El e c t ro n ic
Jo ur
n a l o
f P
r o
b a b il i t y
Vol. 15 (2010), Paper no. 35, pages 1119–1142. Journal URL
http://www.math.washington.edu/~ejpecp/
A new family of mappings of infinitely
divisible distributions related to
the Goldie–Steutel–Bondesson class
Takahiro Aoyama
∗, Alexander Lindner
†and Makoto Maejima
‡Abstract
Let{X(tµ),t ≥ 0}be a Lévy process onRd whose distribution at time 1 is a d-dimensional
in-finitely distribution µ. It is known that the set of all infinitely divisible distributions on Rd,
each of which is represented by the law of a stochastic integralR1 0log
1
td X (µ)
t for some infinitely
divisible distribution onRd, coincides with the Goldie-Steutel-Bondesson class, which, in one
dimension, is the smallest class that contains all mixtures of exponential distributions and is closed under convolution and weak convergence. The purpose of this paper is to study the class of infinitely divisible distributions which are represented as the law ofR1
0
log1
t
1/α
d X(tµ)for generalα >0. These stochastic integrals define a new family of mappings of infinitely divisible distributions. We first study properties of these mappings and their ranges. Then we character-ize some subclasses of the range by stochastic integrals with respect to some compound Poisson processes. Finally, we investigate the limit of the ranges of the iterated mappings .
Key words:infinitely divisible distribution; the Goldie-Steutel-Bondesson class; stochastic inte-gral mapping; compound Poisson process; limit of the ranges of the iterated mappings.
∗Department of Mathematics, Tokyo University of Science, 2641, Yamazaki, Noda 278-8510, Japan. e-mail:
aoyama_takahiro@ma.noda.tus.ac.jp
†Technische Universität Braunschweig, Institut für Mathematische Stochastik, Pockelsstraße 14, D-38106
Braun-schweig, Germany. e-mail: a.lindner@tu-bs.de
‡Department of Mathematics, Keio University, 3-14-1, Hiyoshi, Kohoku-ku, Yokohama 223-8522, Japan. e-mail:
AMS 2000 Subject Classification:Primary 60E07.
1
Introduction
Throughout this paper, L(X) denotes the law of an Rd-valued random variable X and µb(z),z ∈
Rd, denotes the characteristic function of a probability distribution µ on Rd. Also I(Rd)
denotes the class of all infinitely divisible distributions on Rd, Isym(Rd) = {µ ∈ I(Rd) :
µis symmetric onRd}, Iri(Rd) = {µ ∈ I(Rd) :µis rotationally invariant on Rd}, Ilog(Rd) = {µ ∈
I(Rd):R|x|>
1log|x|µ(d x)< ∞}and Ilogm(R
d) ={µ∈I(Rd):R
|x|>1(log|x|)
mµ(d x)<∞}, where
|x|is the Euclidean norm ofx ∈Rd. LetCµ(z),z∈Rd, be the cumulant function ofµ∈I(Rd). That
is, Cµ(z)is the unique continuous function withCµ(0) =0 such thatµb(z) =exp¦Cµ(z)©,z ∈Rd.
Whenµis the distribution of a random variableX, we also writeCX(z):=Cµ(z).
We use the Lévy-Khintchine generating triplet(A,ν,γ)ofµ∈I(Rd)in the sense that
Cµ(z) =−2−1〈z,Az〉+i〈γ,z〉
+
Z
Rd
ei〈z,x〉−1−i〈z,x〉(1+|x|2)−1ν(d x),z∈Rd,
where 〈·,·〉 denotes the inner product inRd, Ais a symmetric nonnegative-definite d×d matrix,
γ∈Rd andν is a measure (called the Lévy measure) on Rd satisfyingν({0}) =0 and R
Rd(|x|2∧
1)ν(d x)<∞.
The polar decomposition of the Lévy measureν ofµ∈I(Rd), with 0< ν(Rd)≤ ∞, is the following:
There exist a measureλon S={ξ∈Rd :|ξ|=1}with 0< λ(S)≤ ∞and a family {νξ:ξ∈S}of
measures on(0,∞)such thatνξ(B)is measurable inξfor eachB∈ B((0,∞)), 0< νξ((0,∞))≤ ∞
for eachξ∈S,
ν(B) =
Z
S λ(dξ)
Z ∞
0
1B(rξ)νξ(d r), B∈ B(Rd\ {0}). (1.1)
Here λ and {νξ} are uniquely determined by ν in the following sense : If λ,{νξ} and λ′,{νξ′}
both have the same properties as above, then there is a measurable function c(ξ) on S such that
0<c(ξ)<∞,λ′(dξ) =c(ξ)λ(dξ)andc(ξ)νξ′(d r) =νξ(d r)forλ-a.e. ξ∈S. The measureνξis a
Lévy measure on(0,∞)forλ-a.e. ξ∈S. We say thatν has the polar decomposition(λ,νξ)andνξ
is called the radial component ofν. (See, e.g., Lemma 2.1 of[3]and its proof.)
Remark 1.1. For µ ∈ Iri(Rd) with generating triplet (A,ν,γ), it is necessary and sufficient that
A U=UAholds for arbitraryd×dorthogonal matrixU,γ=0 andλandνξcan be chosen such that
λis Lebesgue measure andνξis independent ofξ.
Letµ∈I(Rd)and{Xt(µ),t≥0}denote the Lévy process onRd withµas the distribution at time 1.
