Three classes of decomposable distributions

Abstract In this work, we refine the results of Sendov and Shan [New representation theorems for completely monotone and Bernstein functions with convexity properties on their measures, J. Theor. Probab. 28 (2015), 1689–1725] on subordinators obtained by the class of Bernstein functions stable by the Mellin-Euler differential operator I − x d d x I-x\tfrac{\text{d}}{\text{d}x} by giving a stochastic interpretation, proving monotonicity properties of the related distributions and providing additional extensions.


Introduction
Recall that the class of completely monotone functions corresponds to those infinitely differentiable functions The class of Bernstein functions corresponds to non-negative antiderivatives of completely monotone function, i.e. to those functions ϕ represented by: where ≥ q 0 is called the killing rate, ≥ d 0 is called the drift term and μ is a positive measure on ( ∞) 0, which integrates ∧ where we have used the notation l for the nonincreasing function and k for the concave function given by , we sometimes also use the notation¯, In this work, we refine and extend the results obtained by Sendov and Shan [1] on a class of Bernstein functions that we denote here by Θ , and which is defined by , for every 0, 1 . Θ Our contribution in this work consists in the following: (a) The class Θ is demystified by proving the existence of densities enjoying a monotony property for the Lévy measures and for the transition of the associated subordinators, if we restrict ourselves in the subclass of the stabilisators of Θ . We provide a full characterisation by the stochastic point of view. More precisely, a positive and infinitely divisible random variable X has its Bernstein function ∈ ϕ Θ , if, and only if, X is embedded into a subordinator ( ) ≥ X t t 0 , i.e. = X X 1 , and for all ∈ ( ) c 0, 1 , there exists a random variable Z c such that we have the identity in law This decomposability property is linked to the operators θ c defined by Operating on the class and is new in the literature. We denote by RD the class of distributions satisfying (5) and call it the class reverse self-decomposable distributions. Observe that this class is very close (in its formalism) to the famous class SD of self-decomposable ones, formed by those distributions associated with random variables X = satisfying the following: for every ∈ ( ) c 0, 1 , there exists a r.v. Y c such that we have the identity in law  In this paper, we show that Θ 1 is in bijection with the class of subordinators ( ) ≥ X t t 0 , such that for every > t 0, the distribution of X t is of the form As a consequence, we provide a satisfying answer to the open problem raised by Sendov

Some operators of interest
We need to introduce some formalism in order to clarify the characterisation of three classes of distributions that will be studied later on.
For a set of functions ( ∞) → ϕ : 0, , we denote by Δ and δ the subsets Remark 2.2. We have the remarkable facts: (i) The difference operators are obtained from integrals of the differential operators: (ii) The differential operators are obtained from limits of the difference operators: Then, under the last integrability conditions, we have the inversion formulae: 3 Some tools on Bernstein function and infinite divisibility Next result on the stability and closure properties will be used several times in the sequel: See [2] for more details on completely monotone and Bernstein functions. The following two propositions are easy to obtain but useful, and they will be used several times in the sequel: . Then 1. If f is concave, then it is nondecreasing. 2. f is concave if, and only if, ↦ ( / ) x xf x 1 is concave.
The second statement is Lemma 2.2 in [3]. We were not able to find a reference for the first statement, which is probably known in the literature, and we propose this simple proof.
(3) The implication is justified by the facts that is a convex cone, and essentially by (11): since f Θ is a nonincreasing function, then ( )/ f x x Θ 2 is integrable at infinity and necessarily = ( ) / The injectivity of the Laplace transform ensures that the distribution of a positive random variable X is entirely characterised by its Laplace transform 0 0 0 and 1 , 0.
tϕ From now on, we adopt the notation ∼ X X C C , if the distribution of belongs to the class of distributions .
Recall that a random variable X has an infinitely divisible distribution ( ∼ X ID) if for every ∈ * n , there exists sequence of independent and identically distributed random variables … X X , , n nn ,1 , (called the nth folds of X), such that c.f. [4]. Actually, the set infinitely divisible distributions on[ ∞) 0, are in bijection with the class of Bernstein functions null at 0, namely, infinite divisibility of a non-negative r.v. X is entirely characterised by the fact that its cumulant ϕ is a Bernstein function and every non-negative infinitely divisible r.v. X is embedded into a subordinator ( ) ≥ X t t 0 (i.e. an increasing Lévy process, see [5]), this means that  Revisiting the class of self-decomposable distributions A proper subclass of ID is the well-known class SD of self-decomposable distributions, also known as the Lévy class, and introduced by Lévy in 1937. In the specialised literature, the notation is frequent for the class SD (see [6] for instance). This class forms a natural extension of the class of stable laws and its importance stems from the fact that it arises in limit theorems for sums of independent variables. Without using the fact that ⊂ SD ID (frequently required in the literature), self-decomposability could be defined as follows: A reformulation of (12) property in terms of cumulant functions is as follows: where we used the notations (6) and (7). For more account on self-decomposability, we suggest [4] and [8].
For the sake of clarity, we provide a full characterisation of the class SD. With the notations (6) and (7), we have: Theorem 4.2. Let X be a non-negative r.v. with cumulant function ϕ. We have the equivalences and its Lévy measure is of the form , where l is nondecreasing.
As an immediate consequence we get the following.

