ELECTRONIC COMMUNICATIONS in PROBABILITY
A NOTE ON NEW CLASSES OF INFINITELY
DIVISIBLE DISTRIBUTIONS ON
R
dMAKOTO MAEJIMA
Department of Mathematics, Faculty and Science and Technology, Keio University, 3-14-1, Hiyoshi, Kohoku-ku, Yokohama 223-8522, Japan
email: [email protected]
GENTA NAKAHARA
Department of Mathematics, Faculty and Science and Technology, Keio University, 3-14-1, Hiyoshi, Kohoku-ku, Yokohama 223-8522, Japan
email: [email protected]
SubmittedJanuary 3, 2009, accepted in final formAugust 19, 2009 AMS 2000 Subject classification: 60E07
Keywords: infinitely divisible distribution onRd, stochastic integral representation.
Abstract
This paper introduces and studies a family of new classes of infinitely divisible distributions on Rd with two parameters. Depending on parameters, these classes connect the Goldie–Steutel– Bondesson class and the class of generalized type Gdistributions, connect the Thorin class and the class M, and connect the class M and the class of generalized type G distributions. These classes are characterized by stochastic integral representations with respect to Lévy processes.
1
Introduction
Let I(Rd) be the class of all infinitely divisible distributions on Rd. µ(z),b z ∈Rd, denotes the characteristic function of µ∈I(Rd)and|x|denotes the Euclidean norm of x ∈Rd. We use the Lévy-Khintchine triplet(A,ν,γ)ofµ∈I(Rd)in the sense that
b
µ(z) =exp
¨
−2−1〈z,Az〉+i〈γ,z〉+
Z
Rd
ei〈z,x〉−1−i〈z,x〉(1+|x|2)−1ν(d x)
«
, z∈Rd,
whereAis a symmetric nonnegative-definited×dmatrix,γ∈Rd andνis a measure (called the Lévy measure) onRdsatisfying
ν({0}) =0 and
Z
Rd
(|x|2∧1)ν(d x)<∞.
The following polar decomposition is a basic result on the Lévy measure of µ∈I(Rd). Letνbe the Lévy measure of some µ ∈ I(Rd)with 0< ν(Rd)≤ ∞. Then there exist a measure λ on
S={x∈Rd :|x|=1}with 0< λ(S)≤ ∞and a family{ν
ξ:ξ∈S}of measures on(0,∞)such
thatνξ(B)is measurable inξfor eachB∈ B((0,∞)), 0< νξ((0,∞))≤ ∞for eachξ∈S, and
ν(B) =
Z
S
λ(dξ)
Z∞
0
1B(rξ)νξ(d r), B∈ B(Rd\ {0}). (1.1)
Hereλand{νξ}are uniquely determined byνup to multiplication of measurable functionsc(ξ) and 1
c(ξ), respectively, with 0<c(ξ)<∞. We say thatνhas the polar decomposition(λ,νξ)and νξis called the radial component ofν. (See, e.g., Barndorff-Nielsen et al. (2006), Lemma 2.1.)
A real-valued function f defined on(0,∞)is said to be completely monotone if it has derivatives f(n)of all orders and for eachn=0, 1, 2, ...,(−1)nf(n)(r)≥0,r>0. Bernstein’s theorem says that f on(0,∞)is completely monotone if and only if there exists a (not necessarily finite) measureQ on[0,∞)such that f(r) =R
[0,∞)e
−ruQ(du). (See, e.g., Feller (1966), p.439.)
In this paper, we introduce and study the following classes.
Definition 1.1. (The class Jα,β(Rd).) Letα <2andβ >0. We say that µ∈I(Rd)belongs to the
class Jα,β(Rd)ifν=0orν6=0and, in caseν6=0,νξin (1.1) has expression
νξ(d r) =r−α−1gξ(rβ)d r, r>0, (1.2)
where gξ(x)is measurable inξ, is completely monotone in x on(0,∞)λ-a.e.ξ, not identically zero
andlimx→∞gξ(x) =0λ-a.e.ξ.
Remark 1.2. Ifα≤0, then automatically limx→∞gξ(x) =0λ-a.e.ξ, because of the finiteness of
R
|x|>1ν(d x). So, when we consider the classes B(R
d),G(Rd),T(Rd)and M(Rd)appearing later, we
do not have to write this condition explicitly.