For a nonrandom measurable function f on(0,∞), we define a mapping
Φf(µ) =L
Z ∞
0
f(t)d Xt(µ)
, (1.2)
whenever the stochastic integral on the right-hand side is definable in the sense of stochastic
in-tegrals based on independently scattered random measures onRd induced by{X(µ)
t }, as in
Defini-tions 2.3 and 3.1 of Sato[15]. When the support of f is a finite interval (0,a], R∞
0 f(t)d X
(µ)
Ra
0 f(t)d X
(µ)
t , and when the support of f is (0,∞),
R∞
0 f(t)d X
(µ)
t is the limit in probability of
Ra
0 f(t)d X
(µ)
t as a → ∞. D(Φf) denotes the set ofµ∈ I(Rd) for which the stochastic integral in
(1.2) is definable. When we consider the composition of two mappings Φf and Φg, denoted by
Φg◦Φf, the domain ofΦg◦Φf isD(Φg◦Φf) ={µ∈I(Rd):µ∈D(Φf)andΦf(µ)∈D(Φg)}. Once
we define such a mapping, we can characterize a subclass ofI(Rd)as the range ofΦf,R(Φf), say.
In Barndorff-Nielsen et al.[3], they studied the Upsilon mapping
Υ(µ) =L
Z 1
0
log1
td X
(µ)
t
!
, (1.3)
and showed that its rangeR(Υ)is the Goldie–Steutel–Bondesson class,B(Rd), say, that is
Υ(I(Rd)) =B(Rd). (1.4)
It is also known that µ ∈ B(Rd) can be characterized in terms of Lévy measures as follows: A
distributionµ∈I(Rd)belongs toB(Rd)if and only if the Lévy measureν ofµis identically zero or
in caseν6=0,νξin (1.1) satisfies thatνξ(d r) =gξ(r)d r,r>0, wheregξ(r)is completely monotone
inr∈(0,∞)forλ-a.e. ξand measurable inξfor each r>0.
Our purpose of this paper is to generalize (1.3) to
Eα(µ):=L
Z 1
0
log1
t
1/α d X(tµ)
!
for any α > 0, whereE1 = Υ, and investigate R(Eα). We first generalize (1.4) and characterize
Eα(I(Rd)), in the sense of what should replaceB(Rd)for generalα >0. For that, we need a new
class Eα(Rd),α >0. Namely, we say that µ∈ I(Rd)belongs to the class Eα(Rd)if ν =0 or ν 6=0 andνξ in (1.1) satisfies
νξ(d r) =rα−1gξ(rα)d r, r>0,
for some functiongξ(r), which is completely monotone inr∈(0,∞)forλ-a.e.ξ, and is measurable
inξfor eachr>0. Then we will show thatEα(I(Rd)) =Eα(Rd)in Theorem 2.3.
In addition to that, we have two motivations of this generalization of the mapping. On R+, the
Goldie–Steutel–Bondesson classB(R+)is the smallest class that contains all mixtures of exponential
distributions and is closed under convolution and weak convergence. In addition, we denote by
B0(R+)the subclass ofB(R+), where all distributions do not have drift.
It is similarly extended to a class onR, and in Barndorff-Nielsen et al.[3]it was proved thatB(Rd)
in (1.4) is the smallest class of distributions onRd closed under convolution and weak convergence
and containing the distributions of all elementary mixed exponential variables inRd. Here, anRd
-valued random variableU x is called an elementary mixed exponential random variable inRd ifx is
a nonrandom nonzero vector inRd andU is a real random variable whose distribution is a mixture
of a finite number of exponential distributions. The first motivation is to characterize a subclass
of I(Rd) based on a single Lévy process. This type of characterization is quite different from the
characterization in terms of the range of some mappingR(Φf). This type of characterization is also
done by James et al.[6]for the Thorin class. As toB0(R+), we have the following, which is a special
Theorem 1.2. Let Z ={Zt}t≥0 be a compound Poisson process on R+ with Lévy measureνZ(d x) = e−xd x,x >0. Then
B0(R+) =
¨
L
Z ∞
0
h(t)d Zt
, h∈Dom(Z)
«
,
where Dom(Z) is the set of nonrandom measurable functions h for which the stochastic integrals
R∞
0 h(t)d Zt are definable.
We are going to generalize this underlying compound Process Y to other Y with Lévy measure
xα−1e−xαd x,x >0,α >0, and furthermore to the two-sided case.
The second motivation is the following. In Maejima and Sato [9], they showed that the limits of
nested subclasses constructed by iterations of several mappings are identical with the closure of the class of the stable distributions, where the closure is taken under convolution and weak convergence.
We are going to show that this fact is also true forM-mapping, which is defined by
M(µ) =L
Z ∞
0
m∗(t)d Xt(µ)
, µ∈Ilog(Rd),
where m(x) = R∞
0 u −1e−u2
du,x > 0 and m∗(t) is its inverse function in the sense that m(x) =
t if and only if x = m∗(t). This mapping (in the symmetric case) was introduced in Aoyama
et al.[2], as a subclass of selfdecomposable and type G distributions. In Maejima and Sato [9],
limm→∞Mm(Ilogm(Rd)) is not treated, and we want to show that this limit is also equal to the
closure of the class of the stable distributions. For the proof, we need our new mappingE2. Namely,
the proof is based on the fact that
M(µ) = (Φ◦ E2)(µ) = (E2◦Φ)(µ), µ∈Ilog(Rd), (1.5)
whereΦ(µ) =LR∞
0 e
−td X(µ)
t
withD(Φ) =Ilog(Rd).