The class of reverse-decomposable distributions
We propose in this section a new class of infinitely divisible distributions denoted by RD, which is the dual in some sense of the class SD. We shall provide the counterpart of Theorem 4.2 for this class.
In order to clarify the structure of RD, we need several analytic results.

Analytic properties of the classes  θ and  Θ
We start with the following two propositions.
, then the functions ϕ Θ and θ ϕ c are non-negative, nondecreasing and both functions ↦ Proof. Note that the derivative of ( )′ = − ″ ϕ λϕ Θ is non-negative and, by Proposition 3.3(4), we have . We deduce that ϕ Θ is non-negative and nondecreasing. Furthermore, using inversion formula (11), we get the integrability condition for ϕ Θ . The assertion on θ ϕ c is obtained by the representation (8). □ . The following holds: and has no drift term, then ∈ ϕ ; with some nonincreasing function l 0 . The inversion formula (11) gives a representation of type (2) for ϕ: is nonincreasing. Thus, ϕ meets the representation (2) of a Bernstein function.
(2) For every integer ≥ n 1 and real number ≥ λ 0, we have By induction, we see that for every ∈ n , the function θ ϕ c n belongs to , or equivalently The next step is to prove that ( ) = Kλ λ K , for every 0 and some 0.
Using Proposition 3.1 and taking the limit as → ∞ n , the latter will give that ↦ ( ) − ∈ λ ϕ λ Kλ and hence, ∈ ϕ . In order to prove the claim (14), note that for every fixed ≥ λ 0, the increments of sequence The sequence u n , being nondecreasing and bounded by ( ) ϕ λ , is convergent. Then, the function φ is well defined on [ ∞) 0, and satisfies Every ∈ ( ] λ 0, 1 could be written in the form = > λ c x , 0 x , and plugging in (15) with = m 1, we see that the function ( ) = ( ) f x ϕ c x satisfies the iterative functional equation , for every 0, 1 and some 0.
If > λ 1, there exits ∈ p 0 such that ≤ c λ 1 Sendov and Shan [1] showed that if, and only if, is a concave function, and called such measures μ measures with harmonically concave tail. We complete their work and provide the complete characterisation of Θ as follows: Theorem 5.5. Let ϕ be a Bernstein function represented by (1), associated with the Lévy measure μ, and the functions ← μ μ ̅ , be given by (17). Then, we have the equivalence between the following assertions.
(2) ← ( ) μ x is concave; (3) μ has a density function in the form is positive and nondecreasing for every ∈ ( ) c 0, 1 ; Under any of the latter, we have the representation Note that if ∈ ϕ Θ , then certainly ′( ) = +∞ ϕ 0 . Indeed, by point (3) of Theorem 5.5, and because p is nondecreasing, one has With the same argument as for Corollary 4.3, also note that Remark 5.4 and Theorem 5.5 yield:  ( ) ⇒ ( ) is a positive measure. We deduce that μ admits a density function, which we write in the form ( )/ p x x 2 , where p is a non-negative measurable function. Observing that g is concave, differentiable and is represented by Finally, deduce that the measure dp is a positive, i.e. p is nondecreasing.   Due to the linearity of the operator Θ and due to Proposition 3.1, we retrieve the following properties for the class Θ : (i) The class Θ is a convex cone, i.e. if ( ) ∈ ϕ u u U is a family of Θ and ν is a measure on the indexation set U, then, modulo the existence of the integral, we have The class Θ , like , is closed under pointwise limits.
Point (3) in the next theorem shows a nice fact: Proposition 5.9. We have the equivalence between the following assertions.
It remains to show that l a is nonincreasing, but this is easily seen by the fact that and by the differentiation of −l a , which gives the positive measure: The following nice fact gives an additional interest to the class Θ : Proposition 5.10. The operator Δ is a bijection from (resp. Θ ) to (resp. ).
Proof. In [2, Theorem 6.2], it is stated that the operator Δ is a bijection from to . Observe that if ∈ ϕ Θ , then, by Theorem 5.5, there exist ≥ d q , 0 and a nondecreasing function p such that and elementary computations give that meets the representation of a -function as given in [2,Theorem 8.2]. The converse is obtained by reversing the calculus. □

