Three classes of decomposable distributions

Proposition 5.2

If c ∈ (0, 1) and if ϕ ∈ ℬ ℱ , then the functions Θ ϕ and θ c ϕ are non-negative, nondecreasing and both functions x ↦ θ c ϕ ( x ) / x 2 , Θ ϕ ( x ) / x 2 are integrable at ∞ .

Proof

Note that the derivative of ( Θ ϕ ) ′ = − λ ϕ ″ is non-negative and, by Proposition 3.3(4), we have lim λ → 0 + Θ ϕ ( λ ) = ϕ ( 0 ) ≥ 0 . We deduce that Θ ϕ is non-negative and nondecreasing. Furthermore, using inversion formula (11), we get the integrability condition for Θ ϕ . The assertion on θ c ϕ is obtained by the representation (8).□

Proposition 5.3

Let ϕ : [ 0 , ∞ ) → [ 0 , ∞ ) . The following holds:

1. If Θ ϕ ∈ ℬ ℱ and has no drift term, then ϕ ∈ ℬ ℱ ;

2. If θ c ϕ ∈ ℬ ℱ for some c ∈ (0, 1) , then ϕ ∈ ℬ ℱ .

3. If ϕ is nondecreasing and differentiable function on (0, ∞ ) and if θ c n ϕ ∈ C ℱ for some sequence c n ∈ (0, 1) such that c n → 1 , then Θ ϕ ∈ ℬ ℱ .

Proof

1. Since there is no drift term in Θ ϕ , then x ↦ Θ ϕ ( x ) / x 2 is integrable at ∞ , and representation (2) for Θ ϕ gives

   Θ ϕ ( x ) x 2 = ∫ 0 ∞ e − x u ∫ 0 x l 0 ( t ) d t d u , x > 0 ,

   with some nonincreasing function l 0 . The inversion formula (11) gives a representation of type (2) for ϕ :

   ϕ ( λ ) = λ lim u → ∞ ϕ ( u ) u + ∫ 0 ∞ e − λ x l ( x ) , l ( x ) ≔ 1 x ∫ 0 x l 0 ( t ) d t , λ ≥ 0 ,

   and note that x ↦ l ( x ) = ∫ 0 1 l 0 ( x t ) d t 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

   θ c n ϕ ( λ ) = θ c ϕ ( c n − 1 λ ) + c θ c n − 1 ϕ ( λ ) .

   By induction, we see that for every n ∈ ℕ , the function θ c n ϕ belongs to ℬ ℱ , or equivalently

   λ ↦ ϕ ( λ ) − c n ϕ λ c n ∈ ℬ ℱ .

   The next step is to prove that φ ( λ ) = lim n → ∞ c n ϕ ( λ / c n ) is of the form

   (14) φ ( λ ) = K λ , for every λ ≥ 0 and some K ≥ 0 .

   Using Proposition 3.1 and taking the limit as n → ∞ , the latter will give that λ ↦ ϕ ( λ ) − K λ ∈ ℬ ℱ and hence, ϕ ∈ ℬ ℱ . In order to prove the claim (14), note that for every fixed λ ≥ 0 , the increments of sequence u n = ϕ ( λ ) − c n ϕ ( λ / c n ) are of the form

   u n + 1 − u n = c n θ c ϕ λ c n + 1 ≥ 0 .

   The sequence u n , being nondecreasing and bounded by ϕ ( λ ) , is convergent. Then, the function φ is well defined on [0, ∞ ) and satisfies

   (15) φ ( 0 ) = 0 , φ ( c m λ ) = c m φ ( λ ) , for every λ ≥ 0 , m ∈ ℕ .

   Every λ ∈ ( 0 , 1 ] could be written in the form λ = c x , x > 0 , and plugging in (15) with m = 1 , we see that the function f ( x ) = ϕ ( c x ) satisfies the iterative functional equation

   f ( x + 1 ) = c f ( x ) , x > 0 .

   By [9, Theorem 2.4.2, p. 72], the unique solution f is necessarily of the form K c x , with K ≥ 0 . At this stage, we have proved that

   (16) f ( log λ ) = φ ( λ ) = K λ , for every λ ∈ ( 0 , 1 ] and some K ≥ 0 .

   If λ > 1 , there exits p 0 ∈ ℕ such that c p 0 λ ≤ 1 . By (16), and by (15) with m = p 0 , we obtain

   K c p 0 λ = φ ( c p 0 λ ) = c p 0 φ ( λ ) .

   Simplifying the last identity, we get (14).

3. As in the proof of Theorem 4.2, use (9) to get that ( 1 − e − θ c n ϕ ) / ( 1 − c n ) is a sequence of Bernstein functions, then apply Proposition 3.1 and retrieve

Remark 5.4

Recall the classes C ℱ θ and ℬ ℱ Θ of Definition 7.2. Proposition 5.3 shows that actually

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.