The paper is organized as follows. In Section 2, we show several properties of the mappingEα. In
Section 3, we show thatEα(Rd) =Eα(I(Rd)), α >0. This relation has the meaning thatµe∈Eα(Rd)
is characterized by a stochastic integral representation with respect to a Lévy process. Also we characterize Eα(Rd),Eα+(Rd) and Eαsym(Rd) := Eα(Rd)∩Isym(Rd) based on one compound Poisson
distribution onR, whereEα+(Rd) ={µ∈Eα(Rd):µ(Rd\[0,∞)d) =0}. In Section 4, we characterize
Eα0,ri(Rd):={µ∈Eα(Rd):µhas no Gaussian part} ∩Iri(Rd) (1.6)
and certain subclasses ofEα(R1)which correspond to Lévy processes of bounded variation with zero
drift, by (essential improper) stochastic integrals with respect to some compound Poisson processes.
This gives us a new sight of the Goldie–Steutel–Bondesson class inR1. In Section 5, we consider the
compositionΦ◦ Eα, and we apply this composition to show that limm→∞(Φ◦ Eα)m(Ilogm(Rd))is the
closure of the class of the stable distributions as Maejima and Sato[9]showed for other mappings.
Since we will see thatΦ◦ E2=M, we can answer the question mentioned in the second motivation
2
Several properties of the mapping
E
αand the range of
E
αWe start with showing several properties of the mappingEα.
Proposition 2.1. Letα >0. bounded variation with drift
e
where the limit is almost sure.
Proof. (The proof follows along the lines of Proposition 2.4 of Barndorff-Nielsen et al.[3]. However, we give the proof for the completeness of the paper.)
(i) The function f(t) = (logt−1)1/α1(0,1](t) is clearly square integrable, hence the result follows
from Sato[13], see also Lemma 2.3 in Maejima[7].
(ii) By a general result (see Lemma 2.7 and Corollary 4.4 of Sato[12]) and a change of variable,
e
The additional part follows immediately from Theorem 3.15 in Sato[15].
(iii) By (i), we have for eachz∈Rd,
(iv) Apart from minor adjustments, the proof is the same as that of Proposition 2.4 (v) in
Barndorff-Nielsen et al.[3]and hence omitted.
(v) The first equality is clear by duality (e.g. Sato [11], Proposition 41.8). For the second, we
conclude using partial integration (e.g. Sato[12], Corollary 4.9) that for eachs∈(0, 1]it holds
Z 1
But by Proposition 47.11 in Sato [11], applied to each component of X(tµ) separately, it holds
lim supt↓0tǫ−1/2|X(tµ)|=0 a.s. for eachǫ >0, which shows lims↓0Xs(µ)(logs−1)1/α =0, the almost
sure convergence of the integral at 0 and the second equality.
Corollary 2.2. Letα >0. Then a distributionµis symmetric if and only ifEα(µ)is symmetric.
Proof. Note that for a random variable X with the cumulant function CX(z), L(X) is symmetric
Proof (i) (Proof for that Eα(Rd) ⊃ Eα(I(Rd)).) Let µe ∈ Eα(I(Rd)). Then µe =
whereν is the Lévy measure ofµandνξ below is the radial component ofν. Thus, the spherical
componentλe ofνe is equal to the spherical componentλ ofν, and the radial component eνξ ofνe
with the measureQeξbeing defined by
e
Qξ(B) =α
Z ∞
0
1B(x−α)x−ανξ(d x), B∈ B((0,∞)).
We conclude that egξ(·)is completely monotone. Thus,
+
where we have used Fubini’s theorem and the substitutionu=rαt. >From this it is easy to see that
e
ν is a Lévy measure if and only ifR
Sλe(dξ)Qeξ({0}) =0 (which we shall assume without comment
from now on) and
Z
Define further a measureν to have spherical componentλ=λeand radial partsνξ, i.e.
ν(B) =
Thenν is a Lévy measure, since
Z
which is finite by (2.4). Ifµis any infinitely divisible distribution with Lévy measure ν, then part
(i) of the proof shows thatEα(µ)has the given Lévy measureνe, and from the transformation of the
generating triplet in Proposition 2.1 we see thatµ∈I(Rd)can be chosen such thatE
α(µ) =µe.
3
The class
E
α(
R
d)
and its subclasses
Theorem 3.1. For any0< α < β,
Eα(Rd)⊂Eβ(Rd).
Proof.Let 0< α < β. Then ifµ∈Eα(Rd),νξofµis
νξ(d r) =rα−1gξ(rα)d r=rβ−1
gξ(rα/β)β
rβ−α d r =r β−1gξ
(rα/β)β
r(β−α)/ββ d r.
Let
hξ(x) = gξ(x
α/β)
x(β−α)/β.
Note that if g is completely monotone and ψ a nonnegative function such that ψ′ is completely
monotone, then the composition g◦ψ is completely monotone (see, e.g., Feller [5], page 441,
Corollary 2), and if g and f are completely monotone then g f is completely monotone. Thus
gξ(xα/β)is completely monotone and thenhξ(x)is also completely monotone, and we have
νξ(d r) =rβ−1hξ(rβ).
Henceµ∈Eβ(Rd).
In the following, we shall call a class F of distributions inRd closed under scalingif for every Rd
-valued random variableX such thatL(X)∈F it also holds thatL(cX)∈Ffor everyc>0. IfFis a
class of infinitely divisible distributions onRd and satisfies thatµ∈F impliesµs∗∈F for anys>0,
whereµs∗ is the distribution with characteristic function(µb(z))s, we shall call F closed under taking of powers. Recall that a classF of infinitely divisible distributions onRdis calledcompletely closed in the strong sense(abbreviated as c.c.s.s.) if it is closed under convolution, weak convergence, scaling,
taking of powers, and additionally containsµ∗δbfor anyµ∈F andb∈Rd.
