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Let $\{X_t^\mu,t\geq0\}$ be a Levy process on $\mathbb{R}^d$ whose distribution at time $1$ is a $d$-dimensional infinitely distribution $\mu$. It is known that the set of all infinitely divisible distributions on $\mathbb{R}^d$, each of which is represented by the law of a stochastic integral $\int_0^1\!\log(1/t)\,dX_t^\mu$ for some infinitely divisible distribution on $\mathbb{R}^d$, 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 of $\int_0^1\!(\log(1/t))^{1/\alpha}\,dX_t^\mu$ for general $\alpha>0$. 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 characterize 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.

each of which is represented by the law of a stochastic integral 1 0 log 1 t d X (µ) t for some infinitely divisible distribution on d , 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 of 1 0 log 1 t 1/α dX (µ) t for general α > 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 characterize 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 .

Introduction
Throughout this paper, (X ) denotes the law of an d -valued random variable X and µ(z), z ∈ d , denotes the characteristic function of a probability distribution µ on d . Also I( d ) |x| is the Euclidean norm of x ∈ d . Let C µ (z), z ∈ d , be the cumulant function of µ ∈ I( d ).
That is, C µ (z) is the unique continuous function with C µ (0) = 0 such that µ(z) = exp C µ (z) , z ∈ d . When µ is the distribution of a random variable X , we also write C X (z) := C µ (z).
Let µ ∈ I( d ) and {X (µ) t , t ≥ 0} denote the Lévy process on d with µ as the distribution at time 1. For a nonrandom measurable function f on (0, ∞), we define a mapping (1.2) whenever the stochastic integral on the right-hand side is definable in the sense of stochastic integrals based on independently scattered random measures on d induced by {X (µ) t }, as in Definitions 2.3 and 3.1 of Sato [15]. When the support of f is a finite interval (0, a], t , and when the support of f is (0, ∞), ∞ 0 f (t)d X (µ) t is the limit in probability of a 0 f (t)d X (µ) t as a → ∞. D(Φ f ) denotes the set of µ ∈ I( d ) for which the stochastic integral in (1.2) is definable. When we consider the composition of two mappings Φ f and Φ g , denoted by Once we define such a mapping, we can characterize a subclass of I( d ) as the range of Φ f , R(Φ f ), say.
In Barndorff-Nielsen et al. [3], they studied the Upsilon mapping and showed that its range R(Υ) is the Goldie-Steutel-Bondesson class, B( d ), say, that is It is also known that µ ∈ B( d ) can be characterized in terms of Lévy measures as follows: A distribution µ ∈ I( d ) belongs to B( d ) if and only if the Lévy measure ν of µ is identically zero or in case ν = 0, ν ξ in (1.1) satisfies that ν ξ (d r) = g ξ (r)d r, r > 0, where g ξ (r) is completely monotone in r ∈ (0, ∞) for λ-a.e. ξ and measurable in ξ for each r > 0.
In addition to that, we have two motivations of this generalization of the mapping. On + , the Goldie-Steutel-Bondesson class B( + ) is the smallest class that contains all mixtures of exponential distributions and is closed under convolution and weak convergence. In addition, we denote by B 0 ( + ) the subclass of B( + ), where all distributions do not have drift.
It is similarly extended to a class on , and in Barndorff-Nielsen et al. [3] it was proved that B( d ) in (1.4) is the smallest class of distributions on d closed under convolution and weak convergence and containing the distributions of all elementary mixed exponential variables in d . Here, an dvalued random variable U x is called an elementary mixed exponential random variable in d if x is a nonrandom nonzero vector in d and U 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( d ) based on a single Lévy process. This type of characterization is quite different from the characterization in terms of the range of some mapping R(Φ f ). This type of characterization is also done by James et al. [6] for the Thorin class. As to B 0 ( + ), we have the following, which is a special case of Equation (4.18) in Theorem 4.2 as mentioned at the end of Section 4.
where Dom(Z) is the set of nonrandom measurable functions h for which the stochastic integrals We are going to generalize this underlying compound Process Y to other Y with Lévy measure 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 for -mapping, which is defined by . 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], lim m→∞ m (I log m ( d )) 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 mapping 2 . Namely, the proof is based on the fact that The paper is organized as follows. In Section 2, we show several properties of the mapping α . In Section 3, we show that E α ( d ) = α (I( d )), α > 0. This relation has the meaning that µ ∈ E α ( d ) is characterized by a stochastic integral representation with respect to a Lévy process. Also we In Section 4, we characterize and certain subclasses of E α ( 1 ) 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 in 1 . In Section 5, we consider the composition Φ • α , and we apply this composition to show that lim m→∞ (Φ • α ) m (I log m ( d )) is the closure of the class of the stable distributions as Maejima and Sato [9] showed for other mappings.
Since we will see that Φ • 2 = , we can answer the question mentioned in the second motivation above.

