J Electr on i o u r nal o f P c r o ba bility 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 (µ) {X t , t Let ≥ 0} be a Lévy process on d whose distribution at time is a d-dimensional infinitely distribution µ. It is known that the set of all infinitely divisible distributions on d , (µ) each of which is represented by the law of a stochastic integral log 1t dX 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 1/α (µ) of infinitely divisible distributions which are represented as the law of log 1t dX 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 . Key words: infinitely divisible distribution; the Goldie-Steutel-Bondesson class; stochastic integral 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 Braunschweig, 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: maejima@math.keio.ac.jp 1119 AMS 2000 Subject Classification: Primary 60E07. Submitted to EJP on December 12, 2009, final version accepted June 25, 2010. 1120 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 ) denotes the class of all infinitely divisible distributions on d , Isym ( d ) = {µ ∈ I( d ) : µ is symmetric on d }, Iri ( d ) = {µ ∈ I( d ) : µ is rotationally invariant on d }, Ilog ( d ) = {µ ∈ I( d ) : |x|>1 log |x|µ(d x) < ∞} and Ilogm ( d ) = {µ ∈ I( d ) : |x|>1 (log |x|)m µ(d x) < ∞}, where |x| is the Euclidean norm of x ∈ d . Let Cµ (z), z ∈ d , be the cumulant function of µ ∈ I( d ). That is, Cµ (z) is the unique continuous function with Cµ (0) = such that µ(z) = exp Cµ (z) , z ∈ d . When µ is the distribution of a random variable X , we also write CX (z) := Cµ (z). We use the Lévy-Khintchine generating triplet (A, ν, γ) of µ ∈ I( d ) in the sense that Cµ (z) = −2−1 〈z, Az〉 + i〈γ, z〉 ei〈z,x〉 − − i〈z, x〉(1 + |x|2 )−1 ν(d x), z ∈ + d , d where 〈·, ·〉 denotes the inner product in d , A is a symmetric nonnegative-definite d × d matrix, γ ∈ d and ν is a measure (called the Lévy measure) on d satisfying ν({0}) = and d (|x| ∧ 1)ν(d x) < ∞. The polar decomposition of the Lévy measure ν of µ ∈ I( d ), with < ν( d ) ≤ ∞, is the following: There exist a measure λ on S = {ξ ∈ d : |ξ| = 1} with < λ(S) ≤ ∞ and a family {νξ : ξ ∈ S} of measures on (0, ∞) such that νξ (B) is measurable in ξ for each B ∈ ((0, ∞)), < νξ ((0, ∞)) ≤ ∞ for each ξ ∈ S, ∞ ν(B) = λ(dξ) S 1B (rξ)νξ (d r), B ∈ ( d \ {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 < c(ξ) < ∞, λ (dξ) = c(ξ)λ(dξ) and c(ξ)νξ (d r) = νξ (d r) for λ -a.e. ξ ∈ S. The measure νξ is a Lévy measure on (0, ∞) for λ-a.e. ξ ∈ S. We say that ν has the polar decomposition (λ, νξ ) and νξ is called the radial component of ν. (See, e.g., Lemma 2.1 of [3] and its proof.) Remark 1.1. For µ ∈ Iri ( d ) with generating triplet (A, ν, γ), it is necessary and sufficient that A U = UA holds for arbitrary d × d orthogonal matrix U, γ = and λ and νξ can be chosen such that λ is Lebesgue measure and νξ is independent of ξ. (µ) Let µ ∈ I( d ) and {X 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 ∞ Φ f (µ) = (µ) f (t)d X t , (1.2) whenever the stochastic integral on the right-hand side is definable in the sense of stochastic in(µ) tegrals based on independently scattered random measures on d induced by {X t }, as in Defini∞ (µ) tions 2.3 and 3.1 of Sato [15]. When the support of f is a finite interval (0, a], f (t)d X t = 1121 a a ∞ (µ) f (t)d X t , and when the support of f is (0, ∞), (µ) (t)d X t (µ) f (t)d X t is the limit in probability of f as a → ∞. D(Φ f ) denotes the set of µ ∈ I( ) 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 Φ g ◦ Φ f , the domain of Φ g ◦ Φ f is