Recall thatS={ξ∈Rd :|ξ|=1}andµ∈I(Rd)belongs to the classEα(Rd)ifν =0 orν 6=0 and
νξ in (1.1) satisfiesνξ(d r) = rα−1gξ(rα)d r, r > 0, for some function gξ(r), which is completely
monotone inr∈(0,∞)forλ-a.e.ξ, and is measurable inξfor eachr >0. Denote
S+:={ξ= (ξ1, . . . ,ξd)∈S:ξ1, . . . ,ξd≥0}.
Theorem 3.2. Letα >0and Y1(α)and Z1(α)be compound Poisson distributions onRwith Lévy measures νY(α)
1
(d x) = |x|α−1e−xαd x and νZ(α)
1
(d x) = xα−1e−xα1(0,∞)(x)d x, respectively. Then we have the
following.
(i) The class Eα(Rd) is the smallest class of infinitely divisible distributions on Rd which is closed under convolution, weak convergence, scaling, taking of powers and contains each of the distributions
L(Z1(α)ξ)withξ∈S. Further, Eα(Rd)is c.c.s.s.
(ii)The class Eα+(Rd) ={µ∈E
α(Rd):µ(Rd\[0,∞)d) =0}is the smallest class of infinitely divisible distributions onRd which is closed under convolution, weak convergence, scaling, taking of powers and contains each of the distributionsL(Z1(α)ξ)withξ∈S+.
(iii)The class Eαsym(Rd) =Eα(Rd)∩Isym(Rd)is the smallest class of infinitely divisible distributions on
Proof. By the definition it is clear that all the classes under consideration are closed under
con-volution, scaling and taking of powers. The class Eα(Rd) is closed under weak convergence by
Proposition 2.1 (iv) and Theorem 2.3, and hence so areEα+(Rd)andEsym
α (Rd). Further, it is easy to
see that all the given classes contain the specified distributions, since the Lévy measure ofL(Z1(α)ξ)
forξ∈S has polar decomposition λ = δξ and νξ(d r) = rα−1gξ(rα)d r with gξ(r) = e−r, and a
similar argument works for L(Y1(α)ξ). Finally, Eα(Rd) contains all Dirac measures, which shows
that it is c.c.s.s. So it only remains to show that the given classes are the smallest classes among all classes with the specified properties.
(i) LetF be the smallest class of infinitely divisible distributions which is closed under convolution,
weak convergence, scaling, taking of powers and which contains L(Z1(α)ξ) for every ξ ∈ S. As
already shown, this impliesF ⊂ Eα(Rd). Recall from Theorem 2.3 thatEα defines a bijection from
I(Rd)onto E
α(Rd), and letG :=Eα−1(F). ThenG is closed under convolution, weak convergence,
scaling and taking of powers. This follows from the corresponding properties ofF and the definition
ofEαfor the third property, and Proposition 2.1 (ii) and (iv) for the first, fourth and second property,
respectively.
It is easy to see from Proposition 2.1 (ii) that forξ∈S, µξ := Eα−1(Z
(α)
1 ξ) has generating triplet
(A = 0,ν = α−1δξ,γ) for some γ ∈ Rd, so that {X
(µξ)
t } has bounded variation, and its drift is
zero by (2.3) since {XL(Z
(α)
1 ξ)
t } has zero drift. This shows that µξ = L(N1ξ) where {Nt}t≥0 is a
Poisson process with parameter 1/α, and we haveµξ∈G by assumption. Since G is closed under
convolution and scaling this implies thatL(n−1Nnξ) ∈ G for each n∈N and hence E(N1)ξ∈G
by the strong law of large numbers since G is closed under weak convergence. Since E(N1) > 0
and G is closed under taking of powers this shows thatδc ∈ G for all c ∈Rd. HenceG contains
every infinitely divisible distribution with Gaussian part zero and Lévy measureα−1δξ withξ∈S.
Since G is closed under convolution, scaling and taking of powers it also contains all infinitely
divisible distributions with Gaussian part zero and Lévy measures of the formν =Pni=1aiδci with
n ∈ N, ai ≥ 0 and ci ∈ Rd\ {0}. Since every finite Borel measure on Rd is the weak limit of a
sequence of measures of the formPni=1aiδci, it follows from Theorem 8.7 in Sato[11]and the fact
that G is closed under weak convergence that G contains all compound Poisson distributions, and
hence all infinitely divisible distributions by Corollary 8.8 in[11]. This showsG=I(Rd)and hence
F=Eα(Rd)by Theorem 2.3.
(ii) and (iii) follow in analogy to the proof of (i), where for (iii) observe that Eα−1(Y1(α)ξ) has
characteristic triplet(A=0,ν=α−1δξ+α−1δ−ξ,γ=0), so that, by an argument similar to the proof
of (i), every symmetric compound Poisson distribution is inEα−1(F)and hence so every symmetric
infinitely divisible distribution is. HereF is the smallest class of infinitely divisible distributions on
Rd which is closed under convolution, weak convergence, scaling, taking of powers and contains
each of the distributions L(Y1(α)ξ) with ζ ∈ S. Corollary 2.2 and Theorem 2.3 then imply F =
Eαsym(Rd).
Remark 3.3. In the introduction it was mentioned thatB(Rd)is the smallest class of distributions on
Rd closed under convolution and weak convergence and containing the distributions of all
elemen-tary mixed exponential random variables inRd. Theorem 3.2 forα=1 gives a new interpretation
Remark 3.4. Once we are given a mappingEα, we can construct nested classes of Eα(Rd) by the
iteration of the mappingEα, which isEαm = Eα◦ · · · ◦ Eα (m-times composition). It is easy to see
that D(Em
α) = I(Rd) for anym∈N. Then we can characterizeEαm(I(Rd))as the smallest class of
infinitely divisible distributions which is closed under convolution, weak convergence, scaling and
taking of powers and containsEαm(N1ξ)for allξ∈S andN1being a Poisson distribution with mean
1/α. The same proof of Theorem 3.4 works, but we do not go into the details here.
