We denote by F G the cumulative distribution function of G and by F −1 G its right continuous inverse, in the sense of the composition of functions.
3) Formula (104) agrees with, and makes more precise, the result of Bondesson which we recalled in (28) and (30). In particular: 4) Formula (101) may be generalized as follows: Let a, b, c, d ∈ R 4 , with ad − bc = ±1 and let, for x, λ ≥ 0: so that σ(G) be a positive r.v. Then: where λ 0 is a fixed point of σ. The relation (101) corresponds to a = d = 0, b = c = 1. We shall study, in the Appendix, the case where a = d = sinh u, b = c = cosh u (u ≥ 0). [11], Cifarelli and Melilli [9] have obtained the density of D t (G) for t ≥ 1 and James, Lijoi and Prünster [30] have obtained it for t ≤ 1. For t ≤ 1, they obtained:

5) Cifarelli and Regazzini
The proof of this formula (110) hinges upon the knowledge of the density of the r.v. D 1 1 G Yt which is defined by: see (41) and (42) with m = t and m ′ = 1 .
This density equals: which is obtained by inverting its Stieltjes transform.
Other formulae for densities of Dirichlet means may be found in Regazzini, Guglielmi and DiNunno [42]. (105) is obtained from (104) by integrating between 0 and ∞:

6) Formula
The interest of this formula (113) is the following: it allows, in a situation where the law of G is not known but when one knows the laws of D t (G) and D t (1/G) to show that E |log G| < ∞ as soon as E D t (G) −t < ∞ and Formula (104), once multiplied by x ν , and integrated between 0 and ∞, leads to: The formula is also true even when the expectations which appear in this expression are infinite.

7)
The result: with U uniform on [0, 1] see point 3 iii) of Proposition 1.7 is a consequence of (104). Indeed: from (104) applied when replacing G by 1/G : We note that, from (64), the family of the laws of D t (1/G), t ≤ 1 is tight as soon as E(G) < ∞, since E D t we have, by changing λ in 1/λ:

2.2.b
We now show point 3 of Theorem 2.1: i) From formula (101), after multiplying by t and exponentiating, we obtain: Then, taking the Laplace transform of both sides of (116) in the variable λ, we obtain: where K ν denotes the Bessel-McDonald function with index ν and where we have used formula 5.10.25 in [34], p. 108 and 119. We now use (117) to compute f Γt(1/G) by inverting its Stieltjes transform [53].
ii) We now show (107). For this purpose, we shall prove that the two members of (107) admit the same Stieltjes transform with index t. Indeed: after making the change of variables y = 1/x = e −tE(log G) E(e −λΓt(G) ) from (56) whereas: The comparison of (121) and (122) and the injectivity of the Stieltjes transform of index t imply (107).

A complement to the duality theorem
Here again, G denotes a r.v. such that E |log G| < ∞, and we recall that, for any t ≤ 1 Y t denotes a Bernoulli r.v. with parameter t see (41) and (42), with m = t and m ′ = 1 . ) may be expressed in terms of G, as: and when 1/x is in the support of F G . Otherwise replace sin π t F G (1/x) by sin π (1 − t) when x > 0.
2) The following duality formula holds: Then: 4) Equivalently, We note that the right-hand side of (127) depends on t t ∈ [0, 1[ , whereas the left-hand side does not depend on t. = β t,1−t · D t (1/G), the knowledge of the law of D t (1/G) and of that of D t (G), for one t < 1, allow to determine that of G. We shall exploit this fact, in points 2.5 and 2.6 below, to determine the Thorin measure of a Pareto distribution and of a power of a gamma variable.

3)
We note that finding an explicit Thorin measure of an arbitrary GGC is akin to finding the Lévy measure of some infinitely divisible random variable. Bondesson([5], Theorem 4.3.2, p. 61), using inversion techniques, obtains an expression for the Thorin measure, but notes that it seldom yields explicit expressions. On the other hand the use of statement 3) of Theorem 2.3 will often lead to tractable expressions for the Thorin measure.

4)
We shall prove (see Section 3.1.b) that ∧ t is the cumulative distribution function of a r.v. Z t which we shall describe. On the other hand, some trigonometric computations allow to see that ∧ −1 t , the inverse of ∧ t in the sense of composition of functions, equals: 5) Point 1 of Theorem 2.3 is due to James see [28] .