Stochastic interpretation of the class  Θ
Recall that a non-negative r.v. ∼ X ID is embedded into a subordinator ( ) ≥ X t t 0 , i.e. = X X 1 . From this observation we retrieve a simple characterisation of RD: (1) ∼ X RD; (2) ∼ X ID and is embedded into a subordinator ( ) ≥ X t t 0 such that we have the identities in distribution: where Z c t , is a non-negative infinitely divisible random variable independent of X c ; Remark 5.12. In Remark 5.8, we have noted that Θ is a convex cone, i.e. it is stable by mixture of families of Θ by a measure μ. In case where μ is a discrete measure, say = + +⋯+ ν δ δ δ n 1 2 , the latter is stochastically interpreted as follows: Another stochastic interpretation is as follows. By Theorem 5.5, we know that the Lévy measure of a function ∈ ϕ Θ is represented by: nondecreasing. 2 Recall that any Bernstein function ϕ is associated with a subordinator ( ) ≥ X 0 t t , possibly killed with a rate = ( ) q ϕ 0 . The so-called harmonic and harmonic potential measures are defined by: and the exponential functional of the subordinator ( ) ≥ X t t 0 is given by the stochastic integral is also of a particular interest. In [10], it is shown that the measure U is infinitely divisible (in the sense of the convolution), whereas W is not in general, but we have the following: Proposition 5.13. For any Bernstein function ϕ, the following assertions are equivalent: is completely monotone; (2) ↦ ( ( )/ ) λ ϕ λ λ t is completely monotone for all ≥ t 0; is an infinitely divisible distribution; (5) The exponential functional I is such that I log is infinitely divisible; (6) The harmonic potential measure H has a density function of the form Corollary 5.14. Any function ∈ ϕ Θ satisfies the assertions of Proposition 5.13.
Proof. Since The differential operators Ξ and its companion Θ were defined in (2.1) and Θ Ξ Identity .
Observe that the operators Θ and Ξ commute: for some concave function k and it is clear that On the other hand, and . 6 The class  Θ 1 We now introduce a more refined class than Θ , the class Θ 1 , that will provide the behaviour of the transitions of the subordinator behind the involved Bernstein functions.

Definition of  Θ 1 and analytic results
The class Θ 1 has also been considered by Sendov and Shan [1] and denoted there by . We start by some enlightenment on the structure of Θ 1 . The linearity of Θ leads to the following observation that will be used several times.  , and the equivalence between the following assertions: Proof. The inclusion is obtained by (27) and by Proposition 3.1. The inclusion is strict because the identity function belongs to Θ but not to Θ 1 . For the equivalences, use Proposition 3.3: Next result gives a simple sufficient condition for a function to belong to the class Θ 1 : The following result is very close to the requirements on functions in Θ 1 : . The statements on ( ) Y c are justified as follows:   Observe that Corollary 6.9(iv) does not apply on = ϕ tϕ t 1 because ϕ t is not even in Θ : Elementary calculus gives that the associated κ-function in Proposition 6.7 is given by where for every > t 0, p t is nondecreasing. It is then obvious ↦ ( ) ( ) x u x h x , are also nondecreasing.