Remark 1.3. The integrability condition of the Lévy measureRRd(|x|
2∧1)ν(d x)<∞implies that Z∞
0
(r2∧1)r−α−1gξ(rβ)d r<∞, λ-a.e.ξ, (1.3)
so we do not have to assume (1.3) in the definition. It is automatically satisfied. Remark 1.4. The classes Jα,1(Rd),α <2,are studied in Sato (2006b).
Before mentioning our motivation of this study, we state a general result on the relations among the classesJα,β(Rd),α <2,β >0.
Theorem 1.5. (i)Fixα <2and let0< β1< β2. Then Jα,β1(R
d)⊂J
α,β2(R
d).
(ii)Fixβ >0and letα1< α2<2. Then Jα2,β(R
d)⊂J
α1,β(R
d).
Proof.For the proof of (i), we need the following lemma.
Let hξ(x) = gξ(xβ1/β2),x > 0, where g
ξ is the one in (1.2), which is completely monotone on
(0,∞). Sinceψ(x) =xβ1/β2,x>0, has a completely monotone derivative, it follows from Lemma 1.6 that hξ(x)is completely monotone. Supposeµ ∈Jα,β1(Rd)and let gξ be the one in (1.2).
Sincegξ(rβ1) =hξ(rβ2), wherehξis completely monotone as has been just shown above, we have
µ∈Jα,β2(Rd). This proves (i).
To prove (ii), suppose that µ ∈ Jα2,β(Rd). Thenνξ(d r) = r−α2−1gξ(rβ)d r, r > 0, as in (1.2),
wheregξis completely monotone on(0,∞)λ-a.e.ξ. Note that
hξ(x) =x−(α2−α1)/βgξ(x)
is completely monotone, because x−p,p > 0, is completely monotone and the product of two completely monotone functions is also completely monotone. We now have
νξ(d r) =r−α2−1gξ(rβ)d r=r−α1−1hξ(rβ)d r,
and thusµalso belongs toJα1,β(Rd). This proves (ii).
The motivations for studying the classesJα,β(Rd)are the following.
I.The classes connecting the Goldie–Steutel–Bondesson class and the class of generalized typeG distributions.
Let α=−1 and consider the classesJ−1,β(Rd),β >0. A distributionµ∈I(Rd)is said to be of
generalized typeGifνξin (1.2) has expressionνξ(d r) =gξ(r2)d rfor some completely monotone
function gξ on(0,∞), and denote byG(Rd)the class of all generalized typeG distributions on
Rd. LetIsym(Rd) ={µ∈I(Rd) : µis symmetric in the sense thatµ(B) =µ(−B),B∈ B(Rd)}. Remark 1.7. A distributionµ∈G(Rd)∩Isym(Rd)is a so-called type G distribution, which is, in one dimension, a variance mixture of the standard normal distribution with a positive infinitely divisible mixing distribution.
Remark 1.8. G(Rd) =J−1,2(Rd).
Remark 1.9. The Goldie-Steutel-Bondesson class denoted by B(Rd) is J−
1,1(Rd). (For details on
B(Rd), see Barndorff-Nielsen et al. (2006).)
Therefore, by Theorem 1.5 (i) withα=−1, for 1< β <2,
B(Rd)⊂J
−1,β(Rd)⊂G(Rd),
and hence{J−1,β(Rd), 1≤β≤2}is a family of classes of infinitely divisible distributions onRd
connectingB(Rd)andG(Rd)with continuous parameterβ∈[1, 2]. II.The classes connecting the Thorin class and the classM(Rd). Letα=0 and consider the classesJ0,β(Rd),β >0.
Remark 1.10. The Thorin class denoted by T(Rd) is J
0,1(Rd). (For details on T(Rd), see also
Barndorff-Nielsen et al. (2006).)
Remark 1.11. The class M(Rd)is defined by J
0,2(Rd). (The class M(Rd)∩Isym(Rd)is studied in
By Theorem 1.5 (i) withα=0, for 1< β <2,
T(Rd)⊂J0,β(Rd)⊂M(Rd),
and hence {J0,β(Rd), 1≤β ≤2} is a family of classes of infinitely divisible distributions onRd
connectingT(Rd)andM(Rd)with continuous parameterβ∈[1, 2]. III.The classes connecting the classesM(Rd)andG(Rd).