Proof of Theorem 2.3
2.4.a We first prove point 1.
As this point has already been established by James (see [28]), we shall only give a quick proof. We have, from (56):

G Yt
Hence, changing λ in 1 λ and using (101): We then compute f D1 G Y t by inverting its Stieltjes transform: It then suffices to observe that: and: lim as well as: whereas: Then, plugging the values of these different limits in (130), we obtain point 1 of Theorem 2.3.

2.4.d
We now prove point 4 of Theorem 2.3. This result follows by using the generic form of the density of a β t,1−t random variable multiplied by an independent random variable in the particular cases of β t,1−t D t (G) and β t,1−t D t (1/G). The result is completed by applying the identity in (107).

Computation of the Thorin measure of a Pareto r.v.
Here is a first application of Theorem 2.3.

2.5.a
Let m > 0 fixed and: with density: The r.v. X θ is a GGC r.v. (see Bondesson [5], p. 59 ). The rationale of our work is now the following: • We first compute the Thorin measure associated with X θ .
• Then, letting θ converge 1, we shall obtain -as the Thorin measure depends continuously (for the narrow topology) on the law of X θ (see Bondesson,[5])the Thorin measure associated with the r.v. X 1 = γ1 γm , i.e. to a Pareto r.v. with parameter m(m > 0).

2.5.c Thorin measure of γ1
γm (m > 0). By continuity, the r.v. γ1 γm is a (1, G) GGC r.v. and its Thorin measure is the law of G whose cumulative distribution function is obtained by letting θ tend to 1 in (145). To obtain this limit, we shall develop several computations.

iii) Let
Thus, we have: where C m (z) and C m (z) are given by (147) and (149). Hence: Plugging this expression in (145), we obtain: where F is the cumulative distribution function of the Thorin measure associated to the Pareto r.v. γ1 γm , with parameter m > 0.

2.6.b.
We shall now give a more suitable expression of (165). From the relations (162) and (163), we deduce: hence: 1 But, the density of D α (G α ) may be computed from that of D α (1/G α ), thanks to (107): Now from (159) and (158), we obtain: We now note the interesting relationship concerning the law of S α .

