# Some characterizations of multiple selfdecomposability, with extensions and
an application to the Gamma function

Wissem Jedidi Department of Statistics & OR, King Saud University, P.O. Box
2455, Riyadh 11451, Saudi Arabia, , Université de Tunis El Manar, Faculté des
Sciences de Tunis, Département de Mathématiques, Laboratoire d’Analyse
Mathématiques et Applications LR11ES11. 2092 - El Manar I, Tunis, Tunisia
<EMAIL_ADDRESS><EMAIL_ADDRESS>Zbigniew J. Jurek University of
Wroc$\l$aw, Institute of Mathematics, Pl. Grunwaldzki 2/4, 50-384 Wroc$\l$aw,
Poland<EMAIL_ADDRESS>Jumanah Al Romian⋆

###### Abstract

Inspirations for this paper can be traced to Urbanik urb72b ; urbanik1  where
convolution semigroups of multiple decomposable distributions were introduced.
In particular, the classical Gamma $\mathbb{G}_{t}$ and $\log\mathbb{G}_{t}$,
$t>0$, variables are selfdecomposable $\big{(}$i.e. have distributions in
$\mathrm{L}_{0}(\mathds{R})\big{)}$. In fact, we show that
$\log\mathbb{G}_{t}$ is twice selfdecomposable $\big{(}$i.e. have
distributions in $\mathrm{L}_{1}(\mathds{R})\big{)}$ if, and only if, $t\geq
t_{1}$ where $t_{1}$ is an explicit critical value, and this an answer to a
problem raised by Akita & Maejima makoto . Moreover, we provide several new
factorizations of the Gamma function and the Gamma distributions that extend
many known ones in the literature. To this end, we revisit the class of
multiple selfdecomposable distributions, denoted $\mathrm{L}_{n}(\mathds{R})$,
and propose handy tools for its characterization, mainly based on the Mellin-
Euler operator and on the Hadamard fractional integral. Finally, we give a
perspective for the generalization of the class $\mathrm{L}_{n}(\mathds{R})$,
based on linear operators or on stochastic integral representations.

###### Keywords:

Bernstein functions; Difference-differential operators; Factorizations of the
Gamma function; Gamma distributions; Hadamard fractional integral; Kanter’s
factorization; Infinite divisibility; Integral stochastic representation;
Laplace transform; Lévy-Laplace exponents; Lévy processes; Mellin-Euler
differential operator; Multiplicative convolution; Multiple
selfdecomposability; Stable distributions; Spectrally negative Lévy processes;
Subordinators.

In memoriam of Kazimierz Urbanik (February 5, 1930 – May 29, 2005)

##  1 Introduction

The distribution $\mu$ of a real-valued random variable $X$, is said to be
infinitely divisible, and we denote $\mu\in\mathrm{I\,D}(\mathds{R})$ or
$X\sim\mathrm{I\,D}(\mathds{R})$, if

for each natural $n\geq 2$ there exits $\mu_{n}$ an $n$-th fold convolution:
$\;\mu_{n}^{\ast n}=\mu.$

In other terms, $\mu_{n}$ is the distribution of independent and identically
distributed random variables $X_{1,n},\,X_{2,n},\,\ldots,X_{n,n}$ and
$X\stackrel{{\scriptstyle d}}{{=}}X_{1,n}+X_{2,n}+\ldots+X_{n,n}.$ It is known
that every r.v. $X\sim\mathrm{I\,D}(\mathds{R})$ is embedded into a Lévy
process $(Z_{t})_{t\geq 0}$, i.e. a process with independent and stationary
increments, such that $Z_{0}=0$ and $X\stackrel{{\scriptstyle d}}{{=}}Z_{1}$,
in order that the so-called Lévy-Khintchine formula holds:

$X\sim\mathrm{I\,D}(\mathds{R})\Longleftrightarrow\operatorname{\mathbf{E}}[e^{{\rm
i}\,u\,X}]^{t}=\mathbb{E}[e^{{\rm i}\,u\,Z_{t}}]=e^{t\,\Phi(u)},\quad
t>0,\;\;u\in\mathds{R},$ (1)

where $\Phi$ is given by the following expression: for a fixed truncation
function $h$ (i.e. a bounded function such that $\lim_{x\to 0}(h(x)-x)/x^{2}$
exists, for example $h(x)=x{\rm 1\leavevmode\nobreak\ l}_{|x|\leq 1}$ or
$x/(1+x^{2})$, we have

$\Phi(u)={\rm
i}\,\mathrm{a}\,u-\mathrm{b}\,u^{2}+\int_{\mathds{R}\mathbin{\mathchoice{\mspace{-4.0mu}\raisebox{0.8pt}{\rotatebox[origin={c}]{-20.0}{$\displaystyle\smallsetminus$}}\mspace{-4.0mu}}{\mspace{-4.0mu}\raisebox{0.8pt}{\rotatebox[origin={c}]{-20.0}{$\textstyle\smallsetminus$}}\mspace{-4.0mu}}{\mspace{-4.0mu}\raisebox{0.6pt}{\rotatebox[origin={c}]{-20.0}{$\scriptstyle\smallsetminus$}}\mspace{-4.0mu}}{\mspace{-4.0mu}\raisebox{0.45pt}{\rotatebox[origin={c}]{-20.0}{$\scriptscriptstyle\smallsetminus$}}\mspace{-4.0mu}}}\\{0\\}}\big{(}e^{{\rm
i}\,u\,x}-1-{\rm i}\,u\,h(x)\big{)}\pi(dx),$ (2)

where $\mathrm{a}\in\mathds{R},\,\mathrm{b}\geq 0$ are respectively called
drift term and Brownian coefficient. Clearly, if one changes the truncation
function to a new one $g$, the new drift term becomes
$\mathrm{a}_{g}:=\mathrm{a}+\int_{\mathds{R}\mathbin{\mathchoice{\mspace{-4.0mu}\raisebox{0.8pt}{\rotatebox[origin={c}]{-20.0}{$\displaystyle\smallsetminus$}}\mspace{-4.0mu}}{\mspace{-4.0mu}\raisebox{0.8pt}{\rotatebox[origin={c}]{-20.0}{$\textstyle\smallsetminus$}}\mspace{-4.0mu}}{\mspace{-4.0mu}\raisebox{0.6pt}{\rotatebox[origin={c}]{-20.0}{$\scriptstyle\smallsetminus$}}\mspace{-4.0mu}}{\mspace{-4.0mu}\raisebox{0.45pt}{\rotatebox[origin={c}]{-20.0}{$\scriptscriptstyle\smallsetminus$}}\mspace{-4.0mu}}}\\{0\\}}\big{(}g(x)-h(x)\big{)}\pi(dx)$.
The measure $\Pi$ is called the Lévy measure and satisfies the integrability
condition

$\int_{{\mathds{R}\mathbin{\mathchoice{\mspace{-4.0mu}\raisebox{0.8pt}{\rotatebox[origin={c}]{-20.0}{$\displaystyle\smallsetminus$}}\mspace{-4.0mu}}{\mspace{-4.0mu}\raisebox{0.8pt}{\rotatebox[origin={c}]{-20.0}{$\textstyle\smallsetminus$}}\mspace{-4.0mu}}{\mspace{-4.0mu}\raisebox{0.6pt}{\rotatebox[origin={c}]{-20.0}{$\scriptstyle\smallsetminus$}}\mspace{-4.0mu}}{\mspace{-4.0mu}\raisebox{0.45pt}{\rotatebox[origin={c}]{-20.0}{$\scriptscriptstyle\smallsetminus$}}\mspace{-4.0mu}}}\\{0\\}}}(x^{2}\wedge
1)\pi(dx)<\infty.$ (3)

  * •

Spectrally negative Lévy processes on the real line have a particular
interest. They correspond to Lévy processes $(Z_{t})_{t\geq 0}$ with no
positive jumps, viz. such that the associated Lévy measure $\Pi$ in (2) gives
no mass to $(0,\infty)$. Defining

$\mbox{$\Pi$ be the image of $\pi$ by the reflection $x\mapsto-x,\quad$ and
$\quad\chi(x):=-h(-x),\quad\mbox{$h$ given by \eqref{LK}}$},$ (4)

it becomes more handy to use the so-called Lévy-Laplace exponent $\Psi$
instead of the Lévy-Fourier exponent $\Phi$:

$\Psi(\lambda):=\lim_{t\to
0}\frac{\log\operatorname{\mathbf{E}}[e^{\lambda\,Z_{t}}]}{t}=\Phi(-{\rm
i}\lambda)=\mathrm{a}\lambda\,+\mathrm{b}\,\lambda^{2}+\int_{(0,\infty)}\big{(}e^{-\lambda
x}-1+\lambda\,\chi(x)\big{)}\Pi(dx),\quad\lambda\geq 0,$ (5)

where $\Pi$ satisfies the integrability condition (3). The class
$\mathcal{LE}$ of Laplace exponents is defined as the set of functions $\Psi$
of the form (5) with $\chi(x)=x$, i.e.

$\mathcal{LE}:=\big{\\{}\Psi(\lambda)=\mathrm{a}\lambda+\mathrm{b}\lambda^{2}+\int_{(0,\infty)}\big{(}e^{-\lambda
x}-1+\lambda x\big{)}\,\Pi(dx),\quad\lambda\geq 0\big{\\}},$ (6)

where $\mathrm{a}\in\mathds{R}$, $\mathrm{b}\geq 0$, and the Lévy measure
$\Pi$ satisfies integrability condition

$\int_{(0,\infty)}(x^{2}\wedge x)\Pi(dx)<\infty.$ (7)

We define the subclass $\overline{\mathrm{I\,D}}_{-}(\mathds{R})$ and
$\mathrm{I\,D}_{-}(\mathds{R})$ of $\mathrm{I\,D}(\mathds{R})$ as follows:

$X\sim\overline{\mathrm{I\,D}}_{-}(\mathds{R})\;\big{(}\mbox{respectively}\;\mathrm{I\,D}_{-}(\mathds{R})\big{)}\quad\mbox{if}\quad\operatorname{\mathbf{E}}[e^{\lambda
X}]^{t}=\operatorname{\mathbf{E}}[e^{\lambda
Z_{t}}]=e^{t\Psi(\lambda)},\;\;t>0\quad\mbox{and $\Psi$ is as in
$\eqref{ird}\;\,\big{(}$respectively $\Psi\in\mathcal{LE}\big{)}$.}$ (8)

For Lévy processes, the references are numerous, we suggest the books of
Bertoin bert , Kyprianou kyp  or Sato sato . The following proposition
explains why $\overline{\mathrm{I\,D}}_{-}(\mathds{\mathds{R}})$ is the vague
the closure of $\mathrm{I\,D}_{-}(\mathds{R})$:

###### Proposition 1.1

We have the equivalences:

    1. 1)

$X\sim\overline{\mathrm{I\,D}}_{-}(\mathds{\mathds{R}})\Longleftrightarrow
X\stackrel{{\scriptstyle d}}{{=}}\lim_{n}X_{n},\;$ with
$X_{n}\sim\mathrm{I\,D}_{-}(\mathds{R})$.

    2. 2)

$X\sim\mathrm{I\,D}(\mathds{R})\Longleftrightarrow$ there exist two sequences
of independent r.v’s
$X_{n}^{\prime},\,X_{n}^{\prime\prime}\sim\mathrm{I\,D}_{-}(\mathds{R})$ such
that $X\stackrel{{\scriptstyle
d}}{{=}}\lim_{n}(X_{n}^{\prime}-X_{n}^{\prime\prime})$.

From the latter, we see that $X\sim\mathrm{I\,D}(\mathds{R})$ if, and only if
$X=X^{\prime}-X^{\prime\prime}$, where $X^{\prime}$ and $X^{\prime\prime}$ are
independent and
$X^{\prime},\;X^{\prime\prime}\sim\overline{\mathrm{I\,D}}_{-}(\mathds{\mathds{R}})$.

  * •

The class $\mathcal{BF}$ of Bernstein functions is defined by

$\mathcal{BF}:=\big{\\{}\phi(\lambda)=\mathrm{d}\lambda+\int_{(0,\infty)}\big{(}1-e^{-\lambda
x}\big{)}\,\Pi(dx),\quad\lambda\geq 0\big{\\}},$ (9)

where $\mathrm{d}\geq 0$ and the Lévy measure $\Pi$ satisfies the
integrability condition

$\int_{(0,\infty)}(x\wedge 1)\Pi(dx)<\infty.$ (10)

We emphasize that, traditionally, a function of the form $\phi+\mathrm{q}$ is
called Bernstein function, when $\phi\in\mathcal{BF}$ and the addition of so-
called killing term $\mathrm{q}\geq 0$, transforms probability measures into
sub-probability measures. In order to avoid discussions on sub-probability
measures, we only consider the class $\mathcal{BF}$. Observe that a
differentiable function $\Psi$ on $(0,\infty)$ belongs to $\in\mathcal{LE}$
if, and only if, $\operatorname{\mathbf{E}}[Z_{1}]=\Psi^{\prime}(0)$ is finite
and $\Psi^{\prime}-\Psi^{\prime}(0)\in\mathcal{BF}$. Also observe that and if
that $\phi\in\mathcal{BF}$ is equivalent to the fact that $\phi$ is
nonnegative and $\phi^{\prime}$ belongs to the class of completely monotone
functions and this yields $-\phi\in\mathcal{LE}$. The class $\mathcal{CM}$ of
completely monotone functions corresponds to those infinitely differentiable
functions $f:(0,{\infty})\to(0,{\infty})$, such that

$(-1)^{n}f^{(n)}\geq 0,\quad\text{for all $n=0,1,2\cdots\;$}.$

Bernstein characterized the class $\mathcal{CM}$ by

$\mathcal{CM}:=\big{\\{}f(\lambda)=\int_{[0,\infty)}e^{-\lambda
x}\nu(dx),\quad\lambda>0,\quad\mbox{where $\nu$ is some Radon measure on
$[0,\infty)$}\big{\\}}.$ (11)

As for Lévy-Laplace exponents, the class $\mathcal{BF}$ is one-to-one with the
set of infinitely divisible distributions supported by $\mathds{R}_{+}$:

we denote $X\sim\mathrm{I\,D}(\mathds{R}_{+}),\;\;$ if $\;\;X\geq 0$ and
$X\sim\mathrm{I\,D}(\mathds{R})$,

which is equivalent to $X\geq 0$ and $-X\sim\mathrm{I\,D}_{-}(\mathds{R})$. In
this case, the embedding Lévy process $(Z_{t})_{t\geq 0}$ in (1) is a
subordinator (i.e. a Lévy process with increasing paths) and the Lévy-
Khintchine formula takes the form

$X\sim\mathrm{I\,D}(\mathds{R}_{+})\Longleftrightarrow\operatorname{\mathbf{E}}[e^{-\lambda
X}]^{t}=\operatorname{\mathbf{E}}[e^{-\lambda
Z_{t}}]=e^{-t\phi(\lambda)},\;\;t\geq 0,\;\;\lambda\geq
0,\quad\mbox{and}\;\;\phi\in\mathcal{BF}.$ (12)

The class $\mathrm{L}_{0}(\mathds{R})$ of selfdecomposable distributions has
the following description, among others:

$\mbox{we denote}\;\mu\in\mathrm{L}_{0}(\mathds{R}),\quad\mbox{if, for all
$c\in(0,1)$, there exists a measure $\mu_{c}$
s.t.}\;\;\mu=T_{c}\mu\ast\nu_{c}.$ (13)

where, for Borel sets $B,\;T_{c}\mu(B)=\mu(c^{-1}B)$. The latter has several
equivalent formulations that can be found in JV1983  and in the book of Sato
sato :

$\displaystyle X\sim\mathrm{L}_{0}(\mathds{R})$
$\displaystyle\Longleftrightarrow$ $\displaystyle X\stackrel{{\scriptstyle{\rm
d}}}{{=}}c\,X+V_{c},\quad\mbox{where $V_{c}$ is independent of $X$
(necessarily $V_{c}\sim\mathrm{I\,D}(\mathds{R})$),}$
$\displaystyle\Longleftrightarrow$ $\displaystyle
X\sim\mathrm{I\,D}(\mathds{R}),\;\;\mbox{and for each $0<c<1$, and each Borel
set $B\subset\mathds{R}$, the L\'{e}vy measure
satisfies},\;\;\Pi(B)-\Pi(c^{-1}B)\geq 0,$ $\displaystyle\Longleftrightarrow$
$\displaystyle\Pi(dx)=\frac{k(x)}{x}dx,\quad\mbox{where the function $k$ is
non-increasing on $(-{\infty},0)$ and on $(0,{\infty})$}.$ (15)

Observe that

$X\sim\mathrm{L}_{0}(\mathds{R})\;\;\mbox{and}\;\;X\geq
0\Longleftrightarrow\mbox{the $k$-function in \eqref{sdp} is such that
$\;k(x)=0$, for $x<0$}.$

In 1973, Urbanik urb72b ; urbanik1 , introduced a decreasing sequence
$\mathrm{L}_{n}(\mathds{R}),\;n=1,\ldots$, of limiting distributions (for some
specified triangular arrays) contained in the class
$\mathrm{L}_{0}(\mathds{R})$, that is,

$\mathrm{L}_{\infty}(\mathds{R})=\bigcap_{k\geqslant
0}\mathrm{L}_{k}(\mathds{R})\subset\ldots\subset\mathrm{L}_{n}(\mathds{R})\subset\ldots\subset\mathrm{L}_{1}(\mathds{R})\subset\mathrm{L}_{0}(\mathds{R}).$
(16)

Probability measures in $\mathrm{L}_{n}(\mathds{R})$ are called _$n$ -times
selfdecomposable measures._ Urbanik characterized multiple selfdecomposable
distributions by their characteristic functions in urbanik  and (urbanik1, ,
Theorem 1 and 2). Then Urbanik gave in urb72b  the results without detailed
proofs. His proofs in urbanik ; urb ; urb1972  used the Choquet-Krein-Milman
theorem on extreme points in compact convex sets; as a reference for this cf.
Phelps Ph . A probabilistic proof, using random integral representations, is
in JU2 . Urbanik’s description of the classes $L_{n}(\mathds{R})$, in terms of
the convolution factorization is given in (urbanik1, , Proposition 1): with
the convention $\mathrm{L}_{-1}(\mathds{R}):=\mathrm{I\,D}(\mathds{R})$,

$\mu\in\mathrm{L}_{n}(\mathds{R}),\quad\mbox{if, for all $c\in(0,1)$, there
exists $\mu_{c}\in\mathrm{L}_{n-1}(\mathds{R})\;\;$
s.t.}\;\;\mu=T_{c}\mu\ast\nu_{c},$ (17)

For more convenience, we denote

$\mathrm{L}_{n}(\mathds{\mathds{R}_{+}}):=\big{\\{}\mu\in\mathrm{I\,D}(\mathds{R}_{+});\;\mbox{\it
such that}\;\mu\in\mathrm{L}_{n}(\mathds{R})\big{\\}}.$

Note that $\mathrm{I\,D}(\mathds{A})$ and
$\mathrm{L}_{n}(\mathds{A}),\;\mathds{A}=\mathds{R}_{+},\;\mathds{R}$, are
closed (in the weak convergence topology) convolutions semigroups. If
$\mu\in\mathrm{L}_{n}(\mathds{R})\setminus\mathrm{L}_{n+1}(\mathds{R})$, then
we say that $\mu$ is _exactly $n$-times selfdecomposable._ For operator-
selfdecomposability problems, we refer the book of Jurek & Mason JU  and
Meerschaert & Scheffler Meer , for distributional properties of
selfdecomposable distributions we recommend the book of Sato sato . For
analytic properties related to infinitely divisible distributions on the half-
real line, we recommend the book of Schilling, Song & Vondraček SSV  and also
Steutel & van Harn steutel . Several other proofs for the characterization of
$\mathrm{L}_{n}(\mathds{R}_{+})$ are $\mathrm{L}_{n}(\mathds{R})$ are
available and the references are multiple, cf. Berg & Forst berg , Jurek JU2 ;
JU0 ; JU ; JUS , Van Thu van  and also Steutel & van Harn steutel  and the
references therein.

In Section 2, we provide several new decomposability properties for the Gamma
function, the Gamma distributions, and the positive stable distribution. For
this purpose, we consider, for $\alpha\in(0,1)\cup(1,\infty)$ and $t>0$, the
following functions:

$G_{\alpha}(\lambda)=\frac{\Gamma(\lambda)^{\alpha}}{\Gamma(\alpha\lambda)},\;\;\lambda>0\quad\mbox{and}\quad
G_{\alpha,t}(\lambda)=\Gamma(t)^{1-\alpha}\;\alpha^{\alpha\lambda}\,\,\frac{\Gamma(\lambda+t)^{\alpha}}{\Gamma(\alpha\lambda+t)},\;\;\lambda\geq
0,$ (18)

and notice the following relation

$G_{\alpha,t}(\lambda)=\frac{1}{G_{\frac{1}{\alpha},t}(\alpha\lambda)^{\alpha}}.$
(19)

  1. 1.