Several properties of the mapping α and the range of α
We start with showing several properties of the mapping α .
(v) For any µ ∈ I( d ) we also have 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.) 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, we have The additional part follows immediately from Theorem 3.15 in Sato [15].
(iii) By (i), we have for each z ∈ d , Hence we conclude that for each u > 0 and z ∈ d , Hence we see that for each z ∈ d , the function (0, for almost every w ∈ (0, ∞), and by continuity for every w > 0. In particular for w = 1, we (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 each s ∈ (0, 1] it holds But by Proposition 47.11 in Sato [11], applied to each component of X s (log s −1 ) 1/α = 0, the almost sure convergence of the integral at 0 and the second equality.

Corollary 2.2. Let α > 0. Then a distribution µ is symmetric if and only if α (µ) is symmetric.
Proof. Note that for a random variable X with the cumulant function C X (z), (X ) is symmetric if and only if C X (z) = C −X (z). Let X and X have distributions µ and α (µ), respectively. Then where ν is the Lévy measure of µ and ν ξ below is the radial component of ν. Thus, the spherical component λ of ν is equal to the spherical component λ of ν, and the radial component ν ξ of ν satisfies that, for B ∈ ((0, ∞)), with the measure Q ξ being defined by We conclude that g ξ (·) is completely monotone. Thus, for some completely monotone function g ξ . This concludes that µ ∈ E α ( d ).
where g ξ (r) is completely monotone in r and measurable in ξ. For each ξ, there exists a Borel ) (see the proof of Lemma 3.3 in Sato [10]). For ν to be a Lévy measure, it is necessary and sufficient that where we have used Fubini's theorem and the substitution u = r α t. >From this it is easy to see that ν is a Lévy measure if and only if S λ(dξ) Q ξ ({0}) = 0 (which we shall assume without comment from now on) and Define further a measure ν to have spherical component λ = λ and radial parts ν ξ , i.e.
Then ν is a Lévy measure, since which is finite by (2.4). If µ is any infinitely divisible distribution with Lévy measure ν, then part (i) of the proof shows that α (µ) has the given Lévy measure ν, and from the transformation of the generating triplet in Proposition 2.1 we see that µ ∈ I( d ) can be chosen such that α (µ) = µ.

The class E α ( d ) and its subclasses
The first result below shows that the classes E α ( d ) are increasing as α increases.
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 g ξ (x α/β ) is completely monotone and then h ξ (x) is also completely monotone, and we have In the following, we shall call a class F of distributions in d closed under scaling if for every dvalued random variable X such that (X ) ∈ F it also holds that (cX ) ∈ F for every c > 0. If F is a class of infinitely divisible distributions on d and satisfies that µ ∈ F implies µ s * ∈ F for any s > 0, where µ s * is the distribution with characteristic function ( µ(z)) s , we shall call F closed under taking of powers. Recall that a class F of infinitely divisible distributions on d is called completely 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 contains µ * δ b for any µ ∈ F and b ∈ d .
Recall that S = {ξ ∈ d : |ξ| = 1} and µ ∈ I( d ) belongs to the class E α ( d ) if ν = 0 or ν = 0 and ν ξ in (1.1) satisfies ν ξ (d r) = r α−1 g ξ (r α )d r, r > 0, for some function g ξ (r), which is completely monotone in r ∈ (0, ∞) for λ-a.e. ξ, and is measurable in ξ for each r > 0. Denote Proof. By the definition it is clear that all the classes under consideration are closed under convolution, scaling and taking of powers. The class E α ( d ) is closed under weak convergence by Proposition 2.1 (iv) and Theorem 2.3, and hence so are Further, it is easy to see that all the given classes contain the specified distributions, since the Lévy measure of (Z (α) 1 ξ) for ξ ∈ S has polar decomposition λ = δ ξ and ν ξ (d r) = r α−1 g ξ (r α ) d r with g ξ (r) = e −r , and a similar argument works for (Y (α) 1 ξ). Finally, E α ( d ) 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) Let F be the smallest class of infinitely divisible distributions which is closed under convolution, weak convergence, scaling, taking of powers and which contains (Z (α) 1 ξ) for every ξ ∈ S. As already shown, this implies F ⊂ E α ( d ). Recall from Theorem 2.3 that α defines a bijection from I( d ) onto E α ( d ), and let G := −1 α (F ). Then G is closed under convolution, weak convergence, scaling and taking of powers. This follows from the corresponding properties of F and the definition of α for 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 for ξ ∈ S, µ ξ : } has bounded variation, and its drift is Poisson process with parameter 1/α, and we have µ ξ ∈ G by assumption. Since G is closed under convolution and scaling this implies that (n −1 N n ξ) ∈ G for each n ∈ and hence E(N 1 )ξ ∈ G by the strong law of large numbers since G is closed under weak convergence. Since E(N 1 ) > 0 and G is closed under taking of powers this shows that δ c ∈ G for all c ∈ d . Hence G contains every infinitely divisible distribution with Gaussian part zero and Lévy measure α −1 δ ξ with ξ ∈ 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 form ν = n i=1 a i δ c i with n ∈ , a i ≥ 0 and c i ∈ d \ {0}. Since every finite Borel measure on d is the weak limit of a sequence of measures of the form n i=1 a i δ c i , 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 shows G = I( d ) and hence F = E α ( d ) by Theorem 2.3.

