getdoc9736. 134KB Jun 04 2011 12:04:43 AM

ELECTRONIC COMMUNICATIONS in PROBABILITY

A NOTE ON NEW CLASSES OF INFINITELY

DIVISIBLE DISTRIBUTIONS ON

R

d

MAKOTO MAEJIMA

Department of Mathematics, Faculty and Science and Technology, Keio University, 3-14-1, Hiyoshi, Kohoku-ku, Yokohama 223-8522, Japan

email: maejima@math.keio.ac.jp

GENTA NAKAHARA

Department of Mathematics, Faculty and Science and Technology, Keio University, 3-14-1, Hiyoshi, Kohoku-ku, Yokohama 223-8522, Japan

email: genta.nakahara@gmail.com

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)−1Šν(d x)

«

, z∈Rd_, whereAis a symmetric nonnegative-definited×dmatrix,γ∈Rd _{andνis a measure (called the} Lévy measure) onRd_satisfying

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

thatν_ξ(B)is measurable inξfor eachB∈ B((0,∞)), 0< ν_ξ((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{ν_ξ}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)}n_f(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

−ru_{Q(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

ν_ξ(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 lim_x→∞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(R}d_{)appearing later, we}

do not have to write this condition explicitly.

Remark 1.3. The integrability condition of the Lévy measureR_Rd(|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.

Lemma 1.6. (See Feller (1966), p.441, Corollary 2.) Letφbe a completely monotone function on (0,∞)and letψbe a nonnegative function on(0,∞)whose derivative is completely monotone. Then φ(ψ)is completely monotone.

Let h_ξ(x) = g_ξ(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 classesJ_0,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 {J_0,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{J_α,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 {J_α,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 law}µ∈I(Rd)at t=1. We consider the stochastic integrals

Z Gα,β(0)

0

G_α∗_,β(t)d X_t(µ), where Gα,β(0) =

¨

β−1_Γ(−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 integral}R∞

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 Ψ_α,β(µ) =L

Z Gα,β(0)

0

G_α∗_,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

G_α∗_,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 Ψ_α,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 inf_t→∞a(t)/b(t) ≤ lim sup_t→∞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 Φ_ϕ

p(µ) =L

‚Z ∞

0

ϕ_p(t)d X_t(µ)

Œ

.

Suppose thatϕ_pis locally square-integrable with respect to Lebesgue measure on[0,∞)and satisfies (1)ϕ₀(t)≍e−c tas t→ ∞with some c>0,

(2)ϕ_p(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|

p_{µ(d x)<∞}=:I} p(Rd).

(iii)If p=1, thenD_(Φ

ϕ1) ={µ∈I(Rd):

R

Rd|x|µ(d x)<∞ lim_T→∞RT

t0

t−1_{d t}R

|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|

p_{µ(d x)<∞,}R

Rdxµ(d x) =0} =:I0_p(Rd).

(v)If p≥2, thenD_(Φ

ϕp) ={δ0}, whereδ0is the distribution with the total mass at 0.

We apply Proposition 2.2 to our problem. First we note that whenα <2,G_α∗_,β(t)is locally square-integrable with respect to Lebesgue measure on[0,∞).

Theorem 2.3. (Caseα=0.)D_(Ψ

0,β) =Ilog(Rd).

Proof. Note thatt(=Gα,β(u))↑ ∞ if and only ifu(=Gα∗,β(t))↓0, whenα≥0. It is enough to

show that for someC1∈(0,∞),u∼C1e−t ast→ ∞. We have

u e−t =

u

exp{−G_0,β(u)} =exp{G0,β(u) +logu}=exp

¨Z ∞

u

x−1e−xβd x+logu

«

=exp

(

β−1

Z∞

uβ

y−1e−yd y−β−1

Z1 uβ

y−1d y

)

=exp

(

β−1

Z1 uβ

y−1(e−y−1)d y+β−1

Z∞

1

y−1e−yd y

)

→C1,

say, asu↓0. Henceu∼ C1e−t as t → ∞, and the condition (1) of Proposition 2.2 is satisfied.

Thus Proposition 2.2 (i) gives us the assertion. Theorem 2.4. (Caseα∈(0,∞).)

(i)If0< α <1, thenD_(Ψ

α,β) =Iα(Rd).