Theorem 2.1 retrieves Berg, Çetinkayab & Karp’s result (bergkarp, , Theorem
3.13) and gives a stochastic interpretation. The latter and Theorem 2.4
provide the exact range of the parameters $\alpha$ and $t$ for which

$G_{\alpha,t}\;\;\left(\mbox{respecively}\;\;1/G_{\alpha,t}\right)\;\;\mbox{is
the Laplace transforms of a r.v $X_{\alpha,t}$ (respecively $Y_{\alpha,t}$)
with distribution in $\mathrm{I\,D}(\mathds{R}_{+})$ and
$\mathrm{L}_{0}(\mathds{R}_{+})$}.$

  2. 2.

Corollary 30 proves that if $\alpha_{1},\alpha_{2},\ldots,\alpha_{n}\in(0,1)$,
$\sum_{k=1}^{n}\alpha_{k}=1$, and if
$\mathbb{G}_{t},\mathbb{G}_{1,t},\,\ldots,\,\mathbb{G}_{n,t}$, are independent
random variables with Gamma distribution with shape parameter $t>0$, then

$\mathbb{G}_{t}\stackrel{{\scriptstyle{\rm
d}}}{{=}}d({\underline{\alpha}})\;\,\mathbb{G}_{1,t}^{\alpha_{1}}\,\ldots\,\mathbb{G}_{n,t}^{\alpha_{n}}\,\;e^{-X_{\underline{\alpha},t}}\Longleftrightarrow
t\geq 1/2\quad\mbox{and}\quad
d({\underline{\alpha}})\;\,\mathbb{G}_{1,t}^{\alpha_{1}}\,\ldots\,\mathbb{G}_{n,t}^{\alpha_{n}}\stackrel{{\scriptstyle{\rm
d}}}{{=}}\mathbb{G}_{t}\,\;e^{-Y_{\underline{\alpha},t}}\Longleftrightarrow
t<1/2,$

where $d(\underline{\alpha}):=\prod_{k=1}^{n}\alpha_{k}^{\alpha_{k}}$ and all
r.v.’s are assumed to be independent in each side of the identities and
$X_{\underline{\alpha},t},\;Y_{\underline{\alpha},t}\sim\mathrm{I\,D}(\mathds{R}_{+})$.
Analytically, the latter is transcripted in terms of the Gamma function with
the following remarkable fact:

$\lambda\mapsto\left(\frac{\Gamma(\lambda+t)}{d(\underline{\alpha})^{\lambda}\prod_{k=1}^{n}\Gamma(\alpha_{k}\lambda+t)}\right)^{r}\;\mbox{is
completely monotone},$ (20)

if, and only if $t\geq\frac{1}{2}$ and $r>0$ or $t<\frac{1}{2}$ and $r<0$.
This results generalizes several known results in the literature, c.f Alzer &
Berg Alzer , Bertoin & Yor BY , Li & Chen lichen , Mehrez tun , Pestana,
Shanbhag & Sreehari PSS , for instance. We also explore whenever
$X_{\underline{\alpha},t},\;Y_{\underline{\alpha},t}\sim\mathrm{L}_{0}(\mathds{R}_{+})$
or $\mathrm{BO}$, ($\mathrm{BO}$ being the Bondesson class of distributions,
cf. (SSV, , Definition 9.1)).

  3. 3.

We improve Akita & Maejima’s makoto  results who proved that
$\log\mathbb{G}_{t}$ is twice selfdecomposable for any $t\geq 1/2$ and that
there exists a universal constant $t_{1}\in(0,1/2)$ such that the last
property could be extended for $t\in(t_{1},+\infty)$. Using tools developed in
Section 4, we were able in Proposition 2.7 to show that the value of $t_{1}$
is the maximum of an explicit elementary function, and is approximately
0.151649938034.

Section 3 prepares for the results of Section 4 and Lemma 3.4 gives the full
characterization of the class

$\mathcal{M}_{n}:=\big{\\{}k,\;\mbox{s.t. $\;x\mapsto k(e^{x})\;$ is
$n$-monotone on $\mathds{R}$}\big{\\}},$ (21)

cf. Williamson wil  for $n$-monotone functions. Using the so-called Euler-
Mellin differential operator $\Theta=x\;d/dx$ and the so-called Hadamard
fractional integral, we provide the converse of (SSV, , Proposition 1.16) and,
the analog of (SSV, , Theorem 4.11) for the class $\mathcal{M}_{n}$.

In Section 4, mainly in Corollary 4.4, we shall provide a simple proof for the
characterization of the classes $\mathrm{L}_{n}(\mathds{R})$ based on the
class $\mathcal{M}_{n}$ given in (21). Jurek had provided the same
characterization (JU1, , Theorem 6.2 and Theorem 7.1) for infinitely divisible
measures on Banach spaces. Our approach may have the merit to exhibit the
tight link between the Mellin-Euler differential $\Theta$ operator and the
classes $\mathrm{L}_{n}(\mathds{R})$ and suggests that a mechanism producing
new classes of infinitely divisible distribution via linear operators $\Omega$
other than $\Theta$ could be implemented as follows: assume $\Omega$ is a
linear operator on the space of functions $f:[0,\infty)\to[0,\infty)$ which
commute with the dilations, i.e. there exists

$\Omega(x\mapsto f(ux))=\Omega(f)(ux),\quad\mbox{for all}\;x,\;u>0.$

For instance, if $\Omega=\Theta$, then $\Omega(\mathcal{BF})\cap\mathcal{BF}$
is one-to-one with $\mathrm{L}_{n}(\mathds{R}_{+})$ and if $\Omega=I-\Theta$,
then $\Omega(\mathcal{BF})\cap\mathcal{BF}$ is not void. Now, let $\phi$ be a
Bernstein function of the form

$\phi(\lambda)=\mathrm{d}\;\lambda+\int_{(0,\infty)}(1-e^{-\lambda
x})\;\;\frac{k(x)}{x}dx=\mathrm{d}\lambda+\int_{(0,\infty)}(1-e^{-x})\,\tilde{k}(\lambda/x)\,\frac{dx}{x},\quad\lambda\geq
0.$

where $\mathrm{d}\geq 0$ and assume that $\tilde{k}(x):=k(1/x)$ is “good
enough" so that we can swap $\Omega$ and the integral. If
$\Omega^{n},\;n=1,2,\ldots$, is the $n$-th iterate of $\Omega$ and
$\omega:=\Omega(Identity)(1)$, then,

$\Omega^{n}(\phi)(\lambda)=\omega\;\mathrm{d}\;\lambda+\int_{(0,\infty)}(1-e^{-\lambda
x})\;k_{n}(x)\;\frac{dx}{x},\quad\mbox{where}\quad
k_{n}(x):=\Omega^{n}(\tilde{k})(1/x).$

Thus, it is simply seen that $\Omega^{n}(\phi)$ remains a Bernstein function
if $k_{n}$ complies with the non-negativity and integrability conditions
imposed on Lévy measures. With this mechanism, one is able to build new
subclasses of $\mathrm{I\,D}(\mathds{R}_{+})$, eventually nested. When
$\Omega=\Theta$, we obtain in Corollary 4.4 the exact shape of the function
$k_{n}$ (for Bernstein or Lévy-Laplace exponents) of
$X\sim\mathrm{I\,D}(\mathds{R}_{+})$ or $\mathrm{I\,D}_{-}(\mathds{R})$ or
$\overline{\mathrm{I\,D}}_{-}(\mathds{R})$ for which it holds that

$X\sim\mathrm{L}_{n}(\mathds{R}_{+})\;\;\mbox{or}\;\;\mathrm{L}_{n}(\mathds{R}),\quad
n=1,\,2,\ldots,\infty.$

Section 5 improves the last approach with a method based on integral
stochastic representations like (117). All the proofs were postponed to
Section 6 and Section 7 gives some background on the classes
$\mathrm{I\,D}(\mathds{R})$ and $\mathrm{L}_{0}(\mathds{R})$.

##  2 New decomposability properties for the Gamma function, the Gamma
distributions and the positive stable distribution

This section provides new multiple selfdecomposability properties involving
the Gamma function and the Gamma and positive stable distributions.

Recall the Digamma function is defined by
$\psi(t)=\Gamma^{\prime}(t)/\Gamma(t),\;t>0$ and is given by formula 5 p.903
grad :

$\psi(t)=-\gamma+\,\int_{0}^{\infty}\frac{e^{-x}-e^{-tx}}{1-e^{-x}}dx,\quad
t>0,$

where $\gamma$ is the Euler-Mascheroni constant. Since
$\log\Gamma(\lambda)=\int_{1}^{\lambda}\Psi(t)dt,\;\lambda>0$, we recover the
following representations,

$\Gamma(\lambda)=\exp\left\\{-\gamma(\lambda-1)+\int_{0}^{\infty}\\!\\!(e^{-\lambda
u}-e^{-u}-(\lambda-1)ue^{-u})\,\frac{du}{u(1-e^{-u})}\right\\},\quad\lambda>0$

and then,

$\frac{\Gamma(\lambda+t)}{\Gamma(t)}=\exp\left\\{\psi(t)\lambda+\int_{0}^{\infty}\\!\\!(e^{-\lambda
u}-1+\lambda u)\,\frac{e^{-tu}}{u(1-e^{-u})}\;du\right\\},\quad\lambda\geq
0,\;t>0.$ (22)

The $q$-Gamma function $\Gamma_{q}(x),\;x>0$, defined by

$\Gamma_{q}(x)=\left\\{\begin{array}[]{lll}&(1-q)^{1-x}\;\prod_{j=0}^{\infty}\frac{1-q^{j+1}}{1-q^{j+x}},&\quad\mbox{if}\;0<q<1,\\\
&\mbox{and}&\\\
&(q-1)^{1-x}\;q^{x(x-1)/2}\;\prod_{j=0}^{\infty}\frac{1-q^{-(j+1)}}{1-q^{-(j+x)}},&\quad\mbox{if}\;q>1,\end{array}\right.$
(23)

enjoys the basic property: $\lim_{q\to 1-}\Gamma_{q}(x)=\lim_{q\to
1+}\Gamma_{q}(x)=\Gamma(x)$.

From now on, we denote by $\mathbb{G}_{t}$ a random variable with the standard
Gamma distribution with shape parameter $t>0$, which has the density function,
Laplace and Mellin transforms respectively given by

$f_{\mathbb{G}_{t}}(x)=\frac{x^{t-1}}{\Gamma(t)}\,e^{-x},\quad\operatorname{\mathbf{E}}[e^{-\lambda\mathbb{G}_{t}}]=\frac{1}{(1+\lambda)^{t}}\quad\mbox{and}\quad\operatorname{\mathbf{E}}[\mathbb{G}_{t}^{\lambda}]={\Gamma(t+\lambda)\over\Gamma(t)},\quad
x>0,\;\;\lambda>-t.$ (24)

The function $\lambda\mapsto\lambda^{\alpha},\,0<\alpha<1,$ is a generic
example of a Bernstein functions. It is not difficult to derive the
representation

$\lambda^{\alpha}=\frac{\alpha}{\Gamma(1-\alpha)}\int_{0}^{\infty}(1-e^{-\lambda
x})\frac{dx}{x^{\alpha+1}},\quad\lambda\geq 0,$ (25)

and this function is associated to the so-called standard positive stable r.v.
$\mathbb{S}_{\alpha}$ with stability parameter $\alpha\in(0,1)$ via:

$\operatorname{\mathbf{E}}[e^{-\lambda\mathbb{S}_{\alpha}}]=e^{-\lambda^{\alpha}},\quad\lambda\geq
0,\qquad\mbox{and}\qquad\operatorname{\mathbf{E}}[(\mathbb{S}_{\alpha})^{-\lambda}]=\frac{\Gamma(1+\frac{\lambda}{\alpha})}{\Gamma(1+\lambda)},\quad\lambda>-\alpha.$
(26)

The p.d.f. of $\mathbb{S}_{\alpha}$ is not explicit, except for $\alpha=1/2$
where $S_{1/2}\stackrel{{\scriptstyle{\rm d}}}{{=}}1/(4\;\mathbb{G}_{1/2})$,
cf. the monograph of Zolotarev zolo  for more account.

###  2.1 Elementary decomposability properties for Gamma and stable
distributions

Using the representations (22) and (24), we deduce that the functions, given
for $\lambda\geq 0$, by

$\displaystyle\phi_{\mathbb{G}_{t}}(\lambda)$ $\displaystyle=$
$\displaystyle-\log\operatorname{\mathbf{E}}[e^{-\lambda\mathbb{G}_{t}}]=\int_{0}^{\infty}\big{(}1-e^{-\lambda
x}\big{)}\,\frac{k_{1}(x)}{x}\,dx,\quad k_{1}(x)=e^{-x},$
$\displaystyle\Psi_{\mathbb{G}_{t}}(\lambda)$ $\displaystyle=$
$\displaystyle\log\operatorname{\mathbf{E}}[(\mathbb{G}_{t})^{\lambda}]=\int_{0}^{\infty}\big{(}e^{-\lambda
x}-1+\lambda x\big{)}\,\frac{k_{2}(x)}{x}\,dx,\quad
k_{2}(x)=\frac{e^{-tx}}{1-e^{-x}},$ (27)

are respectively in $\mathcal{BF}$ and $\mathcal{LE}$, and that
$k_{1},\,k_{2}$ are non increasing (cf. also (JU10, , p. 98) for $k_{2}$). By
(15), it is immediate that $\mathbb{G}_{t}\sim\mathrm{L}_{0}(\mathds{R}_{+})$
and
$\log\mathbb{G}_{t}\sim\overline{\mathrm{I\,D}}_{-}(\mathds{R})\cap\mathrm{L}_{0}(\mathds{R})$;
cf. Jurek (JU10, , p.98) for the function $k_{2}$, see JV2019  for the Back
driving Lévy process (BDLP)of $\log\mathbb{G}_{t}$, cf. Section for BDLP’s.

Recall that the distribution of positive stable r.v. $\mathbb{S}_{\alpha}$ is
given by (26). As for the r.v. $\mathbb{G}_{t}$, one can deduce from identity
(22), that the functions, given for $\lambda\geq 0$, by

$\displaystyle\phi_{\mathbb{S}_{\alpha}}(\lambda)$ $\displaystyle=$
$\displaystyle-\log\operatorname{\mathbf{E}}[e^{-\lambda\mathbb{S}_{\alpha}}]=t\,\lambda^{\alpha}=\int_{0}^{\infty}\big{(}1-e^{-\lambda
x}\big{)}\,\frac{k_{3}(x)}{x}\,dx,\quad
k_{3}(x)=\frac{\alpha}{\Gamma(1-\alpha)\,x^{\alpha}},$
$\displaystyle\Psi_{\mathbb{S}_{\alpha}}(\lambda)$ $\displaystyle=$
$\displaystyle\log\operatorname{\mathbf{E}}[(\mathbb{S}_{\alpha})^{-\lambda}]=\int_{0}^{\infty}\big{(}e^{-\lambda
x}-1+\lambda x\big{)}\,\frac{k_{4}(x)}{x}\,dx,\quad
k_{4}(x)=\frac{1}{1-e^{-\alpha x}}-\frac{1}{1-e^{-x}}$

are respectively in $\mathcal{BF}$ and $\mathcal{LE}$, and that
$k_{3},\,k_{4}$ are non-increasing. By Corollary 4.4, it is immediate that
$\mathbb{S}_{\alpha}\sim\mathrm{L}_{0}(\mathds{R}_{+})$ and
$-\log\mathbb{S}_{\alpha}\sim\mathrm{I\,D}_{-}(\mathds{\mathds{R}})\cap\mathrm{L}_{0}(\mathds{R})$.

###  2.2 Main results, new decomposability properties

After completing this work, we discovered the recent result of Berg,
Çetinkayab & Karp (bergkarp, , Theorem 3.13) on the function $G_{\alpha,t}$
given in (18), which is equivalent to Theorem 2.1. Our Theorem provides a
stochastic interpretation of (bergkarp, , Theorem 3.13) and which is shown
with a different proof. Furrther, our investigation of the function
$G_{\alpha,t}$ goes beyond by seeking for selfdecomposability properties in
Theorem 2.4, Corollary 30 and Proposition 2.7 below.

###### Theorem 2.1

For the function $G_{\alpha,t},\;t>0$, given in (18), we have the equivalences
between the assertions $(i),\;(ii)$ and $(iii)$ in each of the points 1) and
2).

  1. 1)

    1. (i)

for all $r>0$, the function $(G_{\alpha,t})^{r}\in\mathcal{CM}$;

    2. (ii)

the function $G_{\alpha,t}$ is the Laplace transform of some positive
infinitely divisible random variable $X_{\alpha,t}$:

$G_{\alpha,t}(\lambda)=\operatorname{\mathbf{E}}\left[e^{-\lambda\,X_{\alpha,t}}\right],\quad\lambda\geq
0;$

    3. (iii)

$\alpha\in(0,1)$ and $t\geq 1/2\;$ or $\;\alpha\in(1,\infty)$ and $t<1/2$.

  2. 2)

    1. (i)

the function $(G_{\alpha,t})^{-r}\in\mathcal{CM}$, for all $r>0$;

    2. (ii)

the function $(G_{\alpha,t})^{-1}$ is the Laplace transform of some positive
infinitely divisible random variable $Y_{\alpha,t}$:

$\frac{1}{G_{\alpha,t}(\lambda)}=\operatorname{\mathbf{E}}\left[e^{-\lambda
Y_{\alpha,t}}\right],\quad\lambda\geq 0;$

    3. (iii)

$\alpha\in(0,1)$ and $t<1/2\;$ or $\;\alpha\in(1,\infty)$ and $t\geq 1/2$.

###### Remark 2.2

Equality (19), is equivalent to

$\operatorname{\mathbf{E}}\left[e^{-\lambda
X_{\alpha,t}}\right]=\operatorname{\mathbf{E}}\left[e^{-\alpha\lambda
Y_{1/\alpha,t}}\right]^{\alpha},\quad\lambda\geq 0,$

and this strengthens the fact that $X_{\alpha,t}$ and $Y_{1/\alpha,t}$ are
concomitantly infinitely divisible.

Theorem 2.1 provides the same result as the one of Lv et al. (LV, , Theorem
1.10) for $G_{\alpha,1}$. Using the fact that the pointwise limit of a
sequence of completely monotone functions is also completely monotone, then
using Theorem 2.1 and the fact that the limit

$G_{\alpha}(\lambda)=\lim_{t\to
0}\,\Gamma(t)^{\alpha-1}\,\alpha^{-\alpha\lambda}\,G_{\alpha,t}(\lambda),$

we immediately retrieve the results of Li & Chen (lichen, , Theorem 9) and
also of Alzer & Berg (Alzer, , Theorem 3.5) on the functions $G_{\alpha}$.
Mehrez (tun, , Theorem 1) obtained the same result in the context of the
$q$-analog of the function $G_{\alpha}$, i.e. when it is build with the
$q$-Gamma function defined in (23).

###### Corollary 2.3

We have the following results for the function $G_{\alpha}$ given in (18).

  1. 1)

The function $\lambda\mapsto G_{\alpha}(\lambda)^{r}$, is completely monotone
for all $r>0$, if, and only if, $\;\alpha\in(1,\infty)$;

  2. 2)

The function $\lambda\mapsto G_{\alpha}(\lambda)^{-r}$, is completely monotone
for all $r>0$, if, and only if, $\;\alpha\in(0,1)$.

###### Theorem 2.4

Let $t_{0}=\frac{1}{2}+\frac{1}{2\sqrt{3}}$. It holds that

  1. 1)

$X_{\alpha,t}\sim\mathrm{L}_{0}(\mathds{R}_{+})$ if, and only if,
$\alpha\in(0,1)$ and $t\geq t_{0}$;

  2. 2)

$Y_{\alpha,t}\sim\mathrm{L}_{0}(\mathds{R}_{+})$ if, and only if,
$\alpha\in(1,\infty)$ and $t\geq t_{0}$.