Letβ =2 and consider the classes Jα,2(Rd), α <2. Then, by Theorem 1.5 (ii) withβ = 2, for
−1≤α≤0
M(Rd)⊂Jα,2(Rd)⊂G(Rd),
and hence{Jα,2(Rd),−1≤α≤0}is a family of classes of infinitely divisible distributions onRd connectingM(Rd)andG(Rd)with continuous parameterα∈[−1, 0].
IV.The classes connecting the classesT(Rd)andB(Rd).
Letβ =1 and consider the classes Jα,1(Rd), α <2. Then, by Theorem 1.5 (ii) withβ = 1, for
−1≤α≤0
T(Rd)⊂Jα,1(Rd)⊂B(Rd),
and hence {Jα,1(Rd),−1 ≤ α ≤ 0} is a family of classes of infinitely divisible distributions on Rd connecting T(Rd)andB(Rd) with continuous parameterα ∈[−1, 0]. (This fact is already mentioned in Sato (2006b).)
2
Stochastic integral characterizations for
J
α,β(
R
d)
The purpose of this paper is to characterize the classes Jα,β(Rd)by stochastic integral
represen-tations. For that, we first define mappings fromI(Rd)intoI(Rd)and investigate the domains of those mappings.
We introduce the following functionGα,β(u). Forα <2 andβ >0, let
Gα,β(u) =
Z∞
u
x−α−1e−xβd x, u≥0,
and letG∗α,β(t)be the inverse function ofGα,β(u), that is, t=Gα,β(u)if and only ifu=Gα∗,β(t).
Let{X(tµ)} be a Lévy process onRd with the lawµ∈I(Rd)at t=1. We consider the stochastic integrals
Z Gα,β(0)
0
Gα∗,β(t)d Xt(µ), where Gα,β(0) =
¨
β−1Γ(−αβ−1), ifα <0,
∞, ifα≥0.
As to the definition of stochastic integrals of non-random measurable functions f which are
RT 0 f(t)d X
(µ)
t ,T < ∞,µ ∈ I(Rd), we follow the definition in Sato (2004, 2006a), whose idea
is to define a stochastic integral with respect toRd-valued independently scatted random measure induced by a Lévy process onRd. The improper stochastic integralR∞
0 f(t)d X
(µ)
limit in probability ofRT
0 f(t)d X
(µ)
t asT → ∞whenever the limit exists. See also Sato (2006b).
In the following,L(X)stands for “the law ofX". If we write
Ψα,β(µ) =L
Z Gα,β(0)
0
Gα∗,β(t)d X(tµ)
!
,
thenΨα,β can be considered as a mapping with domainD(Ψα,β)being the class ofµ∈I(Rd)for
whichRGα,β(0)
0 G
∗
α,β(t)d X
(µ)
t is definable.
Theorem 2.1. Ifα <0, thenD(Ψα
,β) =I(Rd).
Proof. By Proposition 3.4 in Sato (2006a), since Gα,β(0) < ∞ for α < 0, if
RGα,β(0)
0
Gα∗,β(t)2d t<∞, thenRGα,β(0)
0 G
∗
α,β(t)d X
(µ)
t is well-defined. Actually,
Z Gα,β(0)
0
G∗α,β(t)2d t=−
Z∞
0
u2d Gα,β(u) =
Z∞
0
u1−αe−uβdu<∞.
To determine the domain of Ψα,β,α ≥ 0, we need the following result by Sato (2006b). In the following, a(t) ∼ b(t) means that limt→∞a(t)/b(t) = 1, a(t) ≍ b(t) means that 0 <
lim inft→∞a(t)/b(t) ≤ lim supt→∞a(t)/b(t) < ∞ and Ilog(Rd) = {µ ∈ I(Rd) : R
Rdlog +
|x|µ(d x)<∞}, where log+|x|= (log|x|)∨0.
Proposition 2.2. (Sato (2006b), Theorems 2.4 and 2.8.)Let p≥0. Denote
Φϕ
p(µ) =L
Z ∞
0
ϕp(t)d Xt(µ)
.