(ii)Ifα=1, thenD_(Ψ

1,β) =I1∗(R d_).

(iii)If1< α <2, thenD_(Ψ

α,β) =Iα0(Rd).

(i v)Ifα≥2, thenD_(Ψ

α,β) ={δ0}.

Proof. (i) and (iii). It is enough to show thatu∼C2t−1/α as t→ ∞for some C2∈(0,∞). We

have, ast→ ∞(equivalentlyu↓0), for someC3∈(0,∞),

u t−1/α=

u

€

Gα,β(u)

Š−1/α=

u

β−1R∞

uβ y−(α/β)−1e−yd y

−1/α∼

u C3u−α

−1/α=C

1/α

3 =:C2,

and the condition (2) of Proposition 2.3 is satisfied. Thus Proposition 2.3 (ii) and (iv) give us the assertions.

(ii). Supposeβ6=1. (The caseβ=1 is proved in Sato (2006b).) We first have G1,β(u) =

Z∞

u

x−2e−xβd x=

Z∞

u

x−2d x+

Z∞

u

x−2(e−xβ

−1)d x

=

Z∞

u

x−2d x+

Z1 u

x−2(e−xβ−1+xβ)du−

Z1 u

x−2+βd x+

Z∞

1

x−2(e−xβ−1)d x =u−1+ (β−1)−1_u−1+β_{+O(1), u↓0.}

Thus

t=G_1,∗_β(t)−1+ (β−1)−1G∗_1,β(t)−1+β+O(1), t→ ∞. Therefore,

G_1,∗_β(t) =t−1+ (β−1)−1_t−1_G∗

1,β(t)

β_+O(t−1_G∗

1,β(t)), t→ ∞. (2.1)

We have shown in (i) and (iii) thatu∼C2t−1/α, but this is also true forα=1. Hence

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)−1_t−β_{+a(t), t→ ∞,}

where

a(t) =

¨

o(t−β), t→ ∞, when 0< β <1, O(t−1_{), t→ ∞, whenβ >1.}

Thus

G_1,∗_β(t) =t−1ψ(t), where

ψ(t):=1+C₂β(β−1)−1_t−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= Ψ_α,β(µ), and note that the mappingΨ_α,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−1_B)s−α−1_e−sβ

ds, B∈ B(Rd\ {0}). (2.3) (2)Ifν 6=0, andν has polar decomposition(λ,ν_ξ), then a polar decomposition ofνe= (λ,e νe_ξ)is given byλe=λandνe_ξ(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

1_s−1_B(x)ν(d x) =

Z∞

0

ν(s−1B)s−α−1e−sβds, which is (2.3).

(2) Next assume thatν6=0 andνhas polar decomposition(λ,ν_ξ). Then, we have

e

ν(B) =

Z∞

0

s−α−1e−sβds

Z S

λ(dξ)

Z∞

0

1s−1_B(rξ)ν_ξ(d r)

=

Z S

λ(dξ)

Z∞

0

νξ(d r)r−1

Z∞

0

(u/r)−α−1e−(u/r)β1_B(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|

α_{µ(d x)<∞. Whenα >0, note that}R Rd|x|

α_{µ(d x)<∞implies}R∞

1 r

α_ν

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 y_{Q(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

α_ν

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 transformationΥ_α,β of Lévy measures as

e

ν= Υ_α,β(ν). Then thisΥ_α,β is a generalized Upsilon transformation discussed in Barndorff-Nielsen et al. (2008) with the dilation measureτ(ds) =s−α−1_e−sβ

ds.

Theorem 2.7. For eachα <2andβ >0, the mappingΨ_α,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 J_α,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, J_α,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.

Theorem 2.8. Letα <1andβ >0. The range of the mappingΨ_α,βequals J_α,β(Rd_{), that is,} Jα,β(Rd) = Ψα,β(D(Ψα,β)).

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 thatΨ_α,β(D_(Ψ

α,β))⊂ Jα,β(Rd). Supposeµ∈D(Ψα,β)and

e

µ= Ψ_α,β(µ), and letν andνebe the Lévy measures ofµandµ, respectively. Thus, ife ν=0, then

e

ν=0 andµe∈Jα,β(Rd). Whenν6=0, it follows from Lemma 2.5 thatµe∈Jα,β(Rd).