In (SSV, , Lemma 92. Theorem 9.7), it is shown that

$\mathrm{ME}\subset\mathrm{BO}\subset\mathrm{I\,D}(\mathds{R}_{+})$

where the Bondesson class $\mathrm{BO}$ is characterized as the smallest class
of probability measures on $(0,{\infty})$ which contains $\mathrm{ME}$, the
class of mixture of exponentials distribution, and which is closed under
convolutions and vague limits. Actually, the class $\mathrm{BO}$ is one-to-one
to the class of complete Bernstein functions, viz. of functions of the form

$\phi(\lambda)=\mathrm{d}\lambda+\int_{(0,{\infty})}\frac{\lambda}{\lambda+x}\frac{\nu(dx)}{x}=\mathrm{d}\lambda+\int_{0}^{\infty}(1-e^{-\lambda
x})\,\mathcal{L}\nu(x)\,dx,$ (28)

where $\quad\mathcal{L}\nu(x)=\int_{(0,{\infty})}e^{-x\;u}\nu(du).$
Furthermore, the representation

$\nu(dx)=x\;\eta(x),\quad\mbox{where $\;\eta:(0,{\infty})\to[0,1],\;$ is
measurable},$ (29)

is equivalent to say that the Bernstein function $\phi$ is associated to a
distribution in $\mathrm{ME}$, cf. (SSV, , Theorem 9.5). As a consequence of
Theorems 2.1 and 2.4, we obtain the following factorizations for the Gamma
distributions.

###### Corollary 2.5

Let $\;t>0$,
$\underline{\alpha}=(\alpha_{1},\alpha_{2},\ldots,\alpha_{n})\in(0,1)^{n}$
such that $\sum_{k=1}^{n}\alpha_{k}=1$,
$d({\underline{\alpha}}):=\prod_{k=1}^{n}\alpha_{k}^{\alpha_{k}}$ and recall
the random variables $X_{\alpha_{k},t}$ and $Y_{\alpha_{k},t}$ given by
Theorem 2.1. Let $\mathbb{G}_{t}$ be a Gamma distributed random variable with
shape parameter $t$ and $\mathbb{G}_{1,t},\,\ldots,\,\mathbb{G}_{n,t}$ denote
independent copies of $\mathbb{G}_{t}$. Assuming that the random variables
involved in the next factorizations in law are independent, we have the
following results.

  1. 1)

If $t\geq 1/2$ , then we have the factorization in law

$\mathbb{G}_{t}\stackrel{{\scriptstyle{\rm
d}}}{{=}}d({\underline{\alpha}})\;\,\mathbb{G}_{1,t}^{\alpha_{1}}\,\ldots\,\mathbb{G}_{n,t}^{\alpha_{n}}\,\;e^{-X_{\underline{\alpha},t}},\quad\mbox{where}\;\;X_{\underline{\alpha},t}:=\sum_{k=1}^{n}X_{\alpha_{k},t}.$
(30)

  2. 2)

If $\;0<t<1/2$ , then we have the factorization in law

$d({\underline{\alpha}})\;\,\mathbb{G}_{1,t}^{\alpha_{1}}\,\ldots\,\mathbb{G}_{n,t}^{\alpha_{n}}\stackrel{{\scriptstyle{\rm
d}}}{{=}}\mathbb{G}_{t}\,\;e^{-Y_{\underline{\alpha},t}},\quad\mbox{where}\;\;Y_{\underline{\alpha},t}:=\sum_{k=1}^{n}Y_{\alpha_{k},t}.$
(31)

  3. 3)

If $\;t\geq t_{0}=\frac{1}{2}+\frac{1}{\sqrt{12}}$, then
$X_{\underline{\alpha},t},\;Y_{\underline{\alpha},t}\sim\mathrm{L}_{0}(\mathds{R}_{+})$.

  4. 4)

We have
$X_{\underline{\alpha},t},\;Y_{\underline{\alpha},t}\sim\mathrm{BO},\;$ if,
and only if, $\;t\geq 1$. In this case, their $(1+t)$-fold convolutions, in
the sense of (12), have distributions in $\mathrm{ME}.$

###### Remark 2.6

1) A direct application of Corollary 30 is

$\log\mathbb{G}_{t}\in\mathrm{I\,D}_{-}(\mathds{\mathds{R}})\cap\mathrm{L}_{0}(\mathds{R}),\;\,\mbox{if}\;\,t>1/2\quad\mbox{and}\quad\log\mathbb{G}_{t}\in\mathrm{I\,D}_{-}(\mathds{\mathds{R}})\cap\mathrm{L}_{1}(\mathds{R}),\;\,\mbox{if}\,\;t>t_{0}.$
(32)

2) The identities (30) and (31) have an interpretation in terms of the Gamma
function by taking the Mellin-transforms in both sides: for all
$(\alpha_{1},\alpha_{2},\ldots,\alpha_{n})\in(0,1)^{n}$,
$\sum_{k=1}^{n}\alpha_{k}=1$ and all $r>0$, we have

$\displaystyle\lambda\mapsto\operatorname{\mathbf{E}}[e^{-\lambda\;X_{\underline{\alpha},t}}]^{r}$
$\displaystyle=$
$\displaystyle\left(\frac{\Gamma(\lambda+t)}{d(\underline{\alpha})^{\lambda}\prod_{k=1}^{n}\Gamma(\alpha_{k}\lambda+t)}\right)^{r}\in\mathcal{CM},\quad\mbox{if
$t\geq\frac{1}{2}$}$ (33)
$\displaystyle\lambda\mapsto\operatorname{\mathbf{E}}[e^{-\lambda\;X_{\underline{\alpha},t}}]^{r}$
$\displaystyle=$
$\displaystyle\left(\frac{d(\underline{\alpha})^{\lambda}\prod_{k=1}^{n}\Gamma(\alpha_{k}\lambda+t)}{\Gamma(\lambda+t)}\right)^{r}\in\mathcal{CM},\quad\mbox{if
$t<\frac{1}{2}$}.$ (34)

The case $t=1$ has been treated by Simon (sim1, , subsection 3.4). The latter
has to be approached to Karp & Prilepkina’s recent result (karp, , Theorem 4),
which states that the function

$\lambda\mapsto\frac{\prod_{k=1}^{p}\Gamma(A_{k}\;\lambda+a_{k})}{\prod_{j=1}^{q}\Gamma(B_{j}\lambda+b_{j})},\quad
A_{k},\;B_{j}>0,\;a_{k},\;b_{j}\geq 0,$

is completely monotone if, and only if,

$\sum_{k=1}^{p}A_{k}=\sum_{j=1}^{q}B_{j},\quad\prod_{k=1}^{p}A_{k}^{A_{k}}=\sum_{j=1}^{q}B_{j}^{B_{j}}\quad\mbox{and}\quad
P(u):=\sum_{j=1}^{p}\frac{e^{-a_{j}\;u/A_{j}}}{1-e^{-a_{j}\;u/A_{j}}}-\sum_{k=1}^{q}\frac{e^{-b_{k}\;u/B_{k}}}{1-e^{-b_{k}\;u/B_{k}}}\geq
0,\quad\mbox{\it for all}\;u>0.$

As we see, our conditions in Corollary 30, for $r=1$, are not expressed in
terms of the function $P$, since we assume $p=1$, $t=a_{k}=b_{k}$ and we
require the stronger property of logarithmic complete monotonicity.

Using iterates of the shift operators $\Delta_{c}f(x)=f(x+c)-f(x)$, Akita &
Maejima (makoto, , Theorem 1) have shown that
$\log\mathbb{G}_{t}\sim\mathrm{L}(\mathds{R})$ if $t>1/2$. In their Remark 2,
they claimed :

“It is possible to extend to this property to $(t_{1},\infty)$ for some
$t_{1}\in(0,1/2)$",

that they evaluated, with numerical calculations, by $t_{1}\leq 0.152$. The
next result and (92) in the proof of Proposition (6.1) below provides the
value of $t_{1}$ as the maximum of an explicit elementary function, a quantity
that could not be computed by hand and that we evaluated by Maple.

###### Proposition 2.7

For every $t>0$ and $\alpha\in(0,1)$, we have
$\log\mathbb{G}_{t}\sim\mathrm{I\,D}_{-}(\mathds{\mathds{R}})\cap\mathrm{L}_{0}(\mathds{R})$
and the identity in law

$\log\mathbb{G}_{t}\stackrel{{\scriptstyle{\rm
d}}}{{=}}\alpha\log\mathbb{G}_{t}+T_{t,\alpha}.$ (35)

Further,
$\log\mathbb{G}_{t}\sim\mathrm{I\,D}_{-}(\mathds{\mathds{R}})\cap\mathrm{L}_{1}(\mathds{R})$,
i.e.
$T_{t,\alpha}\sim\mathrm{I\,D}_{-}(\mathds{\mathds{R}})\cap\mathrm{L}_{0}(\mathds{R})$,
if, and only if $t>t_{1}\approx 0.151649938034$. The r.v. $T_{t,\alpha}$
corresponds to $\;\log\,J_{\alpha,\alpha t}\;$ in (36).

###  2.3 Comments on the factorizations

We start by observing the following:

###### Proposition 2.8

If ${(\alpha_{k})}_{k\geq 1}$ is a sequence of non increasing positive numbers
such that $\sum_{k=1}^{\infty}\alpha_{k}=1$ then the sequences
$X_{\underline{\alpha},t}$ and $Y_{\underline{\alpha},t}$ of Corollary 30
converge in distribution as $n\to\infty$.

Let $\mathbb{B}_{s,t},\,s,t>0$, denotes a Beta-distributed random variable
with probability density function

$\frac{1}{B(s,s+t)}\,x^{s-1}(1-x)^{s+t-1},\quad 0<x<1.$

It is also worth noticing the following facts.

  1. 1.

In a private communication, the first author provided to Bertoin & Yor their
Lemma 1 in BY , which shows a factorization close to (30). This factorization
states that if $\mathbb{S}_{\alpha},\;0<\alpha<1$, is a standard positive
stable r.v. and if $0<\alpha t<s$, then we have the following factorization in
law: with the convention $\mathbb{B}_{t,0}=1$, and

$\operatorname{\mathbf{P}}(\mathbb{S}_{\alpha,s}\in
dx)=\frac{\operatorname{\mathbf{P}}(\mathbb{S}_{\alpha}\in
dx)}{\operatorname{\mathbf{E}}[\mathbb{S}_{\alpha}^{-s}]\;x^{s}},\quad x>0,$

we have

$\mathbb{G}_{t}\stackrel{{\scriptstyle{\rm
d}}}{{=}}\mathbb{G}_{s}^{\alpha}\;\mathbb{J}_{\alpha,s},\qquad\mbox{where}\;\;\mathbb{J}_{\alpha,s}\stackrel{{\scriptstyle{\rm
d}}}{{=}}\frac{\mathbb{B}_{t,\frac{s}{\alpha}-t}}{\left(\mathbb{S}_{\alpha,s}\right)^{\alpha}},\quad\mbox{(on
the r.h.s, the r.v.'s are independent)},$ (36)

  2. 2.

Identity (30) has to be compared with Gordon’s one (gordon, , Theorem 6): if
$p\geq 2$ is an integer, then

$\frac{\mathbb{G}_{pt}}{p}\stackrel{{\scriptstyle{\rm
d}}}{{=}}\left(\mathbb{G}_{t}\,\mathbb{G}_{t+\frac{1}{p}}\,\ldots\mathbb{G}_{t+\frac{p-1}{p}}\right)^{\frac{1}{p}},\quad\mbox{(on
the r.h.s, the r.v.'s are assumed to be independent)}.$ (37)

As an immediate consequence of Gordon’s factorization and the one in (30), we
recover a new independent factorization in law for the Beta distributions:
Using the r.v. $X_{\underline{\alpha},t}$ in (30) and the Gamma-Beta algebra,
we get

$p\,\mathbb{B}_{t,(p-1)t}\stackrel{{\scriptstyle{\rm
d}}}{{=}}e^{-X_{\underline{\alpha},t}}\left(\mathbb{B}_{t,\frac{1}{p}}\,\mathbb{B}_{t,\frac{2}{p}}\ldots\mathbb{B}_{t,\frac{p-1}{p}}\right)^{\frac{1}{p}},\quad\mbox{(on
the r.h.s, the r.v.'s are independent)}.$ (38)

  3. 3.

Let $0<\alpha\leq 1$ and $\beta=1-\alpha$. Motivated by the
selfdecomposability property of the r.v. $\log(\mathbb{S}_{\alpha})$, Pestana,
Shanbhag & Sreehari PSS , explored the structure of the r.v. $V_{\alpha}$
intervening in the so-called Kanter’s identity which involves the exponential
and the positive stable distributions:

$\mathbb{S}_{\alpha}^{-\alpha}\;\stackrel{{\scriptstyle{\rm
d}}}{{=}}\;\mathbb{G}_{1}^{\beta}{e^{-V_{\alpha}}},\quad\mbox{(on the r.h.s,
the r.v.'s are assumed to be independent)}.$ (39)

Using the well known independent factorisation in law
$\mathbb{G}_{1}\stackrel{{\scriptstyle{\rm
d}}}{{=}}\left(\mathbb{G}_{1}/\mathbb{S}_{\alpha}\right)^{\alpha}$, which is
easily justified by

$\operatorname{\mathbf{P}}(\mathbb{G}_{1}^{1/\alpha}>\lambda)=\operatorname{\mathbf{P}}(\mathbb{G}_{1}>\lambda^{\alpha})=e^{-\lambda^{\alpha}}=\operatorname{\mathbf{E}}[e^{-\lambda\mathbb{S}_{\alpha}}]=\operatorname{\mathbf{P}}(\mathbb{G}_{1}>\lambda\mathbb{S}_{\alpha})=\operatorname{\mathbf{P}}\left(\frac{\mathbb{G}_{1}}{\mathbb{S}_{\alpha}}>\lambda\right),\quad\lambda\geq
0,$

then taking $n=2$ and $(\alpha_{1},\alpha_{2})=(\alpha,\beta)$ in (30), we
retrieve:

$\mathbb{G}_{1}\stackrel{{\scriptstyle{\rm
d}}}{{=}}\left(\frac{\mathbb{G}_{1}}{\mathbb{S}_{\alpha}}\right)^{\alpha}\stackrel{{\scriptstyle{\rm
d}}}{{=}}\alpha^{-\alpha}\;\beta^{-\beta}\;\,\mathbb{G}_{1,1}^{\alpha}\;\,\mathbb{G}_{1,2}^{\beta}\,\;e^{-X_{\underline{\alpha},1}}\Longrightarrow\mathbb{S}_{\alpha}^{-\alpha}\;\stackrel{{\scriptstyle{\rm
d}}}{{=}}\;\alpha^{-\alpha}\;\beta^{-\beta}\;\,\mathbb{G}_{1}^{\beta}{e^{-V_{\alpha}}}\Longrightarrow
V_{\alpha}\stackrel{{\scriptstyle{\rm
d}}}{{=}}X_{\underline{\alpha},1}+\alpha\log\alpha+\beta\log\beta,$

where, on the r.h.s, the r.v.’s are assumed to be independent. Observe that

$\operatorname{\mathbf{E}}[e^{-\lambda
V_{\alpha}}]=\frac{\Gamma(1+\lambda)}{\Gamma(1+\lambda\alpha)\Gamma(1+\lambda\beta)},\quad\lambda\geq
0,\quad\mbox{and that}\quad
X_{\underline{\alpha},1}\sim\mathrm{L}_{0}(\mathds{R}_{+})\cap\mathrm{ME},$

constitutes an additional information to Kanter’s factorization (39).

##  3 The Multiplicative convolution, the Euler-Mellin differential operator
$\Theta$ and $n$-monotone functions

To provide a simple characterization of multiple selfdecomposability, we give
some account on Euler-Mellin differential operator $\Theta=x\,d/dx$ and its
relationship with the multiplicative convolution and the concept of
$n$-monotone functions.

We recall that the Mellin convolution (or multiplicative convolution) of two
measures $\mu$ and $\nu$ on $(0,{\infty})$ is defined by:

$\mu\odot\nu(A)=\int_{(0,{\infty})^{2}}{\rm 1\leavevmode\nobreak\
l}_{A}(xy)\mu(dx)\nu(dy),\qquad\mbox{if}\;A\;\mbox{is a Borel set
of}\;(0,{\infty})\,.$

If $\mu$ is absolutely continues with density function $f$, then $\mu\odot\nu$
is the function given by

$\mu\odot\nu(x)=f\odot\nu(x)=\int_{(0,{\infty})}f\left(\frac{x}{y}\right)\frac{\nu(dy)}{y},\quad
x>0.$

Notice that the integrals above may be infinite when $\mu$ (and/or $\nu$) is
not a finite measure.

The Euler-Mellin differential $\Theta$ and its discrete version $\theta_{c}$,
defined by

$\Theta(g)(x)=xg^{\prime}(x)\quad\mbox{and}\quad\theta_{c}(g)(x)=g(x)-g(x/c),\qquad
x,\;c>0.$ (40)

will be needed in the sequel, for an account on $\Theta$ operators, we suggest
Butzer . The iterates of $\Theta$ are denoted by $\Theta^{n},\;n\geq 2$ and
$\theta_{c_{1}}\theta_{c_{2}}\ldots\theta_{c_{n}}$ denotes the composition of
$\theta_{c_{1}},\;\theta_{c_{2}},\ldots,\theta_{c_{n}},\;c_{1},c_{2}\ldots,c_{n}\in(0,1)$.
The following result will be also needed in the sequel.

###### Lemma 3.1

The following is true for every function
$g:(0,\infty)\longrightarrow\mathds{R}$ and $c\in(0,1)\cup(1,\infty)$:

  1. 1)

If $g$ is decreasing and $g(0+)<\infty$, then the Frullani integral

$\int_{0}^{\infty}\frac{g(x)-g(x/c)}{x}\,dx=\int_{0}^{\infty}\frac{\theta_{c}(g)(x)}{x}\,dx,$

equals to $\big{(}g(\infty)-g(0+)\big{)}\log c.$

  2. 2)

Assume further that $g$ is differentiable. Then,

    1. (i)

the function $x\mapsto\theta_{c}(g)(x)/x$ is increasing (resp. decreasing),
for every fixed $c\in(0,1)$, if, and only if, $g$ is convex (resp. concave);

    2. (ii)

the function $x\mapsto\theta_{c}(g)(x)$ is increasing (resp. decreasing), for
every fixed $c\in(0,1)$, if, and only if, $x\mapsto\Theta g(x)$ is decreasing
(resp. increasing).

###### Remark 3.2

Since the operators $\Theta$ and $\theta_{c}$ commute, it is easily seen that
statement $2)(ii)$ in Lemma 3.1 extends to $n$-times differentiable functions
$g$ via the iterates of $\Theta$ and the compositions of the operators
$\theta_{c}$. Then, the two following statements are equivalent:

  1. (i)

The functions $x\mapsto\theta_{c_{1}}\theta_{c_{2}}\ldots\theta_{c_{n}}(g)(x)$
are increasing (resp. decreasing), for every
$c_{1},c_{2}\ldots,c_{n}\in(0,1)$;

  2. (ii)

The function $x\mapsto(-1)^{n}\Theta^{n}(g)(x)\;$ is increasing (resp.
decreasing).

It what follows, our aim is to characterise the class $\mathcal{M}_{n}$
defined in (21).

  * •

Williamson wil  introduced the class of $n$-monotone functions on $(0,\infty)$
that one can extend to functions
$f:(a,\infty)\to\mathds{R},\;a\in[-\infty,\infty)$, by reproducing the same
arguments of Schilling, Song & Vondraček, in (SSV, , Theorem 1.11 and the
discussion p.12 given in case $a=-\infty$). For this, just observe that $f$ is
$n$-monotone on $(a,\infty)$ if, and only, if $f(x+x_{0})$ is $n$-monotone on
$(0,\infty)$ for every $x_{0}>a$. Hence, we will say that $f$ is 1-monotone if
$f(x)\geq 0$ for all $x>a$ and if $f$ is non-increasing and right-continuous.
The function $f$ is $n$-monotone on $(a,\infty)$, $n=2,3,\cdots,$ if it is
$n-2$ times differentiable,

$(-1)^{j}\,f^{(j)}(x)\geq 0,\quad\mbox{for all}\;x>a,\;\;j=0,1,\cdots,n-2,$
(41)

and $(-1)^{n-1}\,f^{(n-2)}$ is non-negative, non-increasing and convex on
$(a,\infty)$.

  * •

Further, with the adaptation of (SSV, , Theorem 1.11), we can affirm that $f$
is $n$-monotone on $(a,\infty)$ if, and only if, $f$ has the representation

$f(x)=\mathrm{c}+\int_{(x,\infty)}(u-x)^{n-1}\,\nu(du),\quad x>a\\\ $ (42)

for some $\mathrm{c}\geq 0$ and some measure $\nu$ on $(a,\infty)$.

  * •

Similarly, $f$ is completely monotone on $\mathds{R}$ if and only if, it is
$n$-monotone on $(a,\infty)$, for all $n\geq 1$, and all $a<0$; and the latter
ensures that $f(x),\;x\in\mathds{R}$, is also represented as in (11), with
some measure $\nu$ on $(0,\infty)$.