1. x μ ̅ ( x ) is concave;

2. μ ← ( x ) is concave;

3. μ has a density function in the form

   μ ( d x ) = p ( x ) x 2 d x , w h e r e p i s n o n d e c r e a s i n g ;

4. θ c μ ← is positive and nondecreasing for every c ∈ (0, 1) ;

5. θ c ϕ ∈ ℬ ℱ , for every c ∈ (0, 1) ;

6. θ c n ϕ ∈ C ℱ , for some sequence c n ∈ (0, 1) such that c n → 1 as n → ∞ ;

7. ϕ ∈ ℬ ℱ Θ ;

8. there exists φ ∈ ℬ ℱ (with drift term equal to 0), s.t. ϕ ( λ ) = λ ∫ λ ∞ φ ( x ) x 2 d x .

Under any of the latter, we have the representation

Corollary 5.6

C ℱ θ = ℬ ℱ θ ∩ { ϕ , s . t . ϕ ( 0 ) = 0 } a n d ℬ ℱ θ = ℬ ℱ Θ ⊂ ℬ ℱ .

Example 5.7

We propose two examples:

1. If ϕ ∈ ℬ ℱ Θ , then λ ↦ ϕ ( λ α ) ∈ ℬ ℱ Θ , for all 0 < α ≤ 1 . Indeed,

   Θ ( λ ↦ ϕ ( λ α ) ) = ( 1 − α ) ϕ ( λ α ) + ( Θ ϕ ) ( λ α ) ,

   and the two functions in r.h.s are in ℬ ℱ since they are obtained by the composition with the stable Bernstein function λ ↦ λ α .

2. Next example is less trivial: the function

To see that, note that for every λ ≥ 0

so that ϕ 0 is represented by

Proof of Theorem 5.5

By Proposition 3.2, observe that x ↦ x μ ̅ ( x ) is concave and non-negative, hence is a nondecreasing function. Thus, the function x ↦ μ ← ( x ) / x ≔ μ ̅ ( 1 / x ) / x is nonincreasing.

( 1 ) ⇔ ( 2 ) : The equivalence is due to [3, Lemma 2.2].

( 1 ) ⇒ ( 3 ) : If x ↦ g ( x ) ≔ x μ ̅ ( x ) is nondecreasing, then

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

obtain from

that

Finally, deduce that the measure dp is a positive, i.e. p is nondecreasing.

( 3 ) ⇒ ( 4 ) : For x > 0 , define

and by the change variable t = s / c , retrieve the representation

Since p is nondecreasing, then μ ̅ c is non-negative and nonincreasing, which is equivalent to the claim on θ c ( μ ← ) .

( 4 ) ⇒ ( 5 ) : Recall the form (2) of ϕ and that μ ̅ c is given in (21). By the change of variable x = y / c , write

Since μ ̅ c , given by (21), is nonincreasing, then θ c ϕ has also the representation (2) of a Bernstein function.

( 5 ) ⇒ ( 6 ) : Just recall that ℬ ℱ ⊂ C ℱ and use the definition of the class ℬ ℱ Θ in 5.1.

( 6 ) ⇒ ( 7 ) : This is point (3) of Proposition 5.3.

( 7 ) ⇔ ( 8 ) : Take d = lim + ∞ ϕ ( u ) / u , φ = Θ ϕ and use inversion formula (11).

( 8 ) ⇒ ( 1 ) : Using representation (2) for Θ ϕ , write

so that

By representation (2), conclude that x ↦ x μ ̅ ( x ) is concave.

For the last assertion, use representations (1) and (2) of ϕ and write

then conclude with (19), (20) and representation (2) for Θ ϕ .□

Remark 5.8

Due to the linearity of the operator Θ and due to Proposition 3.1, we retrieve the following properties for the class ℬ ℱ Θ :

1. The class ℬ ℱ Θ is a convex cone, i.e. if ( ϕ u ) u ∈ U is a family of ℬ ℱ Θ and ν is a measure on the indexation set U, then, modulo the existence of the integral, we have

   λ ↦ ∫ U ϕ u ( λ ) ν ( d u ) ∈ ℬ ℱ Θ .

2. The class ℬ ℱ Θ , like ℬ ℱ , is closed under pointwise limits.

Proposition 5.9

We have the equivalence between the following assertions.

1. ϕ ∈ ℬ ℱ Θ ;

2. λ ↦ ϕ α ( λ ) ≔ ϕ ( λ α ) ∈ ℬ ℱ Θ for every α ∈ (0, 1) ;

3. ϕ is differentiable and for every a > 0 ,

Proof

( 1 ) ⇔ ( 2 ) : It is suffice to write

For the converse, conclude by letting α → 1 − and by the closure property of the class ℬ ℱ Θ in Remark 5.8.

( 1 ) ⇒ ( 3 ) By Theorem 5.5, there is no loss of generality to assume that ϕ ∈ ℬ ℱ Θ has the characteristics 0 , 0 , p ( x ) x 2 d x , with p nondecreasing. Using (2), write

and by (20), it is clear that l a ( x ) ≥ ∫ x ∞ p ( t ) t 2 − p ( x ) x ≥ 0 . 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:

( 3 ) ⇒ ( 1 ) : For the sufficient part, use the closure property of the class ℬ ℱ in Proposition 3.1, and note that Θ ϕ = lim a → 0 + ϕ a / a . □

Proposition 5.10

The operator Δ is a bijection from ℬ ℱ (resp. ℬ ℱ Θ ) to C ℬ ℱ (resp. T ℬ ℱ ).