Next we show that Jα,β(Rd)⊂Ψα,β(D(Ψα,β)). Supposeµe ∈Jα,β(Rd)with the Lévy-Khintchine

triplet (A,eν,e γ). Ife eν =0, thenµe = Ψα,β(µ)for someµ∈D(Ψα,β). Thus, suppose thateν 6=0.

Then, in a polar decomposition (λ,e νe_ξ)of ν, we havee νe_ξ(d r) = r−α−1egξ(rβ)d r, where egξ(v)is

completely monotone in v> 0 eλ-a.e.ξ, and is measurable in ξ. Thus by Bernstein’s theorem, there are measuresQeξon[0,∞)such that

e

gξ(v) =

Z

[0,∞)

e−vuQeξ(du).

In general,Qeξ is a measure on[0,∞), but since limv→∞geξ(v) =0λ-a.e.e ξ,Qeξ does not have a

point mass at 0, and henceQeξis a measure on(0,∞). We see that

e

ν(B) =

Z S

e

λ(dξ)

Z∞

0

1_B(rξ)r−α−1_eg

ξ(rβ)d r (2.6)

=

Z S

e

λ(dξ)

Z∞

0

1B(rξ)r−α−1d r Z∞

0

e−rβuQeξ(du).

SinceR_Rd(|x|2∧1)ν(d x)e <∞, we have

Z S

e

λ(dξ)

Z1 0

r1−αd r

Z∞

1

e−rβuQeξ(du) +

Z S

e

λ(dξ)

Z∞

1

r−α−1d r

Z1 0

e−rβuQeξ(du)<∞.

Hence, we have, by the change of variablesr→vbyrβ_u=v,

Z1 0

r1−αd r

Z∞

1

e−rβuQeξ(du) =

Z∞

1 e

Qξ(du)

Z1 0

r1−αe−rβud r =β−1

Z∞

1

u(α−2)/βQeξ(du)

Zu 0

v−1+(2−α)/βe−vd v≥C4 Z∞

1

u(α−2)/βQeξ(du),

where

C₄=β−1

Z1 0

v−1+(2−α)/βe−vd v∈(0,∞).

Thus _Z

S e

λ(dξ)

Z∞

1

u(α−2)/βQeξ(du)<∞. (2.7)

We also have for anyα <1,

Z∞

1

r−α−1d r

Z1 0

e−rβuQeξ(du) =

Z1 0

e

Qξ(du)

Z∞

1

=β−1

Z 1 0

uα/βQeξ(du)

Z∞

u

v−1−(α/β)e−vd v≥C5 Z1

0

uα/βQeξ(du),

where

C5=β−1 Z∞

1

v−1−(α/β)e−vd v∈(0,∞).

Thus _Z S e λ(dξ) Z1 0

uα/βQeξ(du)<∞. (2.9)

In addition, ifα=0, (2.8) is turned out to be

Z∞

1

r−1d r

Z1 0

e−rβuQe_ξ(du) =β−1

Z1 0

e

Q_ξ(du)

Z1 u

v−1e−vd v ≥(βe)−1

Z1 0

e

Qξ(du)

Z1 u

v−1d v= (βe)−1 Z1

0

(−logu)Qeξ(du).

Thus, whenα=0,

Z S e λ(dξ) Z1 0

(−logu)Qeξ(du)<∞. (2.10)

Furthermore,

Z∞

1

r−α−1d r

Z1 0

e−rβuQ(du)e ≥

Z∞

1

r−α−1e−rβd r

Z1 0

e

Qξ(du) =C6 Z1

0 e

Qξ(du),

where

C6:= Z∞

1

r−α−1e−rβd r∈(0,∞).