By (42), observe that a function $f$ is $n$-monotone on $(0,\infty)$ if, and
only, if it is represented by

$f(x)=\mathrm{c}+\big{(}(1-u)_{+}^{n-1}\odot\mu\big{)}(x),\quad x>0.$

for some $\mathrm{c}\geq 0$ and some measure $\mu\;\big{(}\mbox{take
$\mu(du)=u^{n}\;\nu(du)$ in \eqref{tmo} }\big{)}$.

We will now illustrate to which extent the class of $n$-monotone functions is
intimately related to Euler-Mellin’s operator. The iterates of the usual
differential operator and of $\Theta$ are linked by these relations: if $g$ is
$n$ times differentiable on some interval $I$, then

$x^{n}\,g^{(n)}(x)=\sum_{m=0}^{n}s(n,m)\,\Theta^{m}(g)(x)\qquad\mbox{and}\qquad\Theta^{n}(g)(x)=\sum_{m=0}^{n}S(n,m)\,x^{m}\,g^{(m)}(x),\quad
x\in I,$ (43)

where $s(n,m)$ and $S(n,m)$, $0\leq k\leq n,$ denote the Stirling numbers of
the first and second kind, respectively given by the positive numbers

$s(n,k)=\frac{1}{k!}\frac{d^{n}}{dx^{n}}(\log(x+1))^{k}_{|x=0}\;\;\mbox{and}\;\;S(n,m)=\frac{1}{m!}\frac{d^{n}}{dx^{n}}(e^{x}-1)^{m}_{|x=0},$

cf. Butzer . Notice that $S(n,k)$ is also defined as the number of partitions
of the set $\\{1,\cdots,n\\}$ into exactly $k$ nonempty subsets. It is also
known that $s(n,m)=(-1)^{n-m}[_{m}^{n}]$, where $[_{m}^{n}]$ is the number of
permutations in the symmetric group of order $n$ with exactly $k$ cycles.
Using (43), write

$x^{n}\,(-1)^{n}\,g^{(n)}(x)=\sum_{m=0}^{n}[_{m}^{n}]\,(-1)^{m}\Theta^{m}(g)(x)$

and it is immediate that

$(-1)^{m}\Theta^{m}(g)(x)\geq 0,\quad\forall
m=0,1,\cdots,n\quad\Longrightarrow\quad(-1)^{n}g^{(n)}(x)\geq 0\,.$ (44)

Now, assuming that $g(0)=0$ and $g$ is $n$ times differentiable on
$(0,{\infty})$, consider $h_{n}=(-1)^{n}\Theta^{n}(g)$. If the functions
$h_{m}:=(-1)^{m-1}\Theta(h_{m-1})=(-1)^{m}\Theta^{m}(g)$ are such that
$h_{m}(\infty)=0$ and $h_{m}(u)/u$ is integrable at $\infty$, for all
$m=1,\cdots,n$, then

$h_{n-1}(x)=\int_{x}^{\infty}\frac{h_{n}(u)}{u}du=\int_{0}^{\infty}h_{n}\left(\frac{x}{u}\right){\rm
1\leavevmode\nobreak\ l}_{(0,1]}(u)\,\frac{du}{u}=({\rm 1\leavevmode\nobreak\
l}_{(0,1]}\odot h_{n})(x),\quad x>0,$

then, iterating, we get following inversion formulae

$g=({\rm 1\leavevmode\nobreak\ l}_{(0,1]})^{\odot{n-k}}h_{k}=({\rm
1\leavevmode\nobreak\ l}_{(0,1]})^{\odot{n}}\odot h_{n}.$

Observing that

$\left({\rm 1\leavevmode\nobreak\ l}_{(0,1]}\right)^{\odot
n}(x)=\frac{|\log|^{n}(x)}{n!}{\rm 1\leavevmode\nobreak\ l}_{(0,1]}(x),$ (45)

one retrieves that $g$ is expressed by

$g(x)=\int_{x}^{\infty}\log^{n}\left(\frac{u}{x}\right)\frac{h_{n}(u)}{u}du\quad
x>0.$ (46)

Of course, this discussion is informal, but it illustrates the fact that the
transform $h_{n}\mapsto g$ in (46) is the inverse of the operator
$(-1)^{n}\Theta^{n}$. Hence, for "good" functions $g$, for instance if
$(-1)^{m}\Theta^{m}(g)\geq 0,\;m=1,\ldots,n,\,$ we will have

$g(x)=\int_{x}^{\infty}\log^{n}\left(\frac{u}{x}\right)\,(-1)^{n}\Theta^{n}(g)(u)\,\frac{du}{u}\quad
x>0.$

The transform $h\mapsto g$ in (46), is known as the Hadamard integral of order
$n+1$, cf. (btroj, , (18.43) p. 330) and btroj1 . Further, observing that the
difference operators $\Delta_{c}(f)(x)=f(x+c)-f(x),\;c>0$ are linked to our
$\theta_{d}$ operators by

$\Delta_{c}\big{(}y\mapsto
k(e^{y})\big{)}(x)=k(e^{c}\,e^{x})-k(e^{x})=\theta_{e^{c}}(k)(e^{c+x}),\quad
c>0,$ (47)

and observing that if $k$ is $m$-times differentiable on $(0,{\infty})$, then

$(-1)^{m}\frac{d^{m}}{dx}k(e^{x})=(-1)^{m}\Theta^{m}(k)(e^{x}),\quad
x\in\mathds{R},$ (48)

and it appears natural to introduce the following class:

###### Definition 3.3

Let $n=1,2,\cdots$. A function $k:(0,{\infty})\to(0,{\infty})$, is said to be
$\Theta_{n}$-monotone functions, and we denote $k\in\mathcal{M}_{n}$, if
$x\mapsto k(e^{x})$ is $n$-monotone on $\mathds{R}$. The function $k$ is
$\Theta_{\infty}$-completely monotone if $x\mapsto k(e^{x})$ is completely
monotone.

By (41) and (44), clearly,

$k\in\mathcal{M}_{n}\;\Longleftrightarrow\left\\{\begin{array}[]{lll}&&k_{m}:=(-1)^{m}\Theta^{m}(k)\geq
0,\quad\mbox{for all}\;m=0,1,\cdots,n-2,\\\ &&\mbox{and}\\\ &&\mbox{$k_{n-2}$
is non-negative, non-increasing and $x\mapsto k_{n-2}(e^{x})$ is
convex}\end{array}\right.$ (49)

With these conditions, necessarily $k_{n}$ is also convex and, by (47), it
becomes clear that

$k\in\mathcal{M}_{n}\;\mbox{(resp.
$\mathcal{M}_{\infty}$)}\;\Longrightarrow\;k\;\mbox{is $n$-monotone on
$(0,\infty)$ (resp. completely monotone).}$ (50)

If $k$ is $n$-times differentiable, then by Remark 3.2,

$k\in\mathcal{M}_{n}\;\Longleftrightarrow\;(-1)^{n}\Theta^{n}(k).$ (51)

The implication (50), left to right, was observed in (SSV, , Proposition 1.16)
and the proof there is based on induction on $n$, without formalizing the
class $\mathcal{M}_{n}$.

Finally, after our discussion, adapting (SSV, , Theorem 4.11) for $n$-monotone
functions, using (47) and the $n$-th power multiplicative convolution of the
function ${\rm 1\leavevmode\nobreak\ l}_{(0,1]}$ given by (45), we easily
deduce the full characterization of $\mathcal{M}_{n}$, i.e., the converse to
(SSV, , Proposition 1.16) and the analog of (SSV, , Theorem 4.11) on multiply
monotone functions, for the class $\mathcal{M}_{n}$:

###### Lemma 3.4

Let $k:(0,\infty)\to(0,\infty)$ and $n\geq 1$ . Then, the following assertions
are equivalent.

  1. 1)

$k\in\mathcal{M}_{n}$;

  2. 2)

$\theta_{c_{1}}\ldots\theta_{c_{m}}(k)\geq 0,$ for all $m=1,2,\cdots,n\,$ and
$\,c_{1},\cdots,c_{n}\in(0,1)$;

  3. 3)

$k$ is of the form

$k(x)=\mathrm{c}+\left(\left({\rm 1\leavevmode\nobreak\
l}_{(0,1]}\right)^{\odot\,n-1}\odot\mu\right)(x)=\mathrm{c}+\frac{1}{(n-1)!}\int_{(x,\infty)}\log^{n-1}\left(\frac{y}{x}\right)\,\frac{\mu(dy)}{y},\quad
x>0,$ (52)

where $\mathrm{c}\geq 0$ and the measure $\mu$ is such that
$\int_{1}^{\infty}\log^{n-1}(y)\;\frac{\mu(dy)}{y}<\infty$.

Furthermore, $k\in\mathcal{M}_{\infty}$ if, and only if, it is represented by

$k(x)=\mathrm{c}+\int_{(0,\infty)}\frac{1}{x^{u}}\,\nu(du),\quad x>0,$ (53)

with some finite measure $\nu$ on $(0,\infty)$.

###### Remark 3.5

The integral transform of $\mu$, in (52), is the measure version of the
Hadamard integral of order $n$ in (46).

##  4 Simple characterization of multiple selfdecomposable distributions

By the relations

$\displaystyle\Theta(\phi)(\lambda)$ $\displaystyle=$
$\displaystyle\frac{d}{dc}\left(e^{-\theta_{1/c}(\phi)(\lambda)}\right)|_{c=1}$
$\displaystyle\theta_{1/c}(\phi)(\lambda)$ $\displaystyle=$
$\displaystyle\phi(\lambda)-\phi(c\lambda)=\int_{c\lambda}^{\lambda}\phi^{\prime}(t)dt=\int_{c}^{1}\lambda\phi^{\prime}(s\lambda)ds=\int_{c}^{1}\Theta(\phi)(s\lambda)\frac{ds}{s}$

and the fact that $\mathcal{BF}$ is a closed convex cone, a simple proof for
the characterization of $\mathrm{L}_{0}(\mathds{R}_{+})$ could be provided.
For instance, see Aguech and Jedidi aj , Behme behme  and Mai, Schenk &
Scherer MSS  for the following characterization.

######  Theorem 4.1 (aj ; behme ; MSS )

Let $X$ be a nonnegative r.v. with cumulant function
$\phi(\lambda)=-\log\operatorname{\mathbf{E}}[e^{-\lambda X}],\;\lambda\geq
0.\,$ Recall that $\Theta(\phi)$ and $\theta_{d}(\phi)$ are given by (40).

  1. 1)

If $\theta_{c}(\phi)\in\mathcal{BF}$ for some $c>1$ or if
$\Theta(\phi)\in\mathcal{BF}$, then $\phi\in\mathcal{BF}$.

  2. 2)

The following assertions are equivalent.

    1. (i)

$X\sim\mathrm{L}_{0}(\mathds{R}_{+})$;

    2. (ii)

$\theta_{d}(\phi)$ is a cumulant function, for all $\;d>1$;

    3. (iii)

$\theta_{d}(\phi)\in\mathcal{BF}$, for all $\;d>1$;

    4. (iv)

$\Theta(\phi)\in\mathcal{BF}$;

    5. (v)

$\phi\in\mathcal{BF}$ and is represented by

$\phi(\lambda)=\mathrm{d}\lambda+\int_{0}^{\infty}\big{(}1-e^{-\lambda
x}\big{)}\,\frac{k(x)}{x}\,dx,\quad\lambda\geq{\rm 0}$ (54)

where $\mathrm{d}\geq 0$ and $k$ is a non-increasing function such that
$\int_{0}^{1}k(x)\,dx+\int_{1}^{\infty}\frac{k(x)}{x}\,dx<\infty$.

###### Remark 4.2

Representation (54) explains identity (1). Indeed, we have

$\operatorname{\mathbf{E}}[e^{-\lambda
Y_{c}}]=\frac{\operatorname{\mathbf{E}}[e^{-\lambda
X}]}{\operatorname{\mathbf{E}}[e^{-\lambda
cX}]}=e^{-\theta_{1/c}(\phi)(\lambda)},$

and then, after an elementary change of variable, we obtain the representation

$\theta_{1/c}(\phi)(\lambda)=(1-c)\,\mathrm{d}\,\lambda+\int_{(0,\infty)}(1-e^{-\lambda
x})\frac{\theta_{c}(k)(x)}{x}dx.$ (55)

Since $Y\sim\mathrm{L}_{0}(\mathds{R}_{+})$, then $\theta_{c}(k)(x)/x$ is
necessarily the density function of a Lévy measure.

Using Proposition 1.1 and mimicking the proof of Theorem 4.1, we can state
this proposition without proof:

###### Proposition 4.3

We have
$\mathrm{L}_{0}(\mathds{\mathds{R}})\subset\mathrm{I\,D}(\mathds{\mathds{R}})$
and the following holds true.

  1. 1)

Let $X\sim\overline{\mathrm{I\,D}}_{-}(\mathds{\mathds{R}})$. Then, the
following assertions are equivalent.

    1. (i)

$X\sim\mathrm{L}_{0}(\mathds{R})$;

    2. (ii)

The Lévy-Laplace exponent $\Psi\in\mathcal{LE}$ associated to $X$ satisfies
$\theta_{c}\Psi\in\mathcal{LE}$ for every $c>0$;

    3. (iii)

$\Theta(\Psi)$ has the form

$\Psi(\lambda)=\mathrm{a}\lambda+\mathrm{b}\lambda^{2}+\int_{0}^{\infty}\big{(}e^{-\lambda
x}-1+\lambda\chi(x)\big{)}\frac{k(x)}{x}dx,\quad\lambda\geq 0$ (56)

for some $\mathrm{a}\in\mathds{R},\,\mathrm{b}\geq 0$, some truncation
function $\chi$ as in (4) and some non-increasing function $k$.

  2. 2)

$X\sim\mathrm{I\,D}_{-}(\mathds{\mathds{R}})\cap\mathrm{L}_{0}(\mathds{R})$
if, and only if, its associated Lévy-Laplace exponent $\Psi$ has the form (56)
with $\chi(x)=x$, or equivalently $\Theta(\Psi)\in\mathcal{LE}$.

  3. 3)

Vague closure: $X\sim\mathrm{L}_{0}(\mathds{\mathds{R}})$ if, and only if,
there exist two sequences of independent r.v’s
$X_{n}^{\prime},\,X_{n}^{\prime\prime}\sim\overline{\mathrm{I\,D}}_{-}(\mathds{\mathds{R}})\cap\mathrm{L}_{0}(\mathds{R})$,
such that $X\stackrel{{\scriptstyle
d}}{{=}}\lim_{n}(X_{n}^{\prime}-X_{n}^{\prime\prime})$.

To characterize the class $\mathrm{L}_{n}(\mathds{R})$, it is enough to use
Proposition 4.3 and to focus on the class
$\mathrm{I\,D}_{-}(\mathds{\mathds{R}})\cap\mathrm{L}_{0}(\mathds{R})$ of
real-valued selfdecomposable r.v.’s $X$, having a Lévy-Laplace exponent $\Psi$
of the form

$\Psi(\lambda)=\mathrm{a}\lambda+\mathrm{b}\lambda^{2}+\int_{(0,\infty)}(e^{-\lambda
x}-1+\lambda x)\;\frac{k(x)}{x}dx,\quad\lambda\geq 0.$ (57)

for some $\mathrm{a}\in\mathds{R}$, $\mathrm{b}\geq 0$ and some non-increasing
function $k$ satisfying the condition (7), i.e.:

$\int_{(0,1)}xk(x)dx+\int_{[1,\infty)}k(x)dx<\infty.$

Using the compositions of the Euler-Mellin operators $\Theta,\,\theta_{c}$
given by (40) and right after, and also the class of $\mathcal{M}_{n+1}$ given
by Definition 3.3, multiple selfdecomposability property is simply
characterized as follows:

######  Corollary 4.4 (Characterization of multiple selfdecomposability by the
Euler-Mellin operator)

Let $n\geq 1$.

  1. 1)

Assume one of the following

    1. (a)

$X\sim\mathrm{L}_{0}(\mathds{R}_{+})$ and is associated with a Bernstein
function $\phi$ and with the $k$-function given by (54);

    2. (b)

$X\sim\mathrm{I\,D}_{-}(\mathds{\mathds{R}})\cap\mathrm{L}_{0}(\mathds{R})$
and is associated to a Lévy-Laplace exponent $\Psi$ to the $k$-function given
by (57).

Then, the following assertions are equivalent.

    1. (i)

$X\sim\mathrm{L}_{n}(\mathds{R}_{+})\;\big{(}\mbox{\it
respectively}\;X\sim\mathrm{I\,D}_{-}(\mathds{\mathds{R}})\cap\mathrm{L}_{n}(\mathds{R})\big{)}$;

    2. (ii)

$k\in\mathcal{M}_{n+1}$, i.e., is represented by

$k(x)=\left(\left({\rm 1\leavevmode\nobreak\
l}_{(0,1]}\right)^{\odot\,n-1}\odot\mu\right)(x)=\frac{1}{(n-1)!}\int_{(x,\infty)}\log^{n-1}\left(\frac{y}{x}\right)\,\frac{\mu(dy)}{y},\quad
x>0$ (58)

and $\mu(dx)/x$ is a measure satisfying (10) and the additional integrability
conditions at infinity

$\int_{(1,\infty)}\log^{n+1}(x)\frac{\mu(dx)}{x}<\infty,$ (59)

(respectively satisfying only (3) if
$X\sim\mathrm{I\,D}_{-}(\mathds{\mathds{R}})\cap\mathrm{L}_{0}(\mathds{R})$);

    3. (iii)

For all $\;m=1,\cdots,n+1$ and $d_{1},\cdots,d_{n+1}>1,\;$ the function

$(\theta_{d_{1}}\ldots\theta_{d_{m}})(\phi)\quad\big{(}\mbox{respectively},\;(\theta_{d_{1}}\ldots\theta_{d_{m}})(\Psi)\big{)}$

is the Bernstein function of some r.v.
$Y_{m}\sim\mathrm{I\,D}(\mathds{R}_{+})\;\;\big{(}$respectively, is the
Laplace-exponent of some r.v. $Y_{m}\sim\mathrm{I\,D}_{-}(\mathds{R})\big{)}$;

    4. (iv)

For all $m=1,\cdots,n+1$,
$\;\Theta^{m}(\phi)\in\mathcal{BF}\;\;\big{(}$respectively,
$\;\Theta^{m}(\Psi)\in\mathcal{LE}\big{)}$;

    5. (v)

Let $f=\phi$ (respectively, $\,f=\Psi$). There exists
$Y\sim\mathrm{I\,D}(\mathds{R}_{+})\;\;\big{(}$respectively,
$Y\sim\mathrm{I\,D}_{-}(\mathds{R})\big{)}$ such that we have the
representation

$f(\lambda)=\frac{1}{(n-1)!}\int_{1}^{\infty}g\left(\frac{\lambda}{x}\right)\;\log^{n}(x)\frac{dx}{x},\quad\lambda\geq
0,$ (60)

where $g$ is the Bernstein function (respectively Laplace exponent) of $Y$.

    6. (vi)

$\displaystyle X\stackrel{{\scriptstyle{\rm
d}}}{{=}}\int_{(0,{\infty})}e^{-s^{1/(n+1)}}dZ_{s},\,$ where $Z$ is some
subordinator (respectively, some spectrally negative Lévy process) such that

$\operatorname{\mathbf{E}}[\log^{n+1}(1+Z_{1})]<\infty\quad\mbox{(respectively
$\operatorname{\mathbf{E}}[\log^{n+1}(1+|Z_{1}|)]<\infty$)}.$ (61)

Furthermore, in (v) and (vi), the r.v.’s $Y$ and $Z_{1}$ have the same
distribution.

  2. 2)

$X\sim\overline{\mathrm{I\,D}}_{-}(\mathds{\mathds{R}})\cap\mathrm{L}_{n}(\mathds{R})$
if, and only if, the associated $k$-function and $\mu$-measure satisfy (58)
and (59).