Suppose thatϕpis locally square-integrable with respect to Lebesgue measure on[0,∞)and satisfies (1)ϕ0(t)≍e−c tas t→ ∞with some c>0,
(2)ϕp(t)≍t−1/pas t→ ∞for p∈(0, 1)∪(1,∞),
(3)ϕ1(t)≍t−1as t→ ∞and for some t0>0,c>0andψ(t),ϕ1(t) =t−1ψ(t)for t>t0with R∞
t0 t
−1|ψ(t)−
c|d t<∞. Then
(i)If p=0, thenD(Φϕ
0) =Ilog(R
d).
(ii)If0<p<1, thenD(Φ
ϕp) ={µ∈I(R
d):R
Rd|x|
pµ(d x)<∞}=:I p(Rd).
(iii)If p=1, thenD(Φ
ϕ1) ={µ∈I(Rd):
R
Rd|x|µ(d x)<∞ limT→∞RT
t0
t−1d tR
|x|>txν(d x)exists inR d,R
Rdxµ(d x) =0}=:I
∗
1(Rd).
(i v)If1<p<2, thenD(Φϕ
p) ={µ∈I(R
d):R
Rd|x|
pµ(d x)<∞,R
Rdxµ(d x) =0} =:I0p(Rd).
(v)If p≥2, thenD(Φ
ϕp) ={δ0}, whereδ0is the distribution with the total mass at 0.
Theorem 2.3. (Caseα=0.)D(Ψ
Thus Proposition 2.2 (i) gives us the assertion.
Theorem 2.4. (Caseα∈(0,∞).)
By substituting (2.2) into (2.1), we have
G∗1,β(t) =t−1+C2β(β−1)−1t−1−β+t−1a(t), t→ ∞, =t−1
1+C2β(β−1)−1t−β+a(t), t→ ∞,
where
a(t) =
¨
o(t−β), t→ ∞, when 0< β <1, O(t−1), t→ ∞, whenβ >1.
Thus
G1,∗β(t) =t−1ψ(t), where
ψ(t):=1+C2β(β−1)−1t−β+a(t),
and Z
∞
1
t−1|ψ(t)−1|d t=
Z∞
1
t−1|C2β(β−1)−1t−β+a(t)|d t<∞.
Thus the condition (3) of Proposition 2.2 is satisfied with t0=1 andc=1, and Proposition 2.2
(iii) gives us the assertion (iii). (iv) The same as in Sato (2006b).
We now calculate the Lévy measure ofµe= Ψα,β(µ), and note that the mappingΨα,βis one-to-one.
Lemma 2.5. Letα <2andβ >0. Letµ∈D(Ψ
α,β)andµe= Ψα,β(µ), and letνandνebe the Lévy
measures ofµandµ, respectively.e
(1)We have
e
ν(B) =
Z∞
0
ν(s−1B)s−α−1e−sβ
ds, B∈ B(Rd\ {0}). (2.3)
(2)Ifν 6=0, andν has polar decomposition(λ,νξ), then a polar decomposition ofνe= (λ,e νeξ)is given byλe=λandνeξ(d r) =r−α−1egξ(rβ)d r, where
e
gξ(u) =
Z∞
0
rαe−u/rβνξ(d r). (2.4)
(3)egξin(2.4)satisfies the requirements of gξin(1.2).
Proof.Supposeµ∈D(Ψα
,β)andµe= Ψα,β(µ).
(1) We see that (by using Proposition 2.6 of Sato (2006b)),
e
ν(B) =
ZGα,β(0)
0
d t
Z
Rd
1B(x Gα∗,β(t))ν(d x) =−
Z∞
0
d Gα,β(s)
Z
Rd
1B(xs)ν(d x)
=
Z∞
0
s−α−1e−sβds
Z
Rd
1s−1B(x)ν(d x) =
Z∞
0
ν(s−1B)s−α−1e−sβds,
(2) Next assume thatν6=0 andνhas polar decomposition(λ,νξ). Then, we have
e
ν(B) =
Z∞
0
s−α−1e−sβds
Z
S
λ(dξ)
Z∞
0
1s−1B(rξ)νξ(d r)
=
Z
S
λ(dξ)
Z∞
0
νξ(d r)r−1
Z∞
0
(u/r)−α−1e−(u/r)β1B(uξ)du
=
Z
S
λ(dξ)
Z∞
0
1B(uξ)u−α−1egξ(uβ)du,
whereeλ=λand
e
gξ(u) =
Z ∞
0
rαe−u/rβνξ(d r), (2.5)
which is (2.4). The finiteness ofegξis trivial forα≤0. Forα >0, sinceµ∈D(Ψα,β), we have that
R
Rd|x|
αµ(d x)<∞. Whenα >0, note thatR Rd|x|
αµ(d x)<∞impliesR∞
1 r
αν
ξ(d r)<∞, (see,
e.g. Sato (1999), Theorem 25.3). Hence the integralegξexists.