Thus we have _Z

S e λ(dξ) Z1 0 e

Qξ(d r)<∞. (2.11)

Define

νξ(B) =

Z∞

0

uα/β1B €

u−1/βŠQeξ(du), B∈ B((0,∞)). (2.12)

Then, it follows from (2.7) and (2.9) that

Z S e λ(dξ) Z∞ 0

(r2∧1)νξ(d r) =

Z S e λ(dξ) Z∞ 0

uα/β(u−2/β∧1)Qeξ(du) (2.13)

= Z S e λ(dξ) Z 1 0

uα/βQeξ(du) +

Z∞

1

u(α−2)/βQeξ(du)

<∞. Defineνby

ν(B) = Z S e λ(dξ) Z∞ 0

Then, by (2.13),νis the Lévy measure of some infinitely divisible distributionµ, andµbelongs to

D_(Ψ

α,β)and satisfies

e

ν(B) =

ZGα,β(0)

0

ν((G_α∗_,β(t))−1_{B)d t. (2.15)}

To show (2.15), by (2.6), (2.12) and (2.14), we have

e ν(B) = Z S e λ(dξ) Z∞ 0

1B(rξ)r−α−1d r Z∞

0

e−rβuQeξ(du)

= Z S e λ(dξ) Z∞ 0

1_B(u−1/βsξ)s−α−1e−sβds

Z∞

0

uα/βQeξ(du)

=

Z∞

0

s−α−1e−sβds

Z S e λ(dξ) Z∞ 0

1B(u−1/βsξ)uα/βQeξ(du)

=

Z∞

0

s−α−1e−sβds

Z S e λ(dξ) Z∞ 0

1_B(rsξ)νξ(d r)

=

Z∞

0

s−α−1e−sβds

Z S e λ(dξ) Z∞ 0

1s−1_B(rξ)ν_ξ(d r)

=

Z∞

0

ν(s−1B)s−α−1e−sβds=−

Z∞

0

ν(s−1B)d Gα,β(s)

=

ZGα,β(0)

0

ν((G_α∗_,β(t))−1B)d t. To show thatµ∈D_{(Ψα,β), it is enough to show that}R

|x|>1|x|

α_{ν(d x)<∞, which is if and only}

ifµ∈Iα(Rd), when 0< α <1, and

R

|x|>1log|x|ν(d x)<∞, which is if and only ifµ∈Ilog(R d_),

whenα=0, (see Sato (1999), Theorem 25.3). Note that by (2.12) we see, for any nonnegative measurable function f on(0,∞),

Z∞

0

f(r)ν_ξ(d r) =

Z∞

0

uα/βf(u−1/β_)Qe ξ(du).

Thus if we choose f(r) =I[r>1]rα, whereI[A]is the indicator function of the setA, thenνin (2.14) satisfies that forα >0

Z

|x|>1

|x|α_{ν(d x) =}

Z S e λ(dξ) Z∞ 1

rαν_ξ(d r) =

Z S e λ(dξ) Z1 0 e

Qξ(du)<∞ (2.16)

due to (2.11). Whenα=0,

Z

|x|>1

log|x|ν(d x) =

Z S e λ(dξ) Z∞ 1

logrν_ξ(d r) (2.17)

= Z S e λ(dξ) Z1 0

logu−1/βQeξ(du) =β−1

Z S e λ(dξ) Z1 0

(−logu)Qeξ(du)<∞

Notice again that

ZGα,β(0)

0

G_α∗_,β(t)

2

d t=−

Z∞

0

u2d Gα,β(u) =

Z∞

0

u1−αe−uβdu<∞. DefineAandγby

e

A=

Z Gα,β(0)

0

G_α∗_,β(t)2d t

!

A (2.18)

and

e

γ=

ZGα,β(0)

0

G_α∗_,β(t)d t

γ+ Z Rd x ₁

1+|G∗

α,β(t)x|2

− 1 1+|x|2

ν(d x)

. (2.19) Here we have to check the finiteness of this integral. We first have

ZGα,β(0)

0

G_α∗_,β(t)d t=−

Z∞

0

ud Gα.β(u) =

Z∞

0

u−αe−uβdu<∞,

sinceα <1. Below,C₇,C₈∈(0,∞)are suitable constants. Recallα <1. Whenα6=0, we have

ZGα,β(0)

0

G∗_α,β(t)d t

Z Rd |x| 1 1+|G∗

α,β(t)x|2

− 1 1+|x|2

ν(d x) = Z∞ 0

u−αe−uβdu

Z Rd |x| 1 1+|ux|2−

1 1+|x|2

ν(d x) ≤ Z∞ 0

u−α(1+u2)e−uβdu

Z

Rd

|x|3

(1+|ux|2_)(1+|x|2₎ν(d x)

≤

Z∞

0

u−α(1+u2)e−uβ_du

×

Z

|x|≤1

|x|2ν(d x) +

Z

|x|>1,|ux|≤1

|x|ν(d x) +

Z

|x|>1,|ux|>1

|x| |ux|2ν(d x)

!