Remark 3.4.
Once we are given a mapping α , we can construct nested classes of E α ( d ) by the iteration of the mapping α , which is m α = α • · · · • α (m-times composition). It is easy to see that D( m α ) = I( d ) for any m ∈ . Then we can characterize m α (I( d )) as the smallest class of infinitely divisible distributions which is closed under convolution, weak convergence, scaling and taking of powers and contains m α (N 1 ξ) for all ξ ∈ S and N 1 being a Poisson distribution with mean 1/α. The same proof of Theorem 3.4 works, but we do not go into the details here.

Characterization of subclasses of E α ( d ) by stochastic integrals with respect to some compound Poisson processes
For any Lévy process Y = {Y t } t≥0 on d , denote by L (0,∞) (Y ) the class of locally Y -integrable, real valued functions on (0, ∞) (cf. Sato [15], Definition 2.3), and let Here, following Definition 3.1 of Sato [15], by saying that the (improper stochastic integral) ∞ 0 h(t)d Y t is definable we mean that q p h(t)d Y t converges in probability as p ↓ 0, q → ∞, with the limit random variable being denoted by In this case, ∞ 0 h(t) d Y t is infinitely divisible without Gaussian part and its Lévy measure ν Y,h is given by Recall that E 0,ri α ( d ) = {µ ∈ E α ( d ) : µ has no Gaussian part} ∩ I ri ( d ). The next theorem characterizes E 0,ri α ( d ) as the class of distributions which arise as improper stochastic integrals over (0, ∞) with respect to some fixed rotationally invariant compound Poisson process on d .
Proof. Let µ ∈ E 0,ri α ( d ). By definition and Remark 1.1, the Lévy measure ν of µ has the polar decomposition (λ, ν ξ ) given by and g is independent of ξ and completely monotone. (If µ = δ 0 we define g ξ = 0 and shall also call (λ, ν ξ ) a polar decomposition, even if ν ξ is not strictly positive here). Since g is completely monotone, there exists a Borel measure Q on [0, ∞) such that g( y) = [0,∞) e − y t Q(d t). By (2.4), since ν ξ satisfies ∞ 0 (r 2 ∧ 1)ν ξ (d r) < ∞, we see that Observe that under this condition, we have for each r > 0, Next, observe that since Y (α) is rotationally invariant without Gaussian part, we have by (4.1) that a measurable function h is in Dom(Y (α) ) if and only if where C ∈ (S) and r > 0. Then by (4.4) and (4.5), for every r > 0, where |C| is the Lebesgue measure of C on S. Hence, in order to prove (4.6) and (4.7), it is enough to prove the following: (a) For each Borel measure Q on [0, ∞) satisfying (4.9) there exists a function h ∈ Dom ↓ (Y (α) ) such that there exists a Borel measure Q on [0, ∞) satisfying (4.9) such that (4.12) holds.
To show (a), let Q satisfy (4.9), and denote (4.14) Hence it follows that for every r > 0, by (4.14). The second of these terms is clearly finite by (4.9). To estimate the first, observe that and the first two summands are finite by (4.9), while the last summand is equal to and hence also finite. This shows (4.10) for h and hence (a).
To show (b), let h ∈ Dom(Y (α) ) and assume first that h is nonnegative. Let T : (0, ∞) → (0, ∞] be defined by T (s) = h(s) −α as in (4.13), and consider the image measure T (m 1 ). Define the measure Q on [0, ∞) by Q({0}) = 0 and equality (4.14). Since is automatically infinitely divisible with Lévy measure ν Y,h given by (4.11), we have as in the proof of (a) for every C ∈ (S) and r > 0, In particular, Q must 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 that ( (4.10), and Equation (4.4) and the discussion following it show that has no Gaussian part, gamma part 0 and satisfies ν Y,h = ν Y,h + + ν Y,h − . The corresponding Borel measure Q is given by Q = Q + + Q − , where Q + and Q − are constructed from h + and h − , respectively, completing the proof of (b).