(3) If we put
e
Q(B) =
Z∞
0
rα1B(r−β)νξ(d r),
then it follows that egξ(u) =
R∞
0 e
−u yQ(d y), and thuse eg
ξ is completely monotone by Bernstein’s
theorem. Ifα≤0, then automatically limu→∞egξ(u) =0λ-a.e.ξ, since
∞>
Z
|x|>1 e
ν(d x) =
Z
S
λ(dξ)
Z∞
1
u−α−1egξ(uβ)du.
Whenα >0, sinceR∞
1 r
αν
ξ(d r)<∞, the assertion that limu→∞egξ(u) =0λ-a.e.ξalso follows
from (2.5) by the dominated convergence theorem. The proof of the lemma is thus concluded.
Remark 2.6. (2.3) can be written as, by introducing a transformationΥα,β of Lévy measures as
e
ν= Υα,β(ν). Then thisΥα,β is a generalized Upsilon transformation discussed in Barndorff-Nielsen et al. (2008) with the dilation measureτ(ds) =s−α−1e−sβ
ds.
Theorem 2.7. For eachα <2andβ >0, the mappingΨα,β is one-to-one. The proof is carried out in the same way as for Proposition 4.1 of Sato (2006b).
We are now ready to discuss stochastic integral characterizations of the classesJα,β(Rd), by
show-ing that Jα,β(Rd) is the range of the mapping Ψ
α,β. However, in this paper, we restrict
our-selves to the case α < 1, because in the case 1 ≤ α < 2, Jα,β(Rd) is strictly bigger than the rangeΨα,β(D(Ψα,β))and more deep calculations would be needed. (See, e.g., Sato (2006b) and
Maejima et al. (2009).) Also, the classes appearing in our motivation of introducing the classes Jα,β(Rd)are restricted to the caseα≤0.
Remark 2.9. This theorem is already known for α = −1, 0 and β = 1in Theorems A and C of Barndorff-Nielsen et al. (2006) and forα <1andβ=1in Theorem 4.2 of Sato (2006b).
Proof of Theorem 2.8. We first show thatΨα,β(D(Ψ
=β−1
Then, it follows from (2.7) and (2.9) that
Then, by (2.13),νis the Lévy measure of some infinitely divisible distributionµ, andµbelongs to
whenα=0, (see Sato (1999), Theorem 25.3). Note that by (2.12) we see, for any nonnegative measurable function f on(0,∞),
Z∞ (2.14) satisfies that forα >0
Notice again that
Here we have to check the finiteness of this integral. We first have
=C7+2(1−α)−1
Z
|x|>1
|x|αν(d x)
+
Z
|x|>1
ν(d x)¦−2α−1(1− |x|α) +C
8 ©
=C7+2(1−α)−1
Z
|x|>1
|x|αν(d x)
+2α−1 Z
|x|>1
|x|αν(d x) + (C
8−2α−1) Z
|x|>1
ν(d x)<∞, (2.21)
by (2.16). Whenα=0, since
Z1
1/|x|
u−α−1du=
Z1
1/|x|
u−1du=log|x|,
in (2.20), we have Z
|x|>1
log|x|ν(d x) (2.22)
instead ofR
|x|>1|x|
αν(d x)in (2.21) in the calculation above. The finiteness of (2.22) is assured
by (2.17).
Thusγ can be defined. Hence, if we denote byµ an infinitely divisible distribution having the Lévy-Khintchine triplet(A,ν,γ)above, then by (2.15), (2.18) and (2.19), we see that
e
µ=L
Z Gα,β(0)
0
Gα∗,β(t)d X(tµ)
!
,
concluding thatµe∈Ψα,β(D(Ψα,β)). This completes the proof.
Acknowledgment The authors greatly appreciate the referee’s detailed comments, which im-proved the paper very much.
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