=C₇+

Z

|x|>1

|x|ν(d x)

Z1/|x|

0

u−α(1+u2)e−uβ_du

+

Z

|x|>1

ν(d x)

Z∞

1/|x|

u−α−1(1+u2)e−uβ

du

≤C7+ Z

|x|>1

|x|ν(d x)

Z1/|x|

0

2u−αdu

+

Z

|x|>1

ν(d x)

( Z1

1/|x|

+

Z∞

1 !

u−α−1(1+u2)e−uβ

du

)

≤C7+2(1−α)−1 Z

|x|>1

|x|αν(d x)

+

Z

|x|>1

ν(d x)

(Z1 1/|x|

2u−α−1du+

Z∞

1

u−α−1(1+u2)e−uβdu

)

=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|α_{ν(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|

α_{ν(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

G_α∗_,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.

References

[1] T. Aoyama, M. Maejima and J. Rosi´nski (2008). A subclass of typeGselfdecomposable dis-tributions,J. Theoret. Probab.,21, 14–34. MR2384471 MR2384471

[2] O.E. Barndorff-Nielsen, M. Maejima and K. Sato (2006). Some classes of multivariate in-finitely divisible distributions admitting stochastic integral representations, Bernoulli, 12, 1–33. MR2202318 MR2202318

[3] O.E. Barndorff-Nielsen, J. Rosi´nski and S. Thorbjørnsen, (2008): GeneralΥtransformations. ALEA Lat. Am. J. Probab. Math. Statist. 4, 131-165. MR2421179 MR2421179

[4] W. Feller (1966).An Introduction to Probability Theory and Its Applications, Vol. II, 2nd ed., John Wiley & Sons. MR0210154 MR0210154

[5] M. Maejima, M. Matsui and M. Suzuki (2009). Classes of infinitely divisible distributions on Rd _{related to the class of selfdecomposable distributions, to appear inTokyo J. Math.}

[6] K. Sato (1999). Lévy Processes and Infinitely Divisible Distributions. Cambridge University Press. MR1739520 MR1739520

[7] K. Sato (2004) Stochastic integrals in additive processes and application to semi-Lévy pro-cesses,Osaka J. Math.41, 211–236. MR2040073 MR2040073

[8] K. Sato (2006a). Additive processes and stochastic integrals,Illinois J. Math.50, 825–851. MR2247848 MR2247848

[9] K. Sato (2006b). Two families of improper stochastic integrals with respect to Lévy processes, ALEA Lat. Am. J. Probab. Math. Statist. 147–87. MR2235174 MR2235174

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 thatΨ_α,β(D_(Ψ

α,β))⊂ Jα,β(Rd). Supposeµ∈D(Ψα,β)and

e

µ= Ψ_α,β(µ), and letν andνebe the Lévy measures ofµandµe, respectively. Thus, ifν=0, then

e

ν=0 andµe∈Jα,β(Rd). Whenν6=0, it follows from Lemma 2.5 thatµe∈Jα,β(Rd).

Next we show that Jα,β(Rd)⊂Ψα,β(D(Ψα,β)). Supposeµe ∈Jα,β(Rd)with the Lévy-Khintchine triplet (eA,νe,γ)e. If eν =0, thenµe = Ψα,β(µ)for someµ∈D(Ψα,β). Thus, suppose thateν 6=0. Then, in a polar decomposition (eλ,νe_ξ)of νe, we haveνe_ξ(d r) = r−α−1egξ(rβ)d r, where egξ(v)is completely monotone in v> 0 eλ-a.e.ξ, and is measurable in ξ. Thus by Bernstein’s theorem, there are measuresQeξon[0,∞)such that

e

gξ(v) =

Z

[0,∞)

e−vuQeξ(du).