Next, we assume d = 1 and we ask whether every distribution in : µ has no Gaussian part} can be represented as a stochastic integral with respect to the compound Poisson process Z (α) having Lévy measure ν Z (α) (d x) = x α−1 e −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 E 0 α ( 1 ) which correspond to Lévy processes of bounded variation, but that not every distribution in E 0 α ( 1 ) can be represented in this way. However, every distribution in E 0 α ( 1 ) appears as an essential limit of locally Z (α) -integrable functions. Following Sato [15], Definition 3.2, for a Lévy process Y = {Y t } t≥0 and a locally Yintegrable function h over (0, ∞) we say that the essential improper stochastic integral on (0, ∞) of h with respect to Y is definable if for every 0 < p < q < ∞ there are real constants τ p,q such that q p h(t)d Y t − τ p,q converges in probability as p ↓ 0, q → ∞. We write Dom es (Y ) for the class of all locally Y -integrable functions h on (0, ∞) for which the essential improper stochastic integral with respect to Y is definable, and for each h ∈ Dom es (Y ) we denote the class of distributions arising as possible limits q p h(t)d Y t − τ p,q as p ↓ 0, q → ∞ by Φ h,es (Y ) (the limit is not unique, since different sequences τ p,q may give different limit random variables). As for Dom(Y ), the property of belonging to Dom es (Y ) can be expressed in terms of the characteristic triplet (A Y , ν Y , γ Y ) of Y . In particular, if A Y = 0, then a function h on (0, ∞) is in Dom es (Y ) if and only if h is measurable and (4.1) and (4.2) hold, and in that case Φ h,es (Y ) consists of all infinitely divisible distributions µ with characteristic triplet (A Y,h = 0, ν Y,h , γ), where ν Y,h is given by (4.4) and γ ∈ is arbitrary (cf. [15], Theorems 3.6 and 3.11).
To see that the second inclusion in (4.21) is proper, let µ ∈ E 0 α ( 1 ) with Lévy measure ν being supported on [0, ∞) such that Since ν is supported on [0, ∞), we must have h ≥ 0 Lebesgue almost surely, so that we can suppose that h ≥ 0 everywhere. Then we have from (4.1) and (4.
Together these two equations imply This completes the proof of (4.21).
Proof of Theorem 1.2. This is an immediate consequence of Equation (4.18) since B 0 ( + ) = E

The composition of Φ with α and its application
In this section we study the composition Φ • α . We start with the following proposition.

It then holds
including the equality of the domains. In particular, we have Proof. We first note that D( α ) is independent of the value of α and equals I log ( d ), shown in Theorem 2.3 of [8], (essentially in Theorem 2.4 (i) of [14].) As mentioned right after Equation (1.5), D(Φ) = I log ( d ). Thus it follows from Proposition 5.1 that both Φ • α as well as α • Φ are well defined on I log ( d ) and that they have the same domain. Note that Then, if we are allowed to exchange the order of the integrals by Fubini's theorem, we have and the same calculation can be carried out for In order to assure the exchange of the order of the integrations by Fubini's theorem, it is enough to show that This is Equation (4.5) in Barndorff-Nielsen et al. [3] with the replacement of s by s 1/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 that Φ • α = α • Φ = α . Since 2 = , this shows in particular (5.2).
It is well known that Φ(I log ( d )) = L( d ), the class of selfdecomposable distributions on d . An immediate consequence of Theorem 5.2 is the following.
Next observe that α and hence m+1 α clearly respect convolution. Since L m ( d ) 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) that N α,m ( d ) is closed under convolution and weak convergence, too.
We can now characterize N α,∞ ( d ) as the closure of S( d ) under convolution and weak convergence: In particular, lim m→∞ m (I log m ( d )) = S( d ).
Proof. By (5.6) we have But since each N α,m ( d ) is closed under convolution and weak convergence, so must be the intersection N α,∞ ( d ) = ∞ m=0 N α,m ( d ), and together with = 2 the assertions follow.