4
Characterization of subclasses of
E
α(
R
d)
by stochastic integrals with
respect to some compound Poisson processes
For any Lévy processY ={Yt}t≥0 on Rd, denote byL(0,∞)(Y)the class of locallyY-integrable, real
Here, following Definition 3.1 of Sato [15], by saying that the (improper stochastic integral)
R∞
0 h(t)d Yt is definable we mean that
Rq
ph(t)d Ytconverges in probability as p↓0,q→ ∞, with the
limit random variable being denoted byR0∞h(t)d Yt.
The property of h belonging to Dom(Y) can be characterized in terms of the generating triplet
(AY,νY,γY)ofY and assumptions onh, cf. Sato[15], Theorems 2.6, 3.5 and 3.10. In particular, if
if and only if (4.1) is satisfied, in which caseγY,hin the generating triplet of
R∞
Recall thatEα0,ri(Rd) ={µ∈Eα(Rd):µhas no Gaussian part} ∩Iri(Rd). The next theorem charac-terizesEα0,ri(Rd)as the class of distributions which arise as improper stochastic integrals over(0,∞)
with respect to some fixed rotationally invariant compound Poisson process onRd.
Theorem 4.1. Letα >0and denote by Y(α)={Yt(α)}t≥0 a compound Poisson process onRdwith Lévy measureνY(α)(B) =
R
Sdξ
R∞
0 1B(rξ)r
α−1e−rαd r, equivalently
νY(α)(dξd r) =dξrα−1e−r
α
d r, ξ∈S, r >0 (4.5)
(without drift). Then
Eα0,ri(Rd) =
¨
L
Z ∞
0
h(t)d Yt(α)
: h∈Dom(Y(α))
«
(4.6)
=
¨
L
Z ∞
0
h(t)d Yt(α)
: h∈Dom↓(Y(α))
«
. (4.7)
The function h∈Dom↓(Y(α))in representation(4.7)is uniquely determined byµ∈Eα0,ri(Rd).
Proof. Let µ ∈ Eα0,ri(Rd). By definition and Remark 1.1, the Lévy measure ν of µ has the polar
decomposition(λ,νξ)given by
νξ(d r) =rα−1g(rα)d r, r>0, λ(dξ) =dξ, (4.8)
and g is independent of ξand completely monotone. (Ifµ= δ0 we define gξ =0 and shall also
call (λ,νξ) a polar decomposition, even if νξ is not strictly positive here). Since g is completely
monotone, there exists a Borel measureQon [0,∞) such thatg(y) =R[0,∞)e−y tQ(d t). By (2.4),
sinceνξsatisfies
R∞ 0 (r
2∧1)ν
ξ(d r)<∞, we see that
Q({0}) =0,
Z 1
0
t−1Q(d t)<∞ and
Z ∞
1
t−1−2/αQ(d t)<∞. (4.9)
Observe that under this condition, we have for eachr>0,
νξ([r,∞)) =
Z ∞
r
yα−1g(yα)d y=
Z ∞
0
(αt)−1Q(d t)
Z ∞
r
αt yα−1e−yαtd y
=
Z ∞
0
(αt)−1e−rαtQ(d t).
Next, observe that sinceY(α)is rotationally invariant without Gaussian part, we have by (4.1) that
a measurable functionhis in Dom(Y(α))if and only if
Z ∞
0 ds
Z ∞
0
in which case R0∞h(t)d Yt(α) is infinitely divisible with the generating triplet (AY,h=0,νY,h,γY,h=0)and the Lévy measureνY,his rotationally invariant. SupposeB=C×[r,∞),
whereC ∈ B(S)andr >0. Then by (4.4) and (4.5),
νY,h(B) =νY,h(C×[r,∞)) =
Z ∞
0 ds
Z
S
1C(ξ)dξ
Z ∞
r/|h(s)|
xα−1e−xαd x (4.11)
=α−1|C|
Z ∞
0
e−rα/|h(s)|αds
for everyr >0, where|C|is the Lebesgue measure of C onS. Hence, in order to prove (4.6) and
(4.7), it is enough to prove the following:
(a) For each Borel measureQon[0,∞)satisfying (4.9) there exists a functionh∈Dom↓(Y(α))such
that Z
∞
0
t−1e−rαtQ(d t) =
Z ∞
0
e−rα/|h(s)|αds for everyr>0. (4.12)
(b) For each h∈ Dom(Y(α)) there exists a Borel measure Q on [0,∞) satisfying (4.9) such that
(4.12) holds.
To show (a), letQsatisfy (4.9), and denote
F(x):=
Z
(0,x]
t−1Q(d t), x∈[0,∞),
and by
F←(t) =inf{y≥0 :F(y)≥t}, t∈[0,∞),
its left-continuous inverse, with the usual convention inf;= +∞. Now define
h=hQ:(0,∞)→[0,∞), t7→(F←(t))−1/α.