  3. 3)

We have the same equivalences as in 1) for $n=\infty$, with the following
additional conditions on support and integrability on the $\nu$-measure in the
representation (53) of $k\in\mathcal{M}_{\infty}$:

    1. (i)

$\nu$ is supported by $(0,1)$, in case
$X\in\mathrm{L}_{\infty}(\mathds{R}_{+})$ (respectively $(1,2)$, in case
$X\in\mathrm{I\,D}_{-}(\mathds{R})\cap\mathrm{L}_{\infty}(\mathds{R})$), and
satisfies

$\int_{(0,1)}\frac{1}{x(1-x)}\nu(dx)<\infty\;\;(\mbox{respectively},\;\;\int_{(1,2)}\frac{1}{(x-1)(2-x)}\nu(dx)<\infty).$

The latter is equivalent to the representation

$\phi(\lambda)=\mathrm{d}\lambda+\int_{(0,1)}\lambda^{x}\,\frac{x}{\Gamma(1-x)}\nu(dx)\;\;\big{(}\mbox{respectively},\;\;\Psi(\lambda)=\mathrm{a}\lambda+\mathrm{b}\lambda^{2}+\int_{(1,2)}\lambda^{x}\;\,\frac{x(x-1)}{\Gamma(2-x)}\;\nu(dx)\big{)},$

for some $\mathrm{d},\;\mathrm{b}\geq 0,\;\mathrm{a}\in\mathds{R}$.

    2. (ii)

$\nu$ is supported by $(0,2)$, in case
$X\in\overline{\mathrm{I\,D}}_{-}(\mathds{R})\cap\mathrm{L}_{\infty}(\mathds{\mathds{R}})$,
and satisfies $\displaystyle\int_{(0,2)}\frac{1}{x(2-x)}\nu(dx)<\infty.$

As a straightforward consequence, we recover the general case:

###### Corollary 4.5

Assume $1\leq n\leq\infty$ and $X\sim\mathrm{L}_{0}(\mathds{R})$. Then,
$X\sim\mathrm{L}_{n}(\mathds{R})$ if, and only if $X=X^{+}-X^{-}$, where
$X^{+}$ and $X^{-}$ are independent and have distributions in
$\overline{\mathrm{I\,D}}_{-}(\mathds{\mathds{R}})\cap\mathrm{L}_{n}(\mathds{R})$.

###### Remark 4.6

Formula like the above (58) appeared in Urbanik urb72b ; urbanik1 . The
stochastic integral representation in 1)(vi) of Corollary 4.4 was proved in
Jurek (JU2, , Corollary 2.11).

##  5 Possible extensions of multiple selfdecomposability through integral
stochastic representations

Observe that the integral stochastic representation (117) admits several
extensions as noticed by Jurek & Vervaat JV1983 . As in the proof of (JV1983,
, Proposition 1), we have the following: for $0<a<b$, a function of bounded
variation $h:(a,b]\to\mathds{R}$, a monotone function
$r:(0,\infty)\to(0,\infty)$, and a Lévy process $Z(t)$, the random integral

$X=I^{h,r}_{(a,b]}(Z):=\int_{(a,b]}h(s)dZ_{r(s)},$

is well defined and

$\Phi_{X}(u):=\log\mathbb{E}[e^{{\rm
i}\,u\,X}]=\int_{(a,b]}\log\mathbb{E}[e^{{\rm i}\,u\,\,h(s)Z_{1}}]dr(s),\quad
u\in\mathds{R}.$ (62)

By Jurek JU0 , the Fourier Lévy-exponents $\Phi_{Z(1)}$ and $\Phi_{X}$ are
linked by

$\Phi_{X}(u)=\int_{(a,b]}\Phi_{Z_{1}}(uh(s))dr(s).$

The following may be viewed as a particular of the above scheme. For
subordinators, Maejima mama  and Schilling, Song & Vondraček SSV  had the same
approach: let $f:(0,A)\to(0,B)$ be a strictly decreasing function and
$Z={(Z_{t})}_{t\geq 0}$ be a subordinator with associated Bernstein function
$\phi_{Z}$. Then, the r.v.

$X=\int_{(0,A)}f(s)dZ_{s}$ (63)

is a well-defined r.v. on $[0,\infty]$. Reasoning by Riemann approximation of
the stochastic integral, (SSV, , Lemma 10.1) provides the Laplace
representation

$\operatorname{\mathbf{E}}[e^{-\lambda
X}]=e^{-\phi_{X}(\lambda)},\qquad\phi_{X}(\lambda)=\int_{0}^{A}\phi_{Z}\big{(}\lambda
f(s)\big{)}ds,\quad\lambda\geq 0.$

Since $f$ admits a strictly decreasing inverse $F=(0,B)\to(0,A)$, then, by the
change of variable $s=G(y):=F(1/y)$, we see that $\phi_{X}$ takes the form of
a Mellin convolution

$\phi_{X}(\lambda)=\int_{(1/B,\infty)}\phi_{Z}\left(\frac{\lambda}{y}\right)dG(y).$
(64)

Since $\mathcal{BF}$ is a closed convex cone, then $\phi_{X}\in\mathcal{BF}$
whenever it is a well defined function on $\mathds{R}_{+}$. the latter shows
that $X\sim\mathrm{I\,D}(\mathds{R}_{+})$. Two particular cases arise for the
choice of $f$ in (63):

$\bullet$ Taking a continuous nonnegative r.v. $Y$ independent of $Z$ and
$f(s)=\operatorname{\mathbf{P}}(Y>s)$, we see that $X$ corresponds to the
conditional expectation $X=\operatorname{\mathbf{E}}[Z_{Y}|Z]$ ,and the
integral stochastic representation (117) corresponds to the case where $Z$ has
the standard exponential distribution. In particular, one can choose

$f(s)=\operatorname{\mathbf{E}}[e^{-sY}]=e^{-\phi_{Y}(s)}=\operatorname{\mathbf{P}}(\mathbb{G}_{1}/Y>s),\quad
s>0,$ (65)

where $\mathbb{G}_{1}$ is exponentially distributed and independent of $Z$ and
we are also in the previous situation:
$X=\operatorname{\mathbf{E}}[Z_{\footnotesize\mathbb{G}_{1}/Y}\;|\;Z]$.

$\bullet$ A possible extension of the latter is to take $f$ of the form

$f=e^{-\phi}\quad\mbox{and $\quad\phi:(0,A)\to(-\log B,\infty)$ differentiable
and strictly concave}.$ (66)

Observe that $\phi$ is necessarily increasing and that the inverse function
$\Psi$, of $\phi$, is a differentiable increasing and strictly convex function
on $(0,{\infty})$. Making the change of variable $s=G(y)=\Psi(\log
y),\,y>1/B$, the Bernstein function of $X$ in (64) takes the form

$\phi_{X}(\lambda)=\int_{1/B}^{\infty}\phi_{Z}\left(\frac{\lambda}{y}\right)\Psi^{\prime}(\log
y)\frac{dy}{y},\quad\;\lambda\geq 0.$ (67)

Let $\mathrm{d}_{Z},\,\Pi_{Z}$ denote the drift term and the Lévy measure in
the representation (9) of $\phi_{Z}$. Since $\Psi^{\prime}\geq 0$, then
Fubini-Tonelli applies in (67) and by a change of variable, we obtain the
following Mellin convolution representations:

$\displaystyle\phi_{X}(\lambda)$ $\displaystyle=$
$\displaystyle\mathrm{d}_{Y}\,c_{\tiny\Psi}\lambda+\int_{0}^{\infty}(1-e^{-\lambda
x})\frac{l_{X}(x)}{x}dx,\qquad
c_{\footnotesize\Psi}:=\int_{1/B}^{\infty}\Psi^{\prime}(\log
y)\frac{dy}{y^{2}},$ (68) $\displaystyle l_{X}(x)$ $\displaystyle:=$
$\displaystyle\int_{(x/B,\infty)}\Psi^{\prime}\left(\log\frac{u}{x}\right)\Pi_{Z}(du),\quad
x>0.$ (69)

Assuming (66), the following is worth to be noticed:

  1. (a)

$c_{\footnotesize\Psi}=0$ if $B=\infty$.

  2. (b)

The definiteness of $\phi_{X}$, i.e., the finiteness of the r.v. $X$ given
(63), is guaranteed by the following: if $\mathrm{d}_{Y}>0$, then the
finiteness of the integral $c_{\footnotesize\Psi}$ is required for the
definiteness of $\phi_{X}$ (i.e. $X$ is a well-defined r.v. on
$\mathds{R}_{+}$). Thus, $\phi_{X}$ is a well-defined function on
$\mathds{R}_{+}$ if, and only if the Lévy measure $\Pi_{Z}$ satisfies the
additional integrability condition

$\int_{0}^{\infty}(x\wedge
1)\,l_{X}(x)\,\frac{dx}{x}=\int_{(0,{\infty})}a_{\Psi}(u)\;\Pi_{Z}(du)<\infty,\quad$
(70)

where

$a_{\Psi}(u):=\int_{0}^{Bu}(x\wedge
1)\Psi^{\prime}\left(\log\frac{u}{x}\right)\frac{dx}{x}.$ (71)

Then, with some computation, we obtain that $a_{\Psi}$ is well defined if, and
only if $a_{\Psi}(1)=c_{\footnotesize\Psi}<\infty$. In this case, we have

$a_{\Psi}(u)=\left\\{\begin{array}[]{ll}c_{\footnotesize\Psi}u,&\quad{\rm
if}\quad 0<Bu\leq 1\\\ &\\\ \Psi(\log u)-\Psi(-\log B)+\displaystyle
u\int_{u}^{\infty}\Psi^{\prime}(\log
y)\displaystyle\frac{dy}{y^{2}},&\quad{\rm if}\quad Bu>1.\end{array}\right.$
(72)

  3. (c)

One can release the differentiability assumption on $\phi$ in case $B=1$.
Indeed, the assumption of strict concavity of $\phi$ together with its
positivity, insures that $\phi$ is strictly increasing. Thus, almost
everywhere, $\phi$ is differentiable with a strictly decreasing derivative.
Hence, almost everywhere, $\Psi$ is differentiable and $\Psi^{\prime}>0$.

  4. (d)

One can link (72) with Lemma 6.4 below, through the observation

$x_{0}=\frac{1}{B}\quad\mbox{and}\quad\rho_{h}(dy)=\frac{\Psi^{\prime}(\log
y)}{y^{2}}dy,\quad
y>\frac{1}{B}\quad\Longrightarrow\quad\chi_{h}(u)=a_{\Psi}(u),\quad
u>\frac{1}{B},$ (73)

and notice that $Z_{s},\;s>0,$ has the Lévy measure $s\;\Pi_{Z}$. Then, we
immediately obtain the following consequence which constitutes an improvement
and a simplification of the statement of Sato’s theorem (sato2007, , Theorem
2.6 and Theorem 3.5) in the case of subordinators.

###### Corollary 5.1

Let $X$ be a random variable represented by the stochastic integral in (63).
Then the following assertions are equivalent.

    1. 1)

$X$ is a well defined r.v. on $[0,\infty)$;

    2. 2)

$\operatorname{\mathbf{E}}[a_{\Psi}(Z_{1})\,{\rm 1\leavevmode\nobreak\
l}_{Z_{1}>1/B}]<\infty$

    3. 3)

$\operatorname{\mathbf{E}}[a_{\Psi}(Z_{s})\,{\rm 1\leavevmode\nobreak\
l}_{Z_{s}>1/B}]<\infty$ for all $s>0$;

    4. 4)

$\int_{(1/B,\infty)}a_{\Psi}(u)\;\Pi_{Z}(du)<\infty$.

  5. (e)

The stochastic integral representation (63) for selfdecomposable distributions
corresponds to the case where $\varphi(s)=s$, hence $\Psi^{\prime}\equiv 1$.
In all cases, the function $l_{X}$ is decreasing, because $\Psi^{\prime}$ is
increasing. Thus,

$X\;\mbox{\it is represented by \eqref{choice}, with $f$ as in
\eqref{choice2}}\Longrightarrow X\sim\mathrm{L}_{0}(\mathds{R}_{+}).$ (74)

  6. (f)

By Corollary 4.4, we know that if $X\in\mathrm{L}_{n}(\mathds{R}_{+})$, then
$\phi_{X}$ takes the form

$\phi_{X}(\lambda)=\int_{0}^{\infty}(1-e^{-\lambda x})\frac{k_{X}(x)}{x}dx,$

such that $k\in\mathcal{M}_{n+1}$ and is represented by (52):

$k_{X}(x)=\frac{1}{n!}\int_{(x,\infty)}\log^{n}\left(\frac{y}{x}\right)\,\frac{\mu_{X}(dy)}{y},\quad
x>0.$

Identifying in (65), (68) and (69), we necessarily have there

$k_{X}=l_{X},\quad\frac{\mu_{X}(dy)}{y}=\Pi_{Z}(dy),\quad
B=1\quad\mbox{and}\quad\Psi=\Psi_{Y}\;\,\mbox{s.t.}\;\,\Psi_{Y}^{\prime}(x)=\frac{x^{n}}{n!},\;x>0.$

Since $\Psi_{Y}(0)=0$, then

$\Psi_{Y}(x)=\frac{x^{n+1}}{(n+1)!},\;x>0\quad\mbox{and}\quad\phi_{Y}(s)=\Psi_{Y}^{-1}(s)=\big{(}(n+1)!\,s\big{)}^{\frac{1}{n+1}},\;s>0.$

Observe that $\phi_{Y}\in\mathcal{BF}$, and most of all,
$Y\stackrel{{\scriptstyle{\rm d}}}{{=}}\mathbb{S}_{1/(n+1)}$ the positive
$1/(n+1)$-stable r.v. given by (26). The change of variable,
$s\mapsto\Psi_{Y}(s)=s^{n+1}/(n+1)!$, in (63) with $f$ as in (65), amounts to
change in the clock in the process $Z$ : $Z_{s}\rightsquigarrow
Z_{\Psi_{Y}(s)}$. and we retrieve the following result: there exist a
subordinator $Z$, such that

$X\sim\mathrm{L}_{n}(\mathds{R}_{+})\Longleftrightarrow
X=\int_{0}^{\infty}e^{-\big{(}(n+1)!\;s\big{)}^{1/(n+1)}}\,dZ_{s}=\int_{0}^{\infty}e^{-s}\,dZ_{\Psi_{Y}(s)}.$
(75)

Further, since

$a_{\Psi}(u):=\frac{1}{n!}\int_{0}^{u}(x\wedge
1)\left(\log\frac{u}{x}\right)^{n}\frac{dx}{x}=\frac{1}{(n+1)!}\big{(}u\,{\rm
1\leavevmode\nobreak\ l}_{0<u\leq 1}+\log^{n+1}(u)\,{\rm 1\leavevmode\nobreak\
l}_{u>1}\big{)},$

then Corollary 5.1 insures that $X$ is well defined if, and only if, one of
the equivalent conditions holds

$\int_{[1,\infty)}a_{\Psi}(u)\Pi_{Z}(du)<\infty\Longleftrightarrow\operatorname{\mathbf{E}}[\log^{n+1}(1+Z_{1})]<\infty.$
(76)

Also observe that the change of variable, $s\mapsto s/(n+1)!$ amounts to
change the scale of time in the process $Z$: $Z_{s}\rightsquigarrow
Y_{s}:=Z_{s/(n+1)!}$, we simply express $\mathrm{L}_{n}$ property by:

$X\sim\mathrm{L}_{n}(\mathds{R}_{+})\Longleftrightarrow
X=\int_{0}^{\infty}e^{-s^{1/(n+1)}}\,dY_{s},\quad\mbox{for some subordinator
$Y$.}$

We emphasize that the last discussion could be rephrased identically for
$X\sim\overline{\mathrm{I\,D}}_{-}(\mathds{\mathds{R}})\cap\mathrm{L}_{n}(\mathds{R})$,
and it suggests that a fractional selfdecomposability property of type
$\mathrm{L}_{1/\alpha}(\mathds{R})$, with $\alpha\in(0,1)$ might be
interesting to be formalized, from the stochastic point of view ,since it
would be associated to the fractional version of (75):

$X\sim\mathrm{L}_{1/\alpha}(\mathds{R})\Longleftrightarrow
X=\int_{0}^{\infty}e^{-s^{\alpha}}\,dZ_{s}=\int_{0}^{\infty}\operatorname{\mathbf{E}}[e^{-s\,Y}]\,dZ_{s},$

where, with our notations in (65), $Y\stackrel{{\scriptstyle{\rm
d}}}{{=}}\mathbb{S}_{\alpha}$ the positive $\alpha$-stable r.v. given by (26).
Seeking more involved stochastic and analytical properties for the class of
r.v.’s in (63) will be, hopefully, the scope of future work.

  7. (g)

Theorem 1.1 in BFJ  is closely connected to the class of Bernstein functions
obtained in form (67) and, particularly, investigates whenever the function
$x\mapsto x^{a}\,l_{X}(x),\;a>-1$ is completely monotone. With our approach,
it is immediate that $x\mapsto x^{a}\,l_{X}(x)\in\mathcal{CM}$ for arbitrary
Lévy measures $\Pi_{Z}$, if and only if, in (68),

$B=\infty\quad\mbox{and}\quad\Psi^{\prime}(u)=\Psi_{a}^{\prime}(u):=e^{au-e^{-u}},\;\;u\in\mathds{R}.$

Assuming the latter, we have the representation

$x^{a}\,l_{X}(x)=\int_{(0,\infty)}e^{-\frac{x}{u}}\,u^{a}\,\Pi_{Z}(du),\quad\Pi_{X}(dx)=\,\frac{l_{X}(x)}{x}\,dx,\quad
x>0.$ (77)

For $a=1$ (respectively $a=0$), the shape of the Lévy measure $\Pi_{X}$
corresponds to the well known class $\mathcal{C\\!B}$ complete Bernstein
functions (respectively $\mathcal{T\\!B}$ of Thorin Bernstein functions), cf.
(SSV, , chapters 6, 8). Choosing $\Psi_{a}(-\infty)=0$, observe that the
inverse function of $f_{a}$, given by (66), is provided by

$\Psi_{a}(x)=\int_{e^{-x}}^{\infty}\frac{e^{-u}}{u^{a+1}}du,\quad
x\in\mathds{R},\qquad f_{a}^{-1}(s)=\Psi_{a}(-\log
s)=\int_{s}^{\infty}\frac{e^{-u}}{u^{a+1}}du,\quad s>0.$

Thus,

$\Psi_{a}(+\infty)=\int_{0}^{\infty}\frac{e^{-u}}{u^{a+1}}du<\infty\Longleftrightarrow
a\in(-1,0)\quad\mbox{and}\quad\Psi_{a}(+\infty)=\Gamma(-a).$

Further, using the exponential integral $Ei$ function and (grad, , 8.214
(1,2)), we have

$\Psi_{0}(x)=-Ei(-e^{-x}),\quad
x>0\Longrightarrow\lim_{x\to+\infty}x-\Psi_{0}(x)=\gamma\Longrightarrow\lim_{x\to\infty}\frac{\Psi_{0}(x)}{x}=1,$
(78)

where $\gamma$ is the Euler-Mascheroni constant. If $a>0$

$\Psi_{a}(x)=\int_{e^{-x}}^{1}\frac{e^{-u}}{u^{a+1}}du+\Psi_{a}(0)\geq\frac{e^{ax}-1}{a\,e},\quad
x>0\Longrightarrow\lim_{x\to+\infty}\frac{\Psi_{a}(x)}{x}=+\infty.$ (79)

  8. (h)

For $a\geq 0$, additional investigations on the inverse function $\phi_{a}$ of
$\Psi_{a}$ are feasible. Like $f_{a}$, the function $\phi_{a}$ is also not
explicit, but it certainly increases and satisfies

$\phi_{a}(0+)=-\infty,\quad\phi_{a}(\infty)=+\infty\quad\mbox{and}\quad\phi_{a}(s_{a})=0,\quad\mbox{where}\;\,s_{a}:=\Psi_{a}(0)=\int_{1}^{\infty}\frac{e^{-u}}{u^{a+1}}du,\quad
a\geq 0.$

Further, since

$\phi_{a}^{\prime}(x)=\frac{1}{\Psi_{a}^{\prime}(\phi_{a}(x))}=e^{-a\phi_{a}(x)+e^{-\phi_{a}(x)}}=h_{a}(\phi_{a}(x)),\quad
x>0,\qquad h_{a}(u):=e^{-au+e^{-u}}=\sum_{n=0}^{\infty}e^{-(a+n)u},$

Thus, the function
$x\mapsto\hat{\phi}_{a}(\lambda):={\phi_{a}}(s_{a}+\lambda),\;\lambda\geq 0,$
satisfies

$\hat{\phi}_{a}(0)=0,\quad\hat{\phi}_{a}^{\prime}(0)=\frac{1}{\Psi_{a}^{\prime}(0)}=e\quad\mbox{and}\quad(\hat{\phi}_{a})^{-1}(x)=:\hat{\Psi}_{a}(x)=\Psi_{a}(x)-\Psi_{a}(0)=\int_{e^{-x}}^{1}\frac{e^{-u}}{u^{a+1}}du,\;\;x\geq
0,$

Observing that $h_{a}(u),\;u>0$, is a completely monotone function, then,
using Faa di Bruno’s formula and induction, we retrieve that
${\phi_{a}^{\prime}}_{|(s_{a},\infty)}\in\mathcal{CM}$, hence we have