In general,Qeξ is a measure on[0,∞), but since limv→∞egξ(v) =0λe-a.e.ξ,Qeξ does not have a point mass at 0, and henceQeξis a measure on(0,∞). We see that

e ν(B) =

Z

S

e λ(dξ)

Z∞

0

1_B(rξ)r−α−1egξ(rβ)d r (2.6)

= Z

S

e λ(dξ)

Z∞

0

1B(rξ)r−α−1d r

Z∞

0

e−rβuQeξ(du). SinceR_Rd(|x|

2_{∧1)eν(d x)<∞, we have} Z

S

e λ(dξ)

Z1 0

r1−αd r

Z∞

1

e−rβuQeξ(du) +

Z

S

e λ(dξ)

Z∞

1

r−α−1d r

Z1 0

e−rβuQeξ(du)<∞. Hence, we have, by the change of variablesr→vbyrβ_u=v,

Z1 0

r1−αd r

Z∞

1

e−rβuQeξ(du) =

Z∞

1 e

Qξ(du)

Z1 0

r1−αe−rβud r

=β−1

Z∞

1

u(α−2)/βQeξ(du)

Zu

0

v−1+(2−α)/βe−vd v≥C4

Z∞

1

u(α−2)/βQeξ(du), where

C₄=β−1 Z1

0

v−1+(2−α)/βe−vd v∈(0,∞).

Thus _Z

S

e λ(dξ)

Z∞

1

u(α−2)/βQeξ(du)<∞. (2.7) We also have for anyα <1,

Z∞

1

r−α−1d r

Z1 0

e−rβuQeξ(du) =

Z1 0

e

Qξ(du)

Z∞

1

=β−1 Z 1

0

uα/βQeξ(du)

Z∞

u

v−1−(α/β)e−vd v≥C5 Z1

0

uα/βQeξ(du), where

C5=β−1

Z∞

1

v−1−(α/β)e−vd v∈(0,∞).

Thus _Z

S

e λ(dξ)

Z1 0

uα/βQeξ(du)<∞. (2.9) In addition, ifα=0, (2.8) is turned out to be

Z∞

1

r−1d r

Z1 0

e−rβuQe_ξ(du) =β−1 Z1

0 e

Q_ξ(du) Z1

u

v−1e−vd v

≥(βe)−1 Z1

0 e

Qξ(du)

Z1

u

v−1d v= (βe)−1 Z1

0

(−logu)eQξ(du). Thus, whenα=0,

Z

S

e λ(dξ)

Z1 0

(−logu)eQξ(du)<∞. (2.10) Furthermore,

Z∞

1

r−α−1d r

Z1 0

e−rβuQe(du)≥

Z∞

1

r−α−1e−rβd r

Z1 0

e

Qξ(du) =C6 Z1

0 e

Qξ(du), where

C6:=

Z∞

1

r−α−1e−rβd r∈(0,∞).

Thus we have _Z

S

e λ(dξ)

Z1 0

e

Qξ(d r)<∞. (2.11)

Define

νξ(B) =

Z∞

0

uα/β1B

€

u−1/βŠQeξ(du), B∈ B((0,∞)). (2.12) Then, it follows from (2.7) and (2.9) that

Z

S

e λ(dξ)

Z∞

0

(r2∧1)νξ(d r) =

Z

S

e λ(dξ)

Z∞

0

uα/β(u−2/β∧1)Qeξ(du) (2.13)

= Z

S

e λ(dξ)

Z 1

0

uα/βQeξ(du) +

Z∞

1

u(α−2)/βQeξ(du)

<∞. Defineνby

ν(B) = Z

S

e λ(dξ)

Z∞

0

Then, by (2.13),νis the Lévy measure of some infinitely divisible distributionµ, andµbelongs to D_(Ψ

α,β)and satisfies

e ν(B) =

ZGα,β(0)

0

ν((G_α∗_,β(t))−1_{B)d t. (2.15)}

To show (2.15), by (2.6), (2.12) and (2.14), we have

e ν(B) =

Z

S

e λ(dξ)

Z∞

0

1B(rξ)r−α−1d r

Z∞

0

e−rβuQeξ(du)

= Z

S

e λ(dξ)

Z∞

0

1_B(u−1/βsξ)s−α−1e−sβds

Z∞

0

uα/βQeξ(du)

=

Z∞

0

s−α−1e−sβds

Z

S

e λ(dξ)

Z∞

0

1B(u−1/βsξ)uα/βQeξ(du)

=

Z∞

0

s−α−1e−sβds

Z

S

e λ(dξ)

Z∞

0

1_B(rsξ)νξ(d r)