Thenhis left-continuous, decreasing, and satisfies limt→∞h(t) =0. Denote Lebesgue measure on
(0,∞)bym1, and consider the function
T:(0,∞)→(0,∞], s7→h(s)−α=F←(s). (4.13)
Then (T(m1))|(0,∞), the image measure of m1 under the mapping T, when restricted to (0,∞),
satisfies
(T(m1))|(0,∞)(d t) =t−1Q|(0,∞)(d t). (4.14)
Hence it follows that for everyr>0,
Z
(0,∞)
e−rα/h(s)αm1(ds) =
Z
(0,∞)∩{s:T(s)6=∞}
e−rαT(s)m1(ds)
=
Z
(0,∞)
e−rαt(T(m1))(d t), (4.15)
yielding (4.12). To show (4.10), namely thath∈Dom(Y(α)), observe that
Z ∞
0 ds
Z ∞
0
=
by (4.14). The second of these terms is clearly finite by (4.9). To estimate the first, observe that
Z ∞
and the first two summands are finite by (4.9), while the last summand is equal to
Z 1
and hence also finite. This shows (4.10) forhand hence (a).
To show (b), leth∈Dom(Y(α))and assume first thathis nonnegative. Let T :(0,∞)→(0,∞]be
defined byT(s) =h(s)−α as in (4.13), and consider the image measureT(m1). Define the measure
Q on [0,∞) by Q({0}) = 0 and equality (4.14). Since R∞
0 h(t)d Y
(α)
t is automatically infinitely
divisible with Lévy measureνY,hgiven by (4.11), we have as in the proof of (a) for everyC ∈ B(S)
In particular,Qmust be a Borel measure and (4.12) holds. Since the left hand side of this equation
converges and the right hand side is known to be the tail integral of a Lévy measure, it follows from the proof of (2.4) that (4.9) must hold. Hence we have seen thatL(R0∞h(t)d Yt(α))∈Eα0,ri(Rd)for
h+andh−, respectively, completing the proof of (b).
Define the functions T1,T2 : (0,∞) → (0,∞] by T1(s) := h1(s)−α and T2(s) := h2(s)−α. It then follows from (4.11) that
Z ∞
0
e−rα/|h1(s)|αds=
Z ∞
0
e−rα/|h2(s)|αds<∞
for allr>0, which using the argument of (4.15) can be written as
Z
(0,∞)
e−rαt(T1(m1))(d t) =
Z
(0,∞)
e−rαt(T2(m1))(d t)<∞, r>0. (4.16)
Observe thatT1andT2are left-continuous increasing functions with lims→∞T1(s) =lims→∞T2(s) =
∞. Hence Ti(m1)((0,b]) < ∞ for all b ∈ (0,∞), i = 1, 2, and it follows from (4.16) and the
uniqueness theorem for Laplace transforms of Borel measures on[0,∞)that
(T1(m1))|(0,∞)= (T2(m1))|(0,∞).
In other words we have for everyb∈(0,∞)that
m1({s∈(0,∞):T1(s)≤ b}) =m1({s∈(0,∞):T2(s)≤ b})<∞.
SinceT1and T2 are left-continuous and increasing, this clearly impliesT1=T2and henceh1 =h2,
completing the proof of the uniqueness assertion in representation (4.7).
Next, we assumed=1 and we ask whether every distribution in
Eα0(R1):={µ∈Eα(R1):µhas no Gaussian part}
can be represented as a stochastic integral with respect to the compound Poisson processZ(α)
hav-ing Lévy measureνZ(α)(d x) = xα−1e−x
α
1(0,∞)(x)d x (without drift) plus some constant. We shall
prove that such a statement is true e.g. for those distributions in Eα0(R1)which correspond to Lévy
processes of bounded variation, but that not every distribution inEα0(R1)can be represented in this
way. However, every distribution in Eα0(R1) appears as an essential limit of locally Z(α)-integrable
functions. Following Sato [15], Definition 3.2, for a Lévy process Y = {Yt}t≥0 and a locally Y
-integrable functionhover(0,∞) we say thatthe essential improper stochastic integral on(0,∞) of
h with respect to Y is definableif for every 0< p< q< ∞there are real constants τp,q such that
Rq
p h(t)d Yt−τp,q converges in probability asp↓0, q→ ∞. We write Domes(Y)for the class of all
locallyY-integrable functionshon(0,∞)for which the essential improper stochastic integral with
respect toY is definable, and for eachh∈Domes(Y)we denote the class of distributions arising as
possible limitsRq
ph(t)d Yt−τp,qas p↓0,q→ ∞byΦh,es(Y)(the limit is not unique, since different
sequencesτp,qmay give different limit random variables). As for Dom(Y), the property of belonging
to Domes(Y)can be expressed in terms of the characteristic triplet(AY,νY,γY)ofY. In particular, if
AY =0, then a functionhon(0,∞)is in Domes(Y)if and only ifhis measurable and (4.1) and (4.2)
hold, and in that case Φh,es(Y)consists of all infinitely divisible distributionsµ with characteristic
triplet(AY,h=0,νY,h,γ), whereνY,h is given by (4.4) andγ∈Ris arbitrary (cf.[15], Theorems 3.6
and 3.11).
RecallEα+(R1) ={µ∈Eα(R1):µ((−∞, 0)) =0}and denote
EαBV(R1):={µ∈Eα(R1): {X(tµ)}is of bounded variation},
(i)The class of distributions arising as limits of essential improper stochastic integrals with respect to Z(α)is Eα0(R1):
(iii)Not every distribution in Eα0(R1)can be represented as an improper stochastic integral over(0,∞)
with respect to Z(α)plus some constant. It holds
EαBV(R1)∪Eα0,sym(R1)$
positive and negative parts of h, respectively. Then µis infinitely divisible without Gaussian part
and by (4.4) its Lévy measureνZ,hsatisfies
Then as in the proof of Theorem 4.1,
Z
(0,∞)
e−rαt(αt)−1Qξ(d t) =νZ,h,ξ([r,∞)), r>0, ξ∈ {−1, 1},
andQ1andQ−1satisfy (4.9) and we conclude thatνZ,h,ξ(d r) =rα−1gξ(rα)d rfor completely
mono-tone functionsg1andg−1, so thatΦh,es(Z(α))⊂Eα0(R
1), giving the inclusion “⊃” in equation (4.17).