$\hat{\phi}_{a}^{\prime}=e^{-a\,\hat{\phi}_{a}+e^{-\hat{\phi}_{a}}}\quad\mbox{and}\quad\hat{\phi}_{a}\in\mathcal{BF}.$

Further, by (78) and (79), we have

$\mathrm{d}_{a}:=\lim_{\lambda\to\infty}\frac{\hat{\phi}_{a}(\lambda)}{\lambda}=1,\;\mbox{if}\;a=0\;\;\mbox{and}\;0\;\mbox{otherwise}.$

We deduce that $\hat{\phi}_{a}$ has the drift term $\mathrm{d}_{a}$ and is
associated to a Lévy measure $\Pi_{a}$ and to a subordinator
$Y_{a}={(Y_{a,t})}_{t\geq 0}$ through the representations:

$\hat{\phi}_{a}(\lambda)=\mathrm{d}_{a}\lambda+\int_{(0,{\infty})}(1-e^{-\lambda
x})\Pi_{a}(dx),\quad\lambda\geq 0,$

and

$\hat{\phi}^{\prime}_{a}(\lambda)=\mathrm{d}_{a}+\int_{(0,{\infty})}e^{-\lambda
x}\,x\,\Pi_{a}(dx)=e^{-a\hat{\phi}_{a}(\lambda)+e^{-\hat{\phi}_{a}(\lambda)}}=\sum_{n=0}^{\infty}\frac{e^{-(a+n)\hat{\phi}_{a}(\lambda)}}{n!}=\sum_{n=0}^{\infty}\frac{\operatorname{\mathbf{E}}[e^{-\lambda
Y_{a,a+n}}]}{n!}.$ (80)

Thus, letting $\lambda$ to 0, we get

$\lim_{\lambda\to+\infty}\hat{\phi}^{\prime}_{a}(\lambda)=\mathrm{d}_{0}=1=\sum_{n=0}^{\infty}\frac{1}{n!}\operatorname{\mathbf{P}}(Y_{0,n}=0)=1+\sum_{n=1}^{\infty}\frac{1}{n!}\operatorname{\mathbf{P}}(Y_{0,n}=0),$

which entails

$\sum_{n=1}^{\infty}\frac{1}{n!}\operatorname{\mathbf{P}}(Y_{0,n}=0)=0\Longrightarrow\operatorname{\mathbf{P}}(Y_{a,n}=0)=0,\quad\forall
n=0,1,\ldots\Longrightarrow\operatorname{\mathbf{P}}(Y_{a,t}=0)=0,\quad\forall
t\geq 0.$

Then, applying the Laplace inversion in (80), we get

$x\Pi_{0}(dx)=\sum_{n=1}^{\infty}\frac{1}{n!}\operatorname{\mathbf{P}}(Y_{0,n}\in
dx),\quad x>0.$

Similarly, if $a>0$,

$\displaystyle\mathrm{d}_{a}=0$ $\displaystyle=$
$\displaystyle\sum_{n=0}^{\infty}\frac{1}{n!}\operatorname{\mathbf{P}}(Y_{a,a+n}=0)\Longrightarrow\quad\operatorname{\mathbf{P}}(Y_{a,t}=0)=0,\;\forall
t\geq 0,$ $\displaystyle x\Pi_{a}(dx)$ $\displaystyle=$
$\displaystyle\sum_{n=0}^{\infty}\frac{1}{n!}\operatorname{\mathbf{P}}(Y_{a,a+n}\in
dx),\quad x>0.$

If $N$ is a r.v. with Poisson distribution, with rate parameter equal to 1,
independent of the subordinator $Y_{a}$, then, by (80), we have the closed
forms obtained with the help of the subordinated r.v. $Y_{a,a+N}$: for all
$\lambda\geq 0$,

$\hat{\phi}_{0}^{\prime}(\lambda)=1+e\sum_{n=1}^{\infty}\frac{e^{-1}}{n!}\operatorname{\mathbf{E}}[e^{-\lambda
Y_{0,n}}]=1+e\operatorname{\mathbf{E}}[e^{-\lambda Y_{0,N}}{\rm
1\leavevmode\nobreak\ l}_{N\geq
1}]\Longrightarrow\hat{\phi}_{a}(\lambda)=\lambda+e\;\operatorname{\mathbf{E}}\left[\frac{1-e^{-\lambda
Y_{0,N}}}{Y_{0,N}}{\rm 1\leavevmode\nobreak\ l}_{N\geq 1}\right],$ (81)

and if $a>0$,

$\hat{\phi}_{a}^{\prime}(\lambda)=e\,\sum_{n=0}^{\infty}\frac{e^{-1}}{n!}\operatorname{\mathbf{E}}[e^{-\lambda
Y_{a,a+n}}]=e\,\operatorname{\mathbf{E}}[e^{-\lambda
Y_{a,a+N}}]\Longrightarrow\hat{\phi}_{a}(\lambda)=e\;\operatorname{\mathbf{E}}\left[\frac{1-e^{-\lambda
Y_{a,a+N}}}{Z_{a,a+N}}\right].$ (82)

Additionally, by (80) again, we have that for all $a\geq 0,\,\lambda\geq 0$
and $t>0$,

$\operatorname{\mathbf{E}}[Y_{a,t}e^{-\lambda
Y_{a,t}}]=t\hat{\phi}_{0}^{\prime}(\lambda)e^{-t\hat{\phi}_{a}(\lambda)}=t\sum_{n=0}^{\infty}\frac{\operatorname{\mathbf{E}}[e^{-\lambda
Y_{a,a+t+n}}]}{n!}=e\,t\,\operatorname{\mathbf{E}}[e^{-\lambda Y_{a,a+t+N}}].$
(83)

If $\overline{Y}_{a,t}$ is a version of the size biased distribution of
$Y_{a,t}$, defined for $t>0$, by

$\overline{Y}_{a,t}\;\stackrel{{\scriptstyle{\rm
d}}}{{=}}\;\frac{x\,\operatorname{\mathbf{P}}(Y_{a,t}\in
dx)}{\operatorname{\mathbf{E}}[Y_{a,t}]}=\frac{x\,\operatorname{\mathbf{P}}(Y_{a,t}\in
dx)}{e\,t},$ (84)

then, from (83) we get for $a\geq 0$:

$\operatorname{\mathbf{E}}[e^{-\lambda\overline{Y}_{a,t}}]=\operatorname{\mathbf{E}}[e^{-\lambda
Y_{a,a+t+N}}],\quad\forall\lambda\geq
0\Longleftrightarrow\overline{Y}_{a,t}\;\stackrel{{\scriptstyle{\rm
d}}}{{=}}\;Y_{a,a+t+N}\stackrel{{\scriptstyle{\rm
d}}}{{=}}\;Y_{a,t}+Y_{a,a+N},\quad\forall t>0,$

where in the last identity, the r.v.’s $Y_{a,t}$ and $Y_{a,a+N}$ are assumed
to be independent.

Finally, by (63), we conclude that a nonnegative infinitely divisible r.v. $X$
has a Bernstein function $\phi$ with Lévy measure of the form (77) and $a\geq
0$, if, and only if, for some subordinator $Z$, it has the following
stochastic representation

$X=\int_{(0,\infty)}f(s)dZ_{s},\qquad f(s)=e^{-\phi_{a}(s)},\quad s\geq 0$

and recall that $f$ coincides with

$f(s)=e^{-\hat{\phi}_{a}(s-s_{a})}=\operatorname{\mathbf{E}}[e^{(s_{a}-s)Y_{a,1}}],\quad\mbox{if}\quad
s\geq s_{a}.$

In (SSV, , Chapter 10), several forms of the function $f$ were investigated in
the purpose of characterizing the induced classes of distribution. The cases
$a=0$ and $a=1$ leading to the classes $\mathcal{T\\!B}$ and $\mathcal{C\\!B}$
were also investigated there but no closed-form was proposed for the
corresponding function $f$. For this reason, and because of the remarkable
relations (81), (82), and (84), the distribution of $Y_{a,1}$ would benefit
from being investigated in more detail.

##  6 The proofs

Proof of Proposition 1.1. 1) Consider the Lévy-Laplace exponent $\Psi$ of
$X\sim\overline{\mathrm{I\,D}}_{-}(\mathds{\mathds{R}})$. The Lévy measure
$\pi$ gives no mass to $(0,\infty)$ and its image $\Pi$ given by (4) satisfies
(3). Observe that the truncated Lévy measures
$\Pi_{n}:=\Pi_{|(0,n]},\,n\in\mathds{N}$, satisfies (7), so that the
quantities

$\mathrm{a}_{n}:=\mathrm{a}+\int_{(0,n]}\big{(}\chi(x)-x\big{)}\,\Pi(dx)\quad\mbox{are
finite},$

and each of the functions,

$\Psi_{n}(\lambda):=\mathrm{a}_{n}\lambda\,+\mathrm{b}\,\lambda^{2}+\int_{(0,\infty)}(e^{-\lambda
x}-1+\lambda\,x)\;\Pi_{n}(dx),$ (85)

belongs to $\mathcal{LE}$, hence, is associated with a r.v.
$X_{n}\sim\mathrm{I\,D}_{-}(\mathds{R})$. Since $\Psi=\lim_{n}\Psi_{n}$,
deduce that $X\stackrel{{\scriptstyle d}}{{=}}\lim_{n}X_{n}$. The converse is
obvious because
$\mathrm{I\,D}_{-}(\mathds{R})\subset\overline{\mathrm{I\,D}}_{-}(\mathds{\mathds{R}})$.
Statement 2) is an immediate consequence of 1). $\square$

###  6.1 Additional results, preliminaries for the proofs of the main results

The following technical result will be used in the proof of Lemma 6.3.

###### Proposition 6.1

Let $t\geq 0,\;\alpha\in(0,1)\cup(1,\infty)$ and define for $u>0,$

$\displaystyle e_{\alpha,t}(u)$ $\displaystyle:=$
$\displaystyle\frac{e^{-tu}}{1-e^{-u}}$ (86) $\displaystyle g_{\alpha,t}(u)$
$\displaystyle:=$
$\displaystyle\alpha\,e_{\alpha,t}(u)-e_{\alpha,t}(u/\alpha),$ (87)
$\displaystyle h_{\alpha,t}(u)$ $\displaystyle:=$ $\displaystyle
e_{\alpha,t}(u)-e_{\alpha,t}(u/\alpha).$ (88)

  1. 1)

The following holds for $g_{\alpha,t}$:

    1. (i)

The integral $\int_{0}^{\infty}g_{\alpha,t}(u)du$ is finite if, and only if,
$t>0$. In this case, the integral equals to $-\alpha\log\alpha$.

    2. (ii)

The function $g_{\alpha,t}(u)$ is nonnegative (respectively nonpositive) in
$u$, for every fixed $\alpha<1$ (respectively $\alpha>1$), if, and only if,
$t\geq\frac{1}{2}$.

    3. (iii)

The function $g_{\alpha,t}(u)$ is decreasing (respectively increasing) in $u$,
for every fixed $\alpha\in(0,1)$ (respectively $\alpha>1$), if, and only if,
$t\geq t_{0}=\frac{1}{2}+\frac{1}{2\sqrt{3}}$.

  2. 2)

The function $h_{\alpha,t}(u)$ is decreasing in $u>0$, for every fixed
$\alpha\in(0,1)$ if, and only if, $t\geq t_{1}\simeq 0.151463487259.$

Proof of Proposition 6.1. 1) Since $g_{\alpha,t}(u)=-\alpha
g_{1/\alpha,t}(u/\alpha)$, it is enough to consider the case $\alpha<1$. To
understand the rest of the proof of 1), notice the following expression:

$g_{\alpha,t}(\alpha u)=\frac{1}{u}\left[\frac{\alpha ue^{-t\alpha
u}}{1-e^{-\alpha
u}}-\frac{ue^{-tu}}{1-e^{-u}}\right]=-\frac{\theta_{1/\alpha}(g_{t})(u)}{u},\quad
g_{t}(u)=\frac{ue^{-tu}}{1-e^{-u}}.$ (89)

$1)(i)$ The necessary and sufficient condition $t>0$ is trivial. The value of
the integral is a direct application of 1) in Lemma 3.1.

$1)(ii)$ The necessary part stems from
$\lim_{0+}g_{\alpha,t}(u)=(1-\alpha)(t-\frac{1}{2})$. For the sufficient part,
just notice that $\lim_{+\infty}g_{\alpha,t}(u)=0$ and write

$g_{t}(u)=\frac{ue^{-tu}}{1-e^{-u}}=\frac{ue^{-u/2}}{1-e^{-u}}e^{-u(t-1/2)}=\frac{u/2}{\sinh(u/2)}e^{-u(t-1/2)},$

in order to obtain that $g_{t}$ is decreasing and to conclude that
$g_{\alpha,t}$ is nonnegative.
$1)(iii)$ According to Lemma 3.1 2)$(i)$, we only need to check whenever
$g_{t}$ is convex. Standard calculations lead to

$g_{t}^{\prime\prime}(x)=\frac{g_{t}(x)}{x(e^{x}-1)^{2}}P(x,t),\quad x,\,t>0,$

where

$P(x,t)=x(e^{x}-1)^{2}t^{2}-2t(e^{x}-1)(e^{x}-1-x)+x(e^{x}+1)-2(e^{x}-1),$

and it is clear that the convexity of $g_{t}$ is equivalent to the positivity
of the function $P$. Since the discriminant $\Delta_{P}(x)$ of the polynomial
$t\mapsto P(x,.t)$ equals to

$\Delta_{P}(x)=4(e^{x}-1)^{2}\left[(e^{x}-1)^{2}-x^{2}e^{x}\right]=16e^{x}(e^{x}-1)^{2}\left[\sinh(\frac{x}{2})^{2}-(\frac{x}{2})^{2}\right],$

and is positive, we recover two positive roots:

$t_{\pm}(x)=\frac{1}{x(e^{x}-1)}\left[(e^{x}-1-x)\pm\sqrt{(e^{x}-1)^{2}-x^{2}e^{x}}\right]=\left[\frac{1}{x}-\frac{1}{e^{x}-1}\pm\sqrt{\frac{1}{x^{2}}-\frac{e^{x}}{(e^{x}-1)^{2}}}\right].$

Since

$\left(\frac{1}{x}-\frac{1}{e^{x}-1}\right)^{\prime}=-\frac{(e^{x}-1)^{2}-x^{2}e^{x}}{x^{2}(e^{x}-1)^{2}},\quad\left(\frac{1}{x^{2}}-\frac{e^{x}}{(e^{x}-1)^{2}}\right)^{\prime}=\frac{e^{x}(e^{x}+1)}{(e^{x}-1)^{3}}-\frac{2}{x^{3}},\quad
x>0,$

we see that the functions

$x\mapsto\frac{1}{x}-\frac{1}{e^{x}-1}\quad\mbox{and}\quad
x\mapsto\frac{1}{x^{2}}-\frac{e^{x}}{(e^{x}-1)^{2}},$

are both decreasing, and then so is $x\mapsto t_{+}(x)$. By expansion near
$0$, we obtain

$t_{0}:=\max_{x>0}t_{+}(x)=t_{+}(0+)=\frac{1}{2}+\frac{1}{\sqrt{12}}.$

Since $\min_{x>0}\,t_{-}(x)=0$, we deduce that $P(x,t)$ is positive for every
$x>0$ if, and only if, $t\geq t_{0}$.
2) Observe that

$h_{\alpha,t}(u)=\theta_{\alpha}(e_{t})(u),\quad
e_{t}(u)=e^{-tu}\,e_{0}(u),\quad u>0,$ (90)

and recall that we are looking for the range of $t$ for which we have
$h_{\alpha,t}$ is decreasing. By Lemma 3.4, we have

$\theta_{c_{1}}\theta_{c_{2}}(e_{t})\geq
0,\quad\forall\;c_{1},\,c_{2}\in(0,1)\Longleftrightarrow
e_{t}\in\mathcal{M}_{2}\Longleftrightarrow(-1)^{n}\Theta^{n}(e_{t})\geq
0,\quad\;\mbox{for}\;n=1,2.$ (91)

Using the fact that $e_{t}$ is decreasing, and using (43), it only remains to
check whenever

$\Theta^{2}(e_{t})(x)=x^{2}\,e_{t}^{\prime\prime}(x)+x\,e_{t}^{\prime}(x)=x\,e^{-tu}\big{[}x\,e_{0}(x)\,t^{2}-\big{(}e_{0}(x)+2xe_{0}^{\prime}(x)\big{)}\,t+e_{0}^{\prime}(x)+x\,e_{0}^{\prime\prime}(x)\big{]}\geq
0.$

Since $e_{0}^{\prime}=e_{0}(1-e_{0})$ and
$e_{0}^{\prime\prime}=e_{0}(1-e_{0})(1-2e_{0})$, we find that

$\Theta^{2}(e_{t})(x)\geq 0\Longleftrightarrow
Q(x,t):=xt^{2}-\big{(}1+2x(1-e_{0}(x)\big{)}\,t+\big{(}1-e_{0}(x)\big{)}\big{(}1+x(1-2e_{0}(x)\big{)}\geq
0.$

As in 1), the discriminant $\Delta_{Q}(x)$ of the polynomial $t\mapsto
Q(x,t)$, is given by

$\Delta_{Q}(x)=1-4x^{2}e_{0}(x)\big{(}e_{0}(x)-1\big{)}=1-\frac{4x^{2}e^{x}}{(e^{x}-1)^{2}}=\left(1-\frac{x}{\sinh(x/2)}\right)\left(1+\frac{x}{\sinh(x/2)}\right)$

Paradoxically, things are not as smooth as for $g_{\alpha,t}$: $\Delta_{Q}(x)$
is positive if, and only if, $x$ is bigger than some value $x_{0}$ which can
not be expressed by hand, but could be evaluated by Maple with the value
$x_{0}\sim 4.35463796993$. Hence, for $x>x_{0}$, we recover two positive roots

$t_{\pm}(x)=\frac{1}{2x}\left(1-\frac{2x}{e^{x}-1}\pm\sqrt{1-\frac{4x^{2}e^{x}}{(e^{x}-1)^{2}}}\right)$

and finally, using Maple again, we get

$Q(x,t)\geq 0,\quad\forall\;x>0\Longleftrightarrow t\geq
t_{1}=\max_{x>x_{0}}\,t_{+}(x)\simeq 0.151463487259.$ (92)

$\square$

Let $t>0$, ${\alpha}=(\alpha_{1},\alpha_{2},\ldots,\alpha_{n})\in(0,1)^{n}$.
Recall $e_{t}$ is given by (86) and let the functions

$h_{\underline{\alpha},t}(u)=e_{t}(u)-\sum_{k=1}^{n}e_{t}(u/\alpha_{k}),\quad
u>0.$ (93)

Alzer & Berg (Alzer, , Lemma 2.7) provided a Petrović-type inequality for the
function $e_{1}$:

$\frac{n-1}{2}+\sum_{k=1}^{n}e_{1}(u/\alpha_{k})-e_{1}(u/\sum_{k=1}^{n}\alpha_{k})\geq
0,\quad\mbox{for all $u>0$}.$

We shall provide additional information for the function
$h_{\underline{\alpha},t}$ in Lemma 6.3 below. For this purpose, we need some
preliminary results. We denote by $\delta_{a}$ the Dirac measure in $a$ and by
$\lfloor\,\rfloor$ and $\\{\,\\}$ the integer and the fractional part
functions, respectively.