=

Z∞

0

s−α−1e−sβds

Z

S

e λ(dξ)

Z∞

0

1s−1_B(rξ)ν_ξ(d r)

=

Z∞

0

ν(s−1B)s−α−1e−sβds=−

Z∞

0

ν(s−1B)d Gα,β(s)

=

ZGα,β(0)

0

ν((G_α∗_,β(t))−1B)d t. To show thatµ∈D_{(Ψα,β), it is enough to show that}R

|x|>1|x|

α_{ν(d x)<∞, which is if and only} ifµ∈Iα(Rd), when 0< α <1, and

R

|x|>1log|x|ν(d x)<∞, which is if and only ifµ∈Ilog(R

d_),

whenα=0, (see Sato (1999), Theorem 25.3). Note that by (2.12) we see, for any nonnegative measurable function f on(0,∞),

Z∞

0

f(r)ν_ξ(d r) =

Z∞

0

uα/βf(u−1/β)eQξ(du).

Thus if we choose f(r) =I[r>1]rα, whereI[A]is the indicator function of the setA, thenνin (2.14) satisfies that forα >0

Z

|x|>1

|x|α_{ν(d x) =}

Z

S

e λ(dξ)

Z∞

1

rαν_ξ(d r) = Z

S

e λ(dξ)

Z1 0

e

Qξ(du)<∞ (2.16) due to (2.11). Whenα=0,

Z

|x|>1

log|x|ν(d x) = Z

S

e λ(dξ)

Z∞

1

logrν_ξ(d r) (2.17)

= Z

S

e λ(dξ)

Z1 0

logu−1/βQeξ(du) =β−1

Z

S

e λ(dξ)

Z1 0

(−logu)eQξ(du)<∞ due to (2.10).

Notice again that

ZGα,β(0)

0

G_α∗_,β(t) 2

d t=−

Z∞

0

u2d Gα,β(u) =

Z∞

0

u1−αe−uβdu<∞. DefineAandγby

e

A=

Z Gα,β(0)

0

G_α∗_,β(t)2d t

!

A (2.18)

and

e γ=

ZGα,β(0)

0

G_α∗_,β(t)d t

γ+ Z Rd x ₁

1+|G∗ α,β(t)x|2

− 1

1+|x|2

ν(d x)

. (2.19)

Here we have to check the finiteness of this integral. We first have

ZGα,β(0)

0

G_α∗_,β(t)d t=−

Z∞

0

ud Gα.β(u) =

Z∞

0

u−αe−uβdu<∞,

sinceα <1. Below,C₇,C₈∈(0,∞)are suitable constants. Recallα <1. Whenα6=0, we have

ZGα,β(0)

0

G∗_α,β(t)d t

Z Rd |x| 1 1+|G∗

α,β(t)x|2

− 1

1+|x|2 ν(d x) =

Z∞

0

u−αe−uβdu

Z Rd |x| 1 1+|ux|2−

1 1+|x|2

ν(d x)

≤

Z∞

0

u−α(1+u2)e−uβdu

Z

Rd

|x|3

(1+|ux|2_)(1+|x|2₎ν(d x)

≤

Z∞

0

u−α(1+u2)e−uβ_du

×

Z

|x|≤1

|x|2ν(d x) + Z

|x|>1,|ux|≤1

|x|ν(d x) + Z

|x|>1,|ux|>1

|x| |ux|2ν(d x)

!

=C₇+ Z

|x|>1

|x|ν(d x) Z1/|x|

0

u−α(1+u2)e−uβ_du

+ Z

|x|>1 ν(d x)

Z∞

1/|x|

u−α−1(1+u2)e−uβdu

≤C7+ Z

|x|>1

|x|ν(d x) Z1/|x|

0

2u−αdu

+ Z

|x|>1 ν(d x)

( Z1

1/|x|

+

Z∞

1 !

u−α−1(1+u2)e−uβdu

)

≤C7+2(1−α)−1 Z

|x|>1

|x|αν(d x)

+ Z

|x|>1 ν(d x)

(Z1

1/|x|

2u−α−1du+

Z∞

1

u−α−1(1+u2)e−uβdu

)

=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|α_{ν(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|

α_{ν(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

G_α∗_,β(t)d X(µ)_t

!

,

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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