Now letµ∈Eα0(R1)with Lévy measureν, and define the Lévy measuresν1 andν−1 supported on
[0,∞)by
ν1(B):=ν(B), ν−1(B):=ν(−B), B∈ B((0,∞)). (4.22)
Then
νξ([r,∞)) =
Z ∞
0
(αt)−1e−rαtQξ(d t), r >0, ξ∈ {−1, 1}, (4.23)
for some Borel measures Q1 andQ−1 satisfying (4.9). As in the proof of (a) in Theorem 4.1, we
find nonnegative and decreasing functions h1,h−1 :(0,∞) → [0,∞) such that (4.10) (i.e. (4.1)
withνZ(α) in place ofνY) and (4.12) hold. Sinceh1,h−1 are bounded on compact subintervals of
(0,∞) and since Z(α) has bounded variation, it follows that h1 and h−1 satisfy also (4.2), so that
h1,h−1∈Domes(Z(α))and the Lévy measures ofµe1∈Φh1,es(Z
(α))and e
µ−1∈Φh−1,es(Z
(α))are given
byν1andν−1, respectively. Now define the functionh:(0,∞)→Rby
h(t) =
h1(t−n), t∈(2n, 2n+1], n∈ {1, 2, . . .},
−h−1(t−n−1), t∈(2n+1, 2n+2], n∈ {1, 2, . . .},
h1(t−2−k−1), t∈(2−k, 2−k+2−k−1], k∈ {0, 1, 2, . . .},
−h−1(t−2−k), t∈(2−k+2−k−1, 2−k+1], k∈ {0, 1, 2, . . .}.
(4.24)
Then alsoh∈Domes(Z(α))and anyµe∈Φh,es(Z(α))has Lévy measureν, showing the inclusion “⊂” in equation (4.17).
(ii) Leth∈Dom(Z(α)). ThenR∞
0 h(t)d Z
(α)
t ∈Eα0(R1)by (i). Further, by Theorem 3.15 in Sato[15],
R∞ 0 h(t)d Z
(α)
t is the distribution at time 1 of a Lévy process of bounded variation if and only if
Z ∞
0 ds
Z
R
(|h(s)x| ∧1)νZ(α)(d x)<∞, (4.25)
in which case this Lévy process will have zero drift. SinceL(R0∞h(t)d Zt(α)) has trivially support
contained in[0,∞)ifh≥0, this gives the inclusion “⊃” in (4.18) and (4.19).
Now suppose thatµ∈EαBV,0(R1)with Lévy measureν, defineν1andν−1by (4.22) and choose Borel
measuresQ1andQ−1such that (4.23) holds. Then it can be shown in complete analogy to the proof
leading to (2.4) that forξ∈ {−1, 1},νξsatisfies
R∞
0 (1∧x)νξ(d x)<∞if and only if
Qξ({0}) =0,
Z 1
0
t−1Qξ(d t)<∞ and
Z ∞
1
t−1−1/αQξ(d t)<∞. (4.26)
Forξ∈ {−1, 1}andx ∈[0,∞)defineFξ(x):=
R (0,x]t
−1Q
ξ(d t),hξ= (Fξ←)−1/αandTξ= (hξ)−α=
and (4.25) hold forhξandQξ. By Theorem 3.15 in Sato[15]this then shows thathξ∈Dom(Z(α))
But sinceZ(α)has the generating triplet
this shows (4.21) apart from the fact that the inclusions are proper.
To show that the first inclusion in (4.21) is proper, let µ ∈ Eα0,sym(R1)\EαBV(R1). The latter set
is nonempty since by (4.9) and (4.26) it suffices to find a Borel measure Q on [0,∞) such that
(4.9) holds but R∞
neither symmetric nor of finite variation.
To see that the second inclusion in (4.21) is proper, let µ ∈ Eα0(R1) with Lévy measureν being
supported on[0,∞)such that R1
0 xν(d x) =∞. Suppose there are b∈Randh∈Dom(Z
(α))such
thatµ=L(R0∞h(t)d Zt(α)+b). Sinceνis supported on[0,∞), we must haveh≥0 Lebesgue almost
surely, so that we can suppose thath≥0 everywhere. Then we have from (4.1) and (4.3) that
Together these two equations imply
Proof of Theorem 1.2. This is an immediate consequence of Equation (4.18) since B0(R+) =
E1+,0(R1).
5
The composition of
Φ
with
E
αand its application
Recall thatΦ(µ) =LR∞
tionΦ◦ Eα. We start with the following proposition.
Proposition 5.1. Let α > 0, m ∈ {1, 2, . . .} and µ ∈ I(Rd). Then µ ∈ I
for every measurable nonnegative functionϕ:Rd→[0,∞]. In particular, we have
It then holds
Φ◦ Eα=Eα◦Φ =Nα, (5.1)
including the equality of the domains. In particular, we have
Φ◦ E2=E2◦Φ =M. (5.2)
Proof. We first note that D(Nα) is independent of the value of α and equals Ilog(Rd), shown in
Theorem 2.3 of[8], (essentially in Theorem 2.4 (i) of[14].)