###### Lemma 6.2

Let $\;t>0$,
$\underline{\alpha}=(\alpha_{1},\alpha_{2},\ldots,\alpha_{n})\in(0,1)^{n}$
such that $\sum_{k=1}^{n}\alpha_{k}=1$ and let $\mu_{\underline{\alpha},t}$
and $m_{\underline{\alpha},t}$ be the signed measure and the function defined
by

$\mu_{\underline{\alpha},t}(dx):=\sum_{i=0}^{\infty}\big{(}\delta_{i+t}(dx)-\sum_{k=1}^{n}\delta_{\frac{i+t}{\alpha_{k}}}(dx)\big{)}\quad\mbox{and}\quad
m_{\underline{\alpha},t}(x):=\mu_{\underline{\alpha},t}([0,x)),\;x\geq t.$

Then, $m_{\underline{\alpha},t}$ is positive for every fixed
$\underline{\alpha}$ if, and only if, $t\geq 1$. In this case, we have the
control

$0\leq m_{\underline{\alpha},t}(x)\leq 1+t.$ (94)

Proof.  We treat only the case $n=2$, the case $n\geq 3$ is treated similarly.
By symmetry of the problem, we may assume that $0<\alpha<\beta=1-\alpha<1$ and
take $\underline{\alpha}=(\alpha,\beta)$. Since for every $\gamma>0$,

$\sum_{i=0}^{\infty}\delta_{\frac{i+t}{\gamma}}\big{(}[0,x)\big{)}=\lfloor\gamma
x+1-t\rfloor\,{\rm 1\leavevmode\nobreak\ l}_{(x\geq t/\gamma)},$

then

$\displaystyle m_{\underline{\alpha},t}(x)=\mu_{\underline{\alpha},t}([0,x))$
$\displaystyle=$ $\displaystyle\lfloor x+1-t\rfloor\,{\rm
1\leavevmode\nobreak\ l}_{(x\geq t)}-\lfloor\alpha x+1-t\rfloor\,{\rm
1\leavevmode\nobreak\ l}_{(x\geq t/\alpha)}-\lfloor\beta x+1-t\rfloor\,{\rm
1\leavevmode\nobreak\ l}_{(x\geq t/\beta)}$ (99) $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{ll}0&{\rm if}\;\;0<x<t\\\ \lfloor
x-t\rfloor+1&{\rm if}\;\;t\leq x<t/\beta\\\ \lfloor x-t\rfloor-\lfloor\beta
x-t\rfloor&{\rm if}\;\;t/\beta\leq x<t/\alpha\\\ \lfloor
x-t\rfloor-\lfloor\alpha x-t\rfloor-\lfloor\beta x-t\rfloor-1&{\rm
if}\;\;t/\alpha\leq x.\end{array}\right.$

1) Assume $t\geq 1$. Using the fact that $y-1<\lfloor y\rfloor\leq y$, for all
$y\in\mathds{R},$ the upper bound in (94) easily follows. Because the integer
part function is increasing, we just need to study the positivity of the
function $m_{\underline{\alpha},t}$ restricted to $(t/\alpha,\infty)$. Since
$\lfloor y+z\rfloor\geq\lfloor y\rfloor+\lfloor z\rfloor,\;\forall
y,z\in\mathds{R}$, we deduce that, for every $x>t/\alpha$,

$m_{\underline{\alpha},t}(x)=\lfloor x+1-t\rfloor-\lfloor\alpha x+1-t+\beta
x+1-t\rfloor\\\ =\lfloor x+1-t\rfloor-\lfloor x+2(1-t)\rfloor\geq 0.$

2) If $t<1$, then the function $m_{\underline{\alpha},t}(x)$ could take
negative values if $x>t/\alpha$. Indeed, choose $x$ such that for some
positive integers $k$ and $l$, we have

$k\leq\alpha x-t<k+\frac{1-t}{2}\quad\mbox{and}\quad l\leq\alpha
x-t<l+\frac{1-t}{2},$

so that

$\\{\alpha x-t\\}<\frac{1-t}{2}\quad\mbox{and}\quad\\{\alpha
x-t\\}<\frac{1-t}{2}.$

and then

$\displaystyle m_{\underline{\alpha},t}(x)$ $\displaystyle=$
$\displaystyle\lfloor x-t\rfloor-\lfloor\alpha x-t\rfloor-\lfloor\beta
x-t\rfloor-1=t+\\{\alpha x-t\\}+\\{\beta x-t\\}-1-\\{x-t\\}$
$\displaystyle\leq$ $\displaystyle t+\\{\alpha x-t\\}+\\{\beta x-t\\}-1<0.$

$\square$

This Lemma will be used in the proof of Corollary 30.

###### Lemma 6.3

Let $t_{0}$ be the universal constant of Theorem 2.4. The function
$h_{\underline{\alpha},t}$ defined by (93) with $\sum_{k=1}^{n}\alpha_{k}=1$
satisfies the following:

  1. 1)

If $t\geq 1/2$, then $h_{\underline{\alpha},t}(u)>0,\;$ for all $u>0$.

  2. 2)

If $t\geq t_{0}$, then $h_{\underline{\alpha},t}$ is decreasing.

  3. 3)

The function $u\mapsto h_{\underline{\alpha},t}(u)/u\;$ is completely monotone
if, and only if, $t\geq 1$. In this case, we have the representation

$h_{\underline{\alpha},t}(u)=(1+t)\int_{0}^{\infty}e^{-ux}\,\eta(x)\,dx,\quad
u>0,$

for some measurable function $\eta:(0,\infty)\to[0,1]$.

Proof.  1) and 2) Just notice the expression
$h_{\underline{\alpha},t}(u)=\sum_{k=1}^{n}g_{\alpha_{k},t}$, with
$g_{\alpha_{k},t}$ given by (87), and apply Proposition 6.1.
3) Expanding the terms in $\operatorname{g_{\alpha,t}}$, and using the signed
measure and the positive function $\mu_{\underline{\alpha},t}$ and
$m_{\underline{\alpha},t}$ of Lemma 94, obtain the expression

$\frac{h_{\underline{\alpha},t}(u)}{u}=\frac{1}{u}\sum_{i=0}^{\infty}\big{(}e^{-(i+t)u}-\sum_{k=1}^{n}e^{-\frac{(i+t)}{\alpha_{k}}u}\big{)}=\frac{1}{u}\int_{[t,\infty)}e^{-ux}\mu_{\underline{\alpha},t}(dx)\\\
=\int_{t}^{\infty}e^{-ux}m_{\underline{\alpha},t}(x)\,dx,\quad u>0,$

and conclude with the nonnegativity of $m_{\underline{\alpha},t}$. $\square$

###  6.2 Linking the integrability of infinitely divisible distributions with
their Lévy measure

Let $\mathcal{C}$ be the class of functions $h:[0,\infty)\to[0,\infty)$,
differentiable, such that

$\lim_{x\to\infty}h^{\prime}(x)=0\quad\mbox{there exists $x_{0}\geq 0$ s.t.
$h$ is concave on $[x_{0},\infty)$}.$ (100)

Note that for such functions, there exists a finite positive measure
$\rho_{h}$ on $[x_{0},\infty)$ such that the derivative of $h$ is represented
by

$h^{\prime}(x)=\rho_{h}\big{(}[x,\infty)\big{)},\quad x\geq x_{0}.$ (101)

The following lemma gives an interpretation of the integrability condition
(70) and constitutes a variant of (JUS, , Theorem 2), which is stated with
sub-multiplicative functions $h$.

###### Lemma 6.4

Let $Z$ be nonnegative r.v. with cumulant function
$\phi(\lambda)=-\log\operatorname{\mathbf{E}}[e^{-\lambda Z}]$ and let a
function $h$ in the class $\mathcal{C}$ defined by (100) and associated to the
pair $(x_{0},\rho_{h})$ by (101).

  1. 1)

We have the equivalence:

$\operatorname{\mathbf{E}}[h(Z)]<\infty\Longleftrightarrow\int_{x_{0}}^{\infty}\min\big{(}1,\phi(1/x)\big{)}\;x\;\rho_{h}(dx)<\infty;$
(102)

  2. 2)

Assume $Z\sim\mathrm{I\,D}(\mathds{R}_{+})$ and has characteristics
$(\mathrm{d},\,\Pi)$ in the representation (9) of its Bernstein function
$\phi$, then the following assertions are equivalent.

    1. (i)

$\operatorname{\mathbf{E}}[h(Z)]<\infty$;

    2. (ii)

$\int_{[x_{0},\infty)}\chi_{h}(u)\,\Pi(du)<\infty$, where

$\chi_{h}(u)=\int_{[x_{0},\infty)}(x\wedge
u)\;\rho_{h}(dx)=\left\\{\begin{array}[]{lll}&&x_{0}\;h^{\prime}(x_{0})\;u,\quad\mbox{if}\;\;0<u<x_{0},\\\
&&\\\ &&h(u)-h(x_{0})+x_{0}\;h^{\prime}(x_{0}),\quad\mbox{if}\;\;u\geq
x_{0}.\end{array}\right.$ (103)

    3. (iii)

$\int_{[x_{0},\infty)}\big{(}h(u)-h(x_{0})\big{)}\;\Pi(du)<\infty$.

Proof of  1) Since $h^{\prime}(\infty)=0$, obtain by Tonelli-Fubini’s theorem,
that

$\displaystyle\operatorname{\mathbf{E}}[h(Z)]$ $\displaystyle=$ $\displaystyle
h(0)+\int_{0}^{\infty}h^{\prime}(u)\;\operatorname{\mathbf{P}}(Z>u)\;du$
$\displaystyle=$ $\displaystyle
h(0)+\int_{0}^{x_{0}}h^{\prime}(u)\;\operatorname{\mathbf{P}}(Z>u)\;du+\int_{[x_{0},\infty)}\int_{x_{0}}^{x}\operatorname{\mathbf{P}}(Z>u)\;du\;\rho_{h}(dx),$
$\displaystyle=$ $\displaystyle
h(0)+\int_{0}^{x_{0}}h^{\prime}(y)\;\operatorname{\mathbf{P}}(Z>u)\;du-h^{\prime}(x_{0})\int_{0}^{x_{0}}\operatorname{\mathbf{P}}(Z>u)\;du+\int_{[x_{0},\infty)}\int_{0}^{x}\operatorname{\mathbf{P}}(Z>u)\;du\;\rho_{h}(dx).$

Since the first three terms in last equality are finite, deduce that

$\operatorname{\mathbf{E}}[h(Z)]<\infty\Longleftrightarrow\int_{x_{0}}^{\infty}\int_{0}^{x}\operatorname{\mathbf{P}}(Z>u)\;du\;\,\rho_{h}(dx)<\infty.$
(104)

Now, observe that for all $\lambda\geq 0$,

$(1-e^{-1})\,(\lambda\wedge
1)\leq\left(1-e^{-\lambda}\right)\leq(\lambda\wedge 1).$ (105)

Thus, write for all $x>0$,

$(1-e^{-1})\,x\,(\phi(1/x)\wedge 1)\leq
x\,\left(1-e^{-\phi(1/x)}\right)=\int_{0}^{\infty}e^{-u/x}\;\operatorname{\mathbf{P}}(Z>u)\;du\leq
x\;(\phi(1/x)\wedge 1),$ (106)
$\int_{0}^{\infty}e^{-u/x}\;\operatorname{\mathbf{P}}(Z>u)\;du\geq\int_{0}^{x}e^{-u/x}\operatorname{\mathbf{P}}(Z>u)\;du\geq
e^{-1}\,\int_{0}^{x}\operatorname{\mathbf{P}}(Z>u)\;du,$ (107)

and deduce that

$\displaystyle\int_{0}^{\infty}e^{-u/x}\;\operatorname{\mathbf{P}}(Z>u)\;du$
$\displaystyle=$
$\displaystyle\int_{0}^{x}e^{-u/x}\;\operatorname{\mathbf{P}}(Z>u)\;du+\int_{x}^{\infty}e^{-u/x}\;\operatorname{\mathbf{P}}(Z>u)\;du$
(108) $\displaystyle\leq$
$\displaystyle\int_{0}^{x}\operatorname{\mathbf{P}}(Z>u)\;du+\operatorname{\mathbf{P}}(Z>x)\,\int_{x}^{\infty}e^{-u/x}\;du=\int_{0}^{x}\operatorname{\mathbf{P}}(Z>u)\;du+e^{-1}\;x\;\operatorname{\mathbf{P}}(Z>x)$
$\displaystyle\leq$
$\displaystyle\left(1+e^{-1}\right)\int_{0}^{x}\operatorname{\mathbf{P}}(Z>u)\;du.$

From (106), (107) (108), deduce that

$\frac{e-1}{e+1}\;(\phi(1/x)\wedge
1)\geq\int_{0}^{x}\operatorname{\mathbf{P}}(Z>u)\;du\leq e\;(\phi(1/x)\wedge
1),$

and from equivalence (104), obtain the one in (102).

2) Let $x_{1}:=\inf\\{x>x_{0}\;;\;\phi(1/x)>1\\}>0$. By (102) and by the fact
that

$x\mapsto
x\,\phi(1/x)=\mathrm{d}+\int_{0}^{\infty}e^{-u/x}\Pi(u,\infty)\;du\;\;\mbox{is
non-decreasing and starts from $\lim_{x\to 0+}x\;\phi(1/x)=\mathrm{d}$},$

deduce the following steps:

a) Assume $x_{1}=\infty$, then

$\operatorname{\mathbf{E}}[h(Z)]<\infty\Longleftrightarrow\int_{[x_{0},\infty)}\phi(1/x)\,x\;\rho_{h}(dx)<\infty,$

and there is no problem of integrability of the last expression at 0 if
$x_{0}=0$.

b) Assume $x_{1}<\infty$, then,

$\operatorname{\mathbf{E}}[h(Z)]<\infty\Longleftrightarrow\int_{[x_{0},x_{1})}\,x\;\rho_{h}(dx)+\int_{[x_{1},\infty)}\phi(1/x)\,x\;\rho_{h}(dx)<\infty.$

c) Deduce that in all cases,

$\operatorname{\mathbf{E}}[h(Z)]<\infty\Longleftrightarrow
I(x_{0}):=\int_{[x_{0},\infty)}\phi(1/x)\;x\;\rho_{h}(dx)<\infty.$ (109)

d) By representation (9) of $\phi$ and Tonelli-Fubini’s theorem, deduce the
representation,

$I(x_{0})=\mathrm{d}\,h^{\prime}(x_{0})+\int_{(0,{\infty})}\int_{[x_{0},\infty)}(1-e^{-u/x})\;x\;\rho_{h}(dx)\;\;\Pi(du),$

and from (105), deduce that

$I(x_{0})<\infty\Longleftrightarrow\int_{(0,{\infty})}\chi_{h}(u)\;\Pi(du),\quad\mbox{where
$\chi_{h}(u)$ is given by \eqref{chi}.}$

If $x_{0}>0$, then as a Lévy measure, $\Pi$ always integrates $\chi_{h}$ on
$(0,x_{0})$ and the constants on $[x_{0},\infty)$. The latter gives the
equivalence $(i)\Longleftrightarrow(ii)$. If $x_{0}=0$,
$\chi_{h}(u)=h(u)-h(0)$. Finally, from (109) deduce the equivalence
$(ii)\Longleftrightarrow(iii)$. $\square$

###  6.3 Proofs of the main results

Proof of Theorem 2.1. Using representations (22) for the Gamma function and
(18) for Bernstein functions, then performing an obvious change of variable,
using 1) in Lemma 6.1, and the fact that
$\int_{0}^{\infty}g_{\alpha,t}(x)dx=-\mathrm{d}_{\alpha}$ provided by
Proposition 6.1 1)(i), obtain the following representation for
$\alpha\in(0,1)$ and $t>0$:

$G_{\alpha,t}(\lambda)=\exp\left\\{\mathrm{d}_{\alpha}\lambda+\int_{0}^{\infty}(e^{-\lambda
u}-1+\lambda
u)\,\frac{g_{\alpha,t}(u)}{u}\,du\right\\}=\exp-\left\\{\int_{0}^{\infty}(1-e^{-\lambda
u})\,\frac{g_{\alpha,t}(u)}{u}\,du\right\\},$ (110)

where $\mathrm{d}_{\alpha}$ is given by (18). Thus, since $G_{\alpha,t}(0)=1$
and using (9), retrieve the equivalences in case $\alpha\in(0,1)$:

$\displaystyle\lambda\mapsto\left(G_{\alpha,t}\right)^{r}\in\mathcal{CM},\quad\forall
r>0$ $\displaystyle\Longleftrightarrow$
$\displaystyle\lambda\mapsto\int_{0}^{\infty}(1-e^{-\lambda
u})\,\frac{g_{\alpha,t}(u)}{u}\,du\in\mathcal{BF}\Longleftrightarrow
g_{\alpha,t}\geq 0$ $\displaystyle\Longleftrightarrow$ $\displaystyle
G_{\alpha,t}(\lambda)=\operatorname{\mathbf{E}}[e^{-\lambda
X_{\alpha,t}}],\;\lambda\geq
0,\quad\mbox{and}\;X\sim\mathrm{I\,D}(\mathds{R}_{+}).$

By point 2) in Proposition 6.1, we deduce that $g_{\alpha,t}\geq
0\Longleftrightarrow t\geq 1/2$. The rest of the statements are obtained by
the reflexive relation (19).

$\square$

Proof of Theorem 2.4. By (54), the functions $g_{\alpha,t}(x)/x$ is the
density of the Lévy measure of $X_{\alpha,t},\;\alpha\in(0,1)$. By point
1)(iii) in Proposition 6.1 is nondecreasing if, and only if $t\geq t_{0}$. The
reasoning is analog for $Y_{\alpha,t},\;\alpha>1$. $\square$

Proof of Corollary 30. 1), 2) and 3) are a straightforward consequence of the
Mellin transform representations (33) and of Theorems 2.1 and 2.4.
4) For the claim on $X_{\underline{\alpha},t}$, we only need to use (110), to
observe that

$-\log\operatorname{\mathbf{E}}[e^{-\lambda
X_{\underline{\alpha},t}}]=\int_{0}^{\infty}(1-e^{-\lambda
u})\,\frac{h_{\underline{\alpha},t}(u)}{u}\,du$ (111)

where $h_{\underline{\alpha},t}(u)$ is given by (93) and to apply point 3) of
Lemma 6.3 to finally obtain that

$-\log\operatorname{\mathbf{E}}[e^{-\lambda
X_{\underline{\alpha},t}}]^{1/(1+t)}=\int_{0}^{\infty}\frac{\lambda}{\lambda+x}\frac{\eta(x)}{x}\,dx,\quad\mbox{with}\;\eta(x):=\frac{m_{\underline{\alpha},t}(x)}{(1+t)}\leq
1$

is a Bernstein function that meats the form (28). The assertion for
$Y_{\underline{\alpha},t}$ is shown identically. $\square$

Proof of Proposition 2.7. Using formula (27) and performing the change of
variable $u\to u/\alpha$, we write

${\Gamma(\lambda+t)\over\Gamma(t+\alpha\lambda)}=\exp\left\\{(1-\alpha)\Psi(t)\lambda+\int_{0}^{\infty}(e^{-\lambda
u}-1+\lambda
u)\,\frac{h_{\alpha,t}(u)}{u}\,du\right\\}=\operatorname{\mathbf{E}}[e^{\lambda
T_{t,\alpha}}],\quad\lambda\geq 0$ (112)

where $h_{\alpha,t}$ is the nonnegative function given by (88) and
$T_{t,\alpha}\sim\mathrm{I\,D}_{-}(\mathds{R})$ is such that the identity in
law (35) holds. As a consequence of Proposition 6.1, we obtain that
selfdecomposability of $T_{t,\alpha}$ is equivalent to the decreaseness of
$h_{\alpha,t}$. The latter is also equivalent to $t\geq t_{1}$. $\square$

Proof of Proposition 2.8. Due to the representation (111) for the Bernstein
function of $X_{\underline{\alpha},t}$, it suffices to show that the function
$h_{\underline{\alpha},t}$ given by (93), converges as $n\to\infty$. This not
difficult to obtain, because for $t>0$, the function $u\mapsto
u\,e_{t}(u)=e^{-tu}u/(1-e^{-u})$ is bounded on $(0,{\infty})$ by a constant,
say $C_{t}$. Since $\sum_{k=1}^{n}\alpha_{k}=1$, we have

$e_{t}(u)-h_{\underline{\alpha},t}(u)=\frac{1}{u}\sum_{k=1}^{n}\frac{u}{\alpha_{k}}h_{t}\left(\frac{u}{\alpha_{k}}\right)\;\alpha_{k}\leq\frac{C_{t}}{u},\quad
u>0.$

Thus, for fixed $u,\,t>0$, the bounded and increasing sequence
$e_{t}(u)-h_{\underline{\alpha},t}(u)$ is convergent as $n\to\infty$.
$\square$

Proof of Lemma 3.1. 1) It is enough to consider the case $c\in(0,1)$. By
Tonelli-Fubini’s theorem, we get

$\displaystyle\int_{0}^{\infty}\frac{g(x)-g(x/c)}{x}\,dx$ $\displaystyle=$
$\displaystyle\int_{0}^{\infty}\frac{1}{x}\int_{[x,x/c]}d(-g)(y)\,dx=\int_{(0,\infty)}\int_{cy}^{y}\frac{dx}{x}\;\;d(-g)(y)=\big{(}g(\infty)-g(0+)\big{)}\log
c.$

2) Both assertions $(i)$ and $(ii)$ stem from

$\theta_{c}(g)(x)=-\int_{1}^{1/c}xg^{\prime}(xs)ds=-\int_{1}^{1/c}\Theta(g)(xs)\frac{ds}{s}\quad\mbox{and}\quad\lim_{c\to
1-}\frac{\theta_{c}(g)(x)}{1-c}=-\Theta(g)(x).$