Lemma 2.4
Let 0 < α < 1, and S denote a positive random variable with density (f (y), y > 0) such that: for some C > 0. Then, S is a stable (α) variable; precisely: We postpone the proof of the Lemma for the moment, and we note that, plugging (169) into (165), we obtain: Proof of Lemma 2.4 From (169), we take the Laplace transform of both sides: Denoting φ(λ) = E e −λS , we get: from which we deduce: 3. Explicit examples of GGC variables associated with the (G α , 0 ≤ α ≤ 1) family All the examples discussed in this Section are related to the r.v.'s. (G α , 0 ≤ α ≤ 1) introduced in [4]. Below, we indicate the properties of these r.v.'s which we shall use. We also recall our notation: 3.1.a. For 0 < α < 1, the density f Gα of G α equals: In particular: • for α = 1/2, G 1/2 follows the arc sine law: • for α = 0, G 0 (law) where C is a standard Cauchy variable. In general, for 0 < α < 1 one has: The density f Γ1−α(Gα) of Γ 1−α (G α ) equals: which may be translated as the following identities in law: 3.1.b. We note, for 0 < µ < 1, S µ and S ′ µ two independent copies of positive stable (µ) r.v.'s, i.e.: and, we let: Then see [33] or [8], p. 116 , the density f Zµ of Z µ equals: and we have: or equivalently: We note that the cumulative distribution function F Zµ of Z µ equals: and that its inverse, in the sense of composition of functions, is given by: (see Remark 2.4, point 3).
3.1.c. Although this will not be used in the sequel, we indicate a realization of the r.v. Γ 1−α (G α ) which has been at the start of [4]. Let (R t , t ≥ 0) denote a Bessel process starting from 0, with dimension d = 2(1 − α), with 0 < d < 2, or equivalently 0 < α < 1. Let, for any t > 0: and let e (law) = γ 1 an exponentially distributed r.v., with mean 1, independent from (R t , t ≥ 0). Then: A more general study of quantities such as the RHS of (188), has been developed by M. Winkel [54].
2. Laws of Γ t (1/G 1/2 ) and of D t (1/G 1/2 ). The 3. Wiener-Gamma representation of Γ t (1/G 1/2 ). For any t ≥ 0: Remark 3.2. 1) Theorem 3.1*. has been obtained by Cifarelli and Melilli [9]. It would be possible to prove Theorem 3.1 by first using Theorem 3.1* and then by applying the duality Theorem 2.1. In fact, we shall operate conversely, as we shall first prove Theorem 3.1, then we shall show that Theorem 3.1* may be deduced from it, due to the duality Theorem 2.1.
iv) From Proposition 1.3 and (18), showing point 3 amounts to compute the inverse of the cumulative distribution function of G 1/2 . Now, we have: 3.2.c Proof of Theorem 3.1*.
i) We prove point 1, in two different ways : • A direct proof. We deduce, as a particular case of the beta-gamma algebra, that: where N is a centered Gaussian variable, with variance 1, and e a standard exponential. Hence: Thus: (from the change of variable: hence:  (191) and since E(log G 1/2 ) = −log 4, from (104) and (192). Thus: ii) We now prove point 2 of Theorem 3.1*. Of course, it would be possible to use point 2 of Lemma 3.3 to make this proof, in the same manner that we have used point 1 of this Lemma 3.3 to show point 2 of Theorem 3.1. In fact, we prefer to use formula (104) of the duality Theorem 2.1. We have: Considering now the Laplace transform of the two sides of (214) we obtain: Note that, by taking λ = 0 in (215), we get: (which may also be recovered from the duplication formula for the Gamma function). Hence: The other formulae of Theorem 3.1* are now easily obtained. In particular, since: , we have, from Proposition 1.3: 3.3. Study of the r.v.'s Γ 1−α (G α ) and Γ 1−α (1/G α ), 0 < α < 1 3.3.a. Theorem 3.4. Let, for 0 < α < 1, Γ t (G α ), t ≥ 0 the subordinator characterized by: We then have the following explicit formulae: 3. Wiener-Gamma representation of Γ t (G α ). For every t ≥ 0 and 0 < α < 1: In particular, for t = 1 − α: We note that, for α = 1/2, formula (222) coïncides with (194). The dual version of Theorem 3.4 is: Theorem 3.4*. Let 0 < α < 1 and let Γ t (1/G α ), t ≥ 0 the subordinator characterized by: Then: we note that the law of D 1−α (1/G α ) does not depend on α, and it may be compared with (197): 3. Wiener-Gamma representation of Γ t (1/G α ). For every t ≥ 0: Remark 3.5.
2) We deduce from Proposition 1.6: Of course, we may verify directly that: starting from the formula: see [4], formula 1.19 , then taking the derivative in λ = 0.
2) i) A r.v. Z, which takes values on R, is said to be a Luria-Delbrück r.v. if it satisfies: and M. Möhle [39] determined the density f Z of this r.v.: ii) On the other hand, it is proven in [44], § III, 1.3, p. 1251, with α = k = 1 that there exists a Wald couple (X, H), which is infinitely divisible and such that: • H is positive and X and H are infinitely divisible.

Appendix
Interpolation between the subordinators Γ t (1/G), t ≥ 0 and (γ t , t ≥ 0) We denote, for every u ≥ 0, by σ u : R −→ R the decreasing function defined by: σ u (x) := x sinh u + cosh u x cosh u + sinh u = x tanh u + 1 x + tanh u Since the image of R + by σ u is equal to ]tanh u, coth u], then for every positive r.v. G, and every u > 0, we have: E |log σ u (G)| < ∞. Let Γ t σ u (G) , t ≥ 0 denote the subordinator defined by: Since σ 0 (G) = 1 G and σ ∞ (G) = 1, we have: Thus, the family of subordinators Γ t σ u (G) , t ≥ 0 interpolates, as u describes R + , between Γ t (1/G), t ≥ 0 and (γ t , t ≥ 0). The aim of this appendix is to show that one may compute "explicitly" the Laplace transform, the density, and the Wiener-Gamma representation of the r.v.'s Γ t σ u (G) in terms of those of Γ t (G), t ≥ 0 , for every u ≥ 0.