As mentioned right after Equation (1.5),D(Φ) =Ilog(Rd). Thus it follows from Proposition 5.1 that
bothΦ◦ Eαas well asEα◦Φare well defined onIlog(Rd)and that they have the same domain. Note
Then, if we are allowed to exchange the order of the integrals by Fubini’s theorem, we have
C(Eα◦Φ)(µ)(z) =
In order to assure the exchange of the order of the integrations by Fubini’s theorem, it is enough to
show that Z ∞
This is Equation (4.5) in Barndorff-Nielsen et al.[3]with the replacement ofsbys1/α. Hence, the
proof of (4.5) in Barndorff-Nielsen et al.[3]works also here and concludes (5.4). So, we omit the
detailed calculation. Thus, the calculation in (5.3) is verified, and we have that
and thatΦ◦ Eα=Eα◦Φ =Nα. SinceN2=M, this shows in particular (5.2).
It is well known thatΦ(Ilog(Rd)) = L(Rd), the class of selfdecomposable distributions on Rd. An
immediate consequence of Theorem 5.2 is the following.
Theorem 5.3. Letα >0. Then
Φ(Eα(Rd)∩Ilog(Rd)) =Eα(L(Rd)) =Nα(Ilog(Rd)).
We conclude this section with an application of the relation (5.1) to characterize the limit of certain
subclasses obtained by the iteration of the mappingNα. We need some lemmas. In the following,
Nm
α is defined recursively asNm
+1
α =Nαm◦ Nα.
Lemma 5.4. Letα >0.For m=1, 2, . . ., we have
D(Nm
α ) =Ilogm(Rd) and Nαm= Φ m◦ Em
α =E m α ◦Φ
m.
Proof. By Proposition 5.1, we have µ ∈ Ilogm(Rd) if and only if Eα(µ) ∈ Ilogm(Rd). As shown in
the proof of Lemma 3.8 in [9], we also have that µ ∈ Ilogm+1(Rd) if and only if µ∈ Ilog(Rd) and Φ(µ)∈Ilogm(Rd), and thusD(Φm) =Ilogm(Rd). SinceNα= Φ◦ Eα=Eα◦Φ, we conclude that
µ∈Ilogm+1(Rd)if and only ifµ∈Ilog(Rd)andNα(µ)∈Ilogm(Rd). (5.5)
Now we proveD(Nm
α ) =Ilogm(Rd)inductively. Form=1 this is known, so assume thatD(Nαm) =
Ilogm(Rd)for some m≥1. Ifµ∈D(Nm+1
α ), thenN m+1
α (µ) = N m
α (Nα(µ))is well-defined. Thus,
Nα(µ) ∈D(Nm
α ) = Ilogm(Rd) by assumption, so that µ∈ I
logm+1(Rd)by (5.5). Conversely, if µ ∈ Ilogm+1(Rd), thenµ∈Ilog(Rd)andNα(µ)∈Ilogm(Rd) by (5.5), so thatNm
α (Nα(µ)) is well-defined
by assumption. This showsD(Nm+1
α ) = Ilogm+1(Rd). ThatNαm = Φm◦ Eαm =Eαm◦Φm for everym
then follows easily from (5.1), Proposition 5.1 andD(Φm) =Ilogm(Rd).
Let S(Rd) be the class of all stable distributions on Rd, and for m = 0, 1, . . . denote Lm(Rd) =
Φm+1(I
logm+1(Rd)), L∞(Rd) = ∩∞m=0Lm(Rd),Nα,m(Rd) = Nαm+1(Ilogm+1(Rd)) and Nα,∞(Rd) = ∩∞m=0Nα,m(Rd). Lemma 5.4 implies that Nα,m(Rd) ⊃ Nα,m+1(Rd), so that the family Nα,m, m =
0, 1, . . ., is nested. It is known (cf. Sato[10]) that L∞(Rd) = S(Rd), where the closure is taken
under weak convergence and convolution. In order to show that alsoNα,∞(Rd) =S(Rd), we need
two further lemmas.
Lemma 5.5. Forα >0,Eα maps S(Rd)bijectively onto S(Rd), namely
Eα(S(Rd)) =S(Rd).
This is an immediate consequence of Proposition 2.1 (ii).
Lemma 5.6. Letα >0. For m=0, 1, . . ., Nα,m(Rd)is closed under convolution and weak convergence, and
Proof. By Lemma 5.4,
Nα,m(Rd) =Nαm+1(Ilogm+1(Rd)) = (Eαm+1◦Φm+1)(Ilogm+1(Rd)) =Eαm+1(Lm(Rd)),
henceS(Rd)⊂Nα,m(Rd)by Lemma 5.5 and the fact thatS(Rd)⊂Lm(Rd). Further,
Nα,m(Rd) = (Φm+1◦ Eαm+1)(Ilogm+1(Rd))⊂Φm+1(Ilogm+1(Rd)) =Lm(Rd).
Next observe thatEα and henceEm+1
α clearly respect convolution. Since Lm(Rd) is closed under
convolution and weak convergence (see the proof of Theorem D in[3]), it follows from (5.6) and
Proposition 2.1 (iv) thatNα,m(Rd)is closed under convolution and weak convergence, too.
We can now characterize Nα,∞(Rd) as the closure of S(Rd) under convolution and weak
conver-gence:
Theorem 5.7. Letα >0. It holds
L∞(Rd) =Nα,∞(Rd) =S(Rd).
In particular,
lim
m→∞M m(I
logm(Rd)) =S(Rd).
Proof. By (5.6) we have
S(Rd) = L
∞(Rd)⊃Nα,∞(Rd)⊃S(Rd).
But since eachNα,m(Rd) is closed under convolution and weak convergence, so must be the
inter-sectionNα,∞(Rd) =
T∞
m=0Nα,m(Rd), and together withM =N2the assertions follow.
Acknowledgment
We would like to thank a referee for careful and constructive reading of the paper. Parts of this paper were written while A. Lindner was visiting the Department of Mathematics at Keio University. He thanks for their hospitality and support.
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