$\square$

Proof of Corollary 4.4. 1) Assuming that
$X\sim\mathrm{L}_{0}(\mathds{R}_{+})\;$ (resp.
$X\sim\mathrm{I\,D}_{-}(\mathds{\mathds{R}})\cap\mathrm{L}_{0}(\mathds{R})$),
then, as in (55), the associated Bernstein function $\phi$ represented by (54)
(respectively, the Laplace exponent $\Psi$ represented by (57)), satisfies the
following: if $c_{1},\,c_{2},\cdots,c_{m}\in(0,1)$, and
$c_{m,1}=(1-c_{1})\ldots(1-c_{m}),\;c_{m,2}=(1-c_{1}^{2})\ldots(1-c_{m}^{2})$,
then the representations

$\displaystyle\theta_{1/c_{1}}\theta_{1/c_{2}}\ldots\theta_{1/c_{m}}(\phi)(\lambda)$
$\displaystyle=$ $\displaystyle
c_{m,1}\,\mathrm{d}\,\lambda+\int_{0}^{\infty}(1-e^{-\lambda
y})\frac{\theta_{c_{1}}\dots\theta_{c_{m}}(k)(x)}{x}dx$
$\displaystyle\theta_{1/c_{1}}\theta_{1/c_{2}}\ldots\theta_{1/c_{m}}(\Psi)(\lambda)$
$\displaystyle=$ $\displaystyle
c_{m,1}\,\mathrm{a}\,\lambda+c_{m,2}\,\mathrm{b}\,\lambda^{2}+\int_{0}^{\infty}(e^{-\lambda
x}-1+\lambda x)\frac{\theta_{c_{1}}\dots\theta_{c_{m}}(k)(x)}{x}dx$

are straightforward and Lemma 3.4 clarifies the equivalences between
$(i),\;(ii),(iii)$ and $(iv)$. Due to the integrability condition (10)
(respectively (3)), we see that in the representation (52) of $k$, necessarily
$\mathrm{c}=0$ and, after some calculus, we see that the $\mu$-measure should
satisfy

$\int_{(0,{\infty})}(x^{l}\wedge
1)\frac{k(x)}{x}dx=\int_{(0,{\infty})}a_{n,l}(y)\frac{\mu(dy)}{y}<\infty,$
(113) $a_{n,l}(y):=\;\frac{1}{n!}\int_{0}^{y}(x^{l}\wedge
1)\log^{n}\left(\frac{y}{x}\right)\frac{dx}{x}\;=\;\left\\{\begin{array}[]{lll}\frac{y^{l}}{l^{n+1}},&\;\mbox{if}&x<1\\\
&&\\\ \frac{\log^{n+1}(y)}{(n+1)!}+\frac{y^{l}}{n!}\int_{\log
y}^{\infty}z^{n}e^{-lz}dz,&\;\mbox{if}&x\geq 1\,,\end{array}\right.$ (114)

with $l=1$ if $X\sim\mathrm{L}_{n}(\mathds{R}_{+})$ and $l=2$ if
$X\sim\overline{\mathrm{I\,D}}_{-}(\mathds{\mathds{R}})\cap\mathrm{L}_{n}(\mathds{R})$.
Observing that $a_{n,l}(y)\sim\log^{n+1}(y)/(n+1)!$, as $y\to\infty$, we
recover the condition (59) which is satisfied if
$X\sim\mathrm{I\,D}_{-}(\mathds{\mathds{R}})\cap\mathrm{L}_{n}(\mathds{R})$
due to (7) and to the fact that $\lim_{y\to+\infty}a_{n,2}(y)/y=0$. The
equivalence with $(iv)$ and $(v)$ are due to (75) in the case that
$X\sim\mathrm{L}_{n}(\mathds{R}_{+})$. The proof is identical in the case that
$X\sim\mathrm{I\,D}_{-}(\mathds{\mathds{R}})\cap\mathrm{L}_{n}(\mathds{R})$.
The equivalence with $(vi)$ is provided by (75) and (76).

2) The assertion is immediate due to the decomposition and the approximation
observed in point 3) of Proposition 4.3, taking into account (113).

3) The first assertion is evident from point 1). The conditions on the support
of $\mu$ are due to (3), (10) and to Lemma 3.4, which read as follows: taking
$l=0$ if $X\sim\mathrm{L}_{\infty}(\mathds{R}_{+})$ (respectively $l=1$ if
$X\sim\mathrm{I\,D}_{-}(\mathds{R})\cap\mathrm{L}^{\infty}(\mathds{R})$), then

$\int_{0}^{\infty}x^{l}(x\wedge
1)\frac{k(x)}{x}dx=\int_{(0,{\infty})}\left(\int_{0}^{1}x^{l-u}dx+\int_{1}^{\infty}\frac{1}{x^{u-l+1}}dx\right)\nu(du)<\infty$

if, and only if, $support(\nu)=(0,1)$ if $l=0$ (respectively
$support(\nu)=(1,2)$), and then necessarily

$\int_{(0,1)}\frac{1}{u(1-u)}\nu(du)<\infty,\;\;\mbox{if $l=0$
(respectively}\;\int_{(1,2)}\frac{1}{(u-1)(2-u)}\nu(du)<\infty,\;\;\mbox{if
$l=0$)}.$

$\square$

##  7 Some background on infinitely divisible and selfdecomposable
distributions

An equivalent definition for the infinite divisibility of a probability
measure $\mu$ is, that there exists a real row-wise independent random
variables $\xi_{n,k}$, $k=1,2,\ldots,k_{n}$, $\;n\geq 1$,
$\;k_{n}\nearrow\infty$, satisfying the infinitesimally condition

$\lim_{n\to\infty}\max_{1\leq k\leq k_{n}}P(|\xi_{n,k}|>\epsilon)\to
0,\quad\mbox{for each $\epsilon>0$,}$

and such that we have the limit in distribution

$\xi_{n,1}+\xi_{n,2}+\ldots+\xi_{n,k_{n}}\stackrel{{\scriptstyle
d}}{{\longrightarrow}}\mu.$ (115)

Conversely, any infinitely divisible distribution can be obtained in the
scheme (115); see Gnedenko & Kolmogorov GK  or Loève love  or Parthasarathy
Par1967 , for multi-dimensional spaces. Similarly, the distribution $\mu$ is
selfdecomposable, if there exists two sequences $a_{n}>0,\;b_{n}\in\mathds{R}$
and an infinitesimal triangular array of independent real random variables of
the form $\xi_{n,k}:=a_{n}\;Z_{k}$, such that

$a_{n}\;(Z_{1}+Z_{2}+\ldots+Z_{n})+b_{n}\stackrel{{\scriptstyle
d}}{{\longrightarrow}}\mu.$ (116)

The terminology of selfdecomposability is due to the fact that the limiting
distribution $\mu$ corresponds to an infinitely divisible r.v. $X$ that
satisfies (13). The most general case of limits (116), where the $Z_{k}$’s are
Banach space-valued random variables and the $a_{k}$’s are chosen from a group
of bounded linear operators on a Banach space in question, was studied in
Jurek JU1 . In (JU1, , Theorem 3.1 and Section 4), taking reals as a Banach
space $\mathds{B}$, the multiplication by positive scalars as linear operators
and all probability measures as a set $Q$, it is shown that

$\mu\in\mathrm{L}_{m}(\mathds{B}),\;\;\mbox{if, and only if, \eqref{zn} holds
and}\;\;Z_{k}\sim\mathrm{L}_{m-1}(\mathds{B}),\;k\geq 1,\;m\geq 1.$

Conversely, each selfdecomposable distribution can be obtained via the
limiting scheme (116). It might be worth to remembers that all
selfdecomposable ate absolutely continuous with respect to Lebesgue measure,
cf. (JV1983, , Section 3.8, p.162). Additionally, if we assume that the r.v.’s
$Z_{1},\;Z_{2},\ldots,Z_{n},\ldots$ have the same distribution, then we get,
at the limit, the class $\mathrm{S}(\mathds{R})$ of _stable distributions_ on
$\mathds{R}$, see the monograph of Zolotarev zolo  or (26) for stable
distributions. Furthermore, if $a_{n}:=(\sigma\sqrt{n})^{-1}$ where
$\sigma^{2}$ is the variance of $Z_{k}$ and
$b_{n}=-n\operatorname{\mathbf{E}}[Z_{1}]$, then in (116) we get Central Limit
Theorem, i.e., $\mu$ is the standard normal distribution $N(0,1)$.

From the above way of reasoning we have the inclusions:

$\mbox{(normal
distributions)}\;\subsetneq\mathrm{S}(\mathds{R})\subsetneq\mathrm{L}_{0}(\mathds{R})\subsetneq\mathrm{I\,D}(\mathds{R}).$

The class $\mathrm{L}_{0}(\mathds{R})$ is quite large and contains among
others $\chi^{2}$, Fisher, gamma, log-gamma, etc; see Jurek JU10 .

In (JU2, , Corollary 2.11), Jurek showed that when taking his operator equal
to the identity, we have $X\sim\mathrm{L}_{m}(\mathds{R}),\;\ m\geq 0$, if,
and only if, there exists a Lévy process ${(Y_{t})}_{t\geq 0}$, such that
$\mathbb{E}[\log^{m+1}(1+|Y_{1}|)]<\infty$ and such that we have the integral
stochastic representation

$X\stackrel{{\scriptstyle{\rm d}}}{{=}}\int_{0}^{\infty}e^{-t}dY_{r(t)},\quad
r(t)=\frac{t^{m+1}}{(m+1)!}.$ (117)

Note that these integral representations allow descriptions of classes
$\mathrm{L}_{m}(\mathds{R})$ in terms of characteristic functions, see (JU2, ,
Theorem 3.1), as Urbanik urb1972 ; urbanik1  obtained by the extreme point
method. For $m=0$, the constructed Lévy process $Y$ in (117) is such that

$Y(t+s)-Y(t)=e^{-s}V_{e^{-t}}+V_{e^{-s}},\quad s,\;t>0,\quad\mbox{where
$V_{c}$ is given by \eqref{xcy}},$

and the random integral characterization of selfdecomposable distributions is
from Jurek & Vervaat (JV1983, , pp. 252-253). The process $Y$ is coined as the
_background driving Lévy process_ of $X$, in short, BDLP. Other constructions
of BDLP’s for selfdecomposable random variables are given in Jeanblanc, Pitman
& Yor JPY .

Declarations of interest: none.
Acknowledgment: The authors are grateful to Cyril Banderier form Paris-Nord,
who kindly helped them with Maple numerical computations of the universal
constant $t_{1}$ in (92). Warm thanks to the Polish team of the department of
mathematics of Wroc$\l$aw for their welcome to the first author during his
several visits, they were a major source of inspiration.

## References

  * (1) Aguech, R.; Jedidi, W.: New characterizations of completely monotone functions and Bernstein functions, a converse to Hausdorff’s moment characterization theorem. _Arab. J. Math. Sci._ 25(1), pp. 57–82, 2019.
  * (2) Alzer, H.; Berg, C.: Some classes of completely monotonic functions, II. Ramanujan J. 11, pp. 225–248, 2006.
  * (3) Akita, K.; Maejima, M.: On certain self-decomposable self-similar processes with independent increments. _Stat. Probabil. Lett._ 59, pp. 53–59, 2002.
  * (4) Berg, C.; Çetinkaya, A.; Karp, D.: Completely monotonic ratios of basic and ordinary Gamma functions. _Aequat. Math._ 95, pp. 569–588, 2021.
  * (5) Behme, A.: Exponential functionals of Lévy processes with jumps. ALEA 12, 375–397, (2015).
  * (6) Berg, C.; Forst, G.: Multiply self–decomposable probability measures on $\mathds{R}_{+}$ and $\mathds{Z}_{+}$. _Z. Wahrscheinlichkeitstheor. verw. Geb._ , 62, 147–163 (1983).
  * (7) Bertoin, J.: _Lévy processes_. Cambridge University Press, Cambridge 1996.
  * (8) Bertoin, J.; Yor, M.: On subordinators, self-similar Markov processes and some factorizations of the exponential variable. Elect. Comm. in Probab. 6, pp. 95–106 (2001).
  * (9) Bridaa, S.; Fourati, S.; Jedidi, W.: Some new classes and techniques in the theory of Bernstein functions. Accepted for publication in “A Lifetime of Excursions Through Random Walks and Lévy Processes, A Volume in Honour of Ron DoneyâÄôs 80th Birthday", _Progress in Probability_ , Springer Series 2020.
  * (10) Butzer, P.L.; Jansche, S.: A Direct approach to the Mellin transform. J. Fourier Anal. 3(4), pp. 325–376, 1997.
  * (11) Butzer, P.L.; Kilbas, A.A.; Trujillo, J.J. : Fractional calculus in the Mellin setting and Hadamard-type fractional integrals. J. Math. Anal. Appl. 269, pp. 1–27, 2002.
  * (12) Gnedenko, B. V.; Kolmogorov, A. N.: Limit Distributions for Sums of Independent Random Variables. Addison-Wesley, Reading, MA. 1968, 2nd ed.
  * (13) Gordon, L.: A Stochastic approach to the Gamma function. Am. Math. Mon. 101(9), pp. 858–865, 1994.
  * (14) Gradshteyn , I.S.; Ryzhik, I.M.: Table of Integrals, Series, and Products. Seventh Edition, Elsevier 2007.
  * (15) Jeanblanc, M.; Pitman, J.; Yor, M.: Self-similar processes with independent increments associated with Lévy and Bessel processes, _Stoch. Process. Their Appl._ 100, pp. 223–231, 2002.
  * (16) Jeanblanc, M.; Yor, M.; Chesney, M.: _Mathematical Methods for Financial Markets_. Springer Finance (1), Springer-Verlag London 2009.
  * (17) Jedidi, W.: Stable processes, mixing, and distributional properties, Part II. _Teor. Veroyatn. Primen._ 53 (1), 81–105, 2009.
  * (18) Jurek, Z.J.: Limit distributions and one-parameter groups of linear operators on Banach spaces. _J. Multivar. Anal._ 13(4), pp. 578–604, 1983.
  * (19) Jurek, Z.J.: The classes $\mathrm{L}_{m}(Q)$ of probability measures on Banach spaces. _Bull. Acad. Polon. Sci._ 31, 51–62, 1983.
  * (20) Jurek, Z.J.: Self-decomposability: an exception or a rule? _Annales Univer. M. Curie-Sklodowska, Lublin-Polonia_ , vol. LI, Sectio A, pp. 93–107, 1997.
  * (21) Jurek, Z.J.: The random integral representation conjecture: a quarter of a century later. _Lith. Math. J._ 51(3), pp. 362–369, 2011.
  * (22) Jurek, Z.J.: Background driving distribution functions and series representation for log-gamma selfdecomposable random variables, 2019. https://arxiv.org/abs/1904.04160
  * (23) Jurek, Z.J; Mason, J.D.: Operator-Limit Distributions in Probability Theory. John Wiley and Sons, Chichester, 1993.
  * (24) Jurek, Z.J; Smalara, J.: On integrability with respect to infinitely divisible measures. _Bull. Acad. Polon. Sci. Mat._ 29, 179–185, 1981.
  * (25) Jurek, Z.J.; Vervaat, W.: An integral representation for selfdecomposable Banach space valued random variables. _Probab. Theory Relat. Fields_ 62(2), 247–262, 1983.
  * (26) Kanter, M.: Stable densities under change of scale and total variation inequalities. Ann. Probability 3(4), pp. 697–707, 1975.
  * (27) D. B. Karp, E. G. Prilepkina: Completely monotonic gamma ratio and infinitely divisible H-function of Fox. Comput. Methods Funct. Theory 16, 135–153, 2016.
  * (28) Kumar A.; Schreiber, B. M.: Characterization of subclasses of class $L$-probability Distributions. Ann. Prob. 6(2), pp. 279–293, 1978.
  * (29) A. E. Kyprianou, Fluctuations of Lévy Processes with Applications, Introductory Lectures, Second Edition, Springer-Verlag, Berlin Heidelberg, 2006.
  * (30) Li A.J., Chen C.P.: Some completely monotonic functions involving the gamma and Polygamma functions. _J. Korean Math. Soc._ 45(1), pp. 273–287, 2008.
  * (31) Loève, M.: Ranking limit problem. Proceedings of the 3rd Berkeley symposium on mathematical statistics and probability 2, pp. 177–194, 1956.
  * (32) Lv, Y.P.; Sun, T.C. Chu, Y.M: Necessary and sufficient conditions for a class of functions and their reciprocals to be logarithmically completely monotonic J. Inequal. Appl. 36, pp. 1–8, 2011.
  * (33) Mai, J.F; Schenk, S.; Scherer, M.: Two Novel Characterizations of Self-Decomposability on the Half-Line. J. Theor. Probab. 30, 365–383, 2017.
  * (34) Maejima M.: _Classes of Infinitely Divisible Distributions and Examples_. In: Lévy Matters V, Lect. Notes in Math. 2149, Springe, pp. 1–65, 2015.
  * (35) Mehrez, K.: Logarithmically completely monotonic functions related the $\mathrm{q}$–Gamma and the $\mathrm{q}$-digamma functions with applications. Electron. J. Math. Anal. Appl. 6(1), pp. 174–182, 2018.
  * (36) Meerschaert, M.M.; Scheffler, H.P.: Limits Distributions for Sums of Independent Random Vectors, J. Wiley & Sons, New York, 2001.
  * (37) Parthasarathy, K. R.: On the imbedding of an infinitely divisible distribution in a one-parameter convolution semigroup. Teor. Veroyatnost. i Primenen. 12(3), pp. 426–432, 1967.
  * (38) Pestana, D.; Shanbhag, D. N.; Sreehari, M.: Some further results in infinite divisibility. _Math. Proc. Camb. Philos. Soc._ 82, pp. 289–295, 1977.
  * (39) Phelps, R. R.: _Lectures on Choquet’s Theorem_ , Princeton, 1966.
  * (40) Samko S. G.; Kilbas, A.A.; Marichev, O.I.: Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach Science Publishers. Switzerland; Philadelphia, Pa., USA 1993.
  * (41) Sato, K.: Class L of multivariate distributions of its subclasses. _J. Multivariate Anal_. 10, pp. 207–232, 1980.
  * (42) Sato, K. I.: Lévy processes and Infinitely Divisible Distributions. Cambridge Studies in Advanced Mathematics, 68. Cambridge University Press, Cambridge, 1999.
  * (43) Sato, K.: Transformations of infinitely divisible distributions via improper stochastic integrals. Alea 3, 67–110, 2007.
  * (44) Sato, K.: _Fractional Integrals and Extensions of Self-Decomposability_. In: Lecture Notes in Math. (Springer), 2001, Lévy Matters I, pp. 1–91, 2010.
  * (45) Schilling, R.L.; Song, R.; Vondraček, Z.: Bernstein functions. Theory and Applications. Second edition. de Gruyter Studies in Mathematics, 37. Walter de Gruyter & Co., Berlin, 2012.
  * (46) T. Simon: Moment problems related to Bernstein functions, Ann. Fac. Sci. Toulouse Math., Série 6, 29(3), pp. 577–594, 2020.
  * (47) Steutel, F.W., Van Harn, K.: Infinite Divisibility of Probability Distributions on the Real Line. Marcel Dekker, 2004.
  * (48) Urbanik, K.: A representation of self-decomposable distributions. Bull. Acad. Polon. Sci. Ser. Sci. Math. Astronom. Phys. 16, pp. 209–214, 1968.
  * (49) Urbanik, K.: Self-decomposable probability distributions on $\mathds{R}^{m}$. Zastos. Mat. 10, pp. 91–97, 1969.
  * (50) Urbanik, K.: Lévy’s probability measures on euclidean spaces, Studia Math. 44, pp. 119–148, 1972.
  * (51) Urbanik, K.: Slowly varying sequences of random variables, _Bull. de L’Acad. Polon. Sciences_ , Ser. Math. Astr. Phys., vol. XX, 8, pp. 679–682, 1972.
  * (52) Urbanik, K.: Limit Laws for Sequences of Normed Sums Satisfying Some Stability Conditions. In: Multivariate Analysis-III (P Krishnaiah, ed.), pp. 225–237. Academic Press, New York, 1973.
  * (53) Van Thu, N.: An alternative approach to multiple self-decomposable probability measures on Banach spaces, _Probab. Theory Rel. Fields_ 72, pp. 35–54, 1986.
  * (54) Williamson, R.E.: Multiply monotone functions and their Laplace transforms. Duke Math. J. 23(2), pp. 189–207, 1956.
  * (55) Zolotarev, V.M.: _One Dimensional Stable Laws_. Translations of Mathematical Monographsvol, 65. American Mathematical Society, Providence 1986.