D(Φ g ◦ Φ f ) = {µ ∈ I( d ) : µ ∈ D(Φ f ) and Φ f (µ) ∈ D(Φ g )}. Once we define such a mapping, we can characterize a subclass of I( d ) as the range of Φ f , R(Φ f ), say. d In Barndorff-Nielsen et al. [3], they studied the Upsilon mapping Υ(µ) = (µ) log d X t t , and showed that its range R(Υ) is the Goldie–Steutel–Bondesson class, B( Υ(I( d )) = B( d ). (1.3) d ), say, that is (1.4) 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, νξ in (1.1) satisfies that νξ (d r) = gξ (r)d r, r > 0, where gξ (r) is completely monotone in r ∈ (0, ∞) for λ-a.e. ξ and measurable in ξ for each r > 0. Our purpose of this paper is to generalize (1.3) to α (µ) := log t 1/α (µ) dXt for any α > 0, where = Υ, and investigate R( α ). We first generalize (1.4) and characterize d )), in the sense of what should replace B( d ) for general α > 0. For that, we need a new α (I( class Eα ( d ), α > 0. Namely, we say that µ ∈ I( d ) belongs to the class Eα ( d ) if ν = or ν = and νξ in (1.1) satisfies νξ (d r) = r α−1 gξ (r α )d r, r > 0, for some function gξ (r), which is completely monotone in r ∈ (0, ∞) for λ-a.e. ξ, and is measurable in ξ for each r > 0. Then we will show that α (I( d )) = Eα ( d ) in Theorem 2.3. 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 ( + ) the subclass of B( + ), where all distributions 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 d valued 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 ( + ), 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. 1122 Theorem 1.2. Let Z = {Z t } t≥0 be a compound Poisson process on e−x d x, x > 0. Then + with Lévy measure ν Z (d x) = ∞ B ( +) = h ∈ Dom(Z) , , h(t)d Z t where Dom(Z) is the set of nonrandom measurable functions h for which the stochastic integrals ∞ h(t)d Z t are definable. We are going to generalize this underlying compound Process Y to other Y with Lévy measure α x α−1 e−x d 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 for -mapping, which is defined by ∞ (µ) m∗ (t)d X t (µ) = , µ ∈ Ilog ( d ), ∞ where m(x) = u−1 e−u du, x > 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→∞ m (Ilogm ( 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 . Namely, the proof is based on the fact that (µ) = (Φ ◦ where Φ(µ) = ∞ −t e (µ) dXt )(µ) =( ◦ Φ)(µ), with D(Φ) = Ilog ( d µ ∈ Ilog ( d ), (1.5) ). The paper is organized as follows. In Section 2, we show several properties of the mapping α . In Section 3, we show that Eα ( d ) = α (I( d )), α > 0. This relation has the meaning that µ ∈ Eα ( d ) is characterized by a stochastic integral representation with respect to a Lévy process. Also we sym characterize Eα ( d ), Eα+ ( d ) and Eα ( d ) := Eα ( d ) ∩ Isym ( d ) based on one compound Poisson distribution on , where Eα+ ( d ) = {µ ∈ Eα ( d ) : µ( d \[0, ∞)d ) = 0}. In Section 4, we characterize Eα0,ri ( d ) := {µ ∈ Eα ( d ) : µ has no Gaussian part} ∩ Iri ( d ) (1.6) and certain subclasses of Eα ( ) 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 . In Section 5, we consider the composition Φ ◦ α , and we apply this composition to show that limm→∞ (Φ ◦ α )m (Ilogm ( 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 ◦ = , we can answer the question mentioned in the second motivation above. 1123 Several properties of the mapping and the range of α We start with showing several properties of the mapping Proposition 2.1. Let α > 0. (i) α (µ) can be defined for any µ ∈ |Cµ (z(log t −1 1/α ) d I( α α. ) and is infinitely divisible, and we have )| d t < ∞ and C Cµ (z log t −1 )1/α d t, (z) = α (µ) z∈ d . (ii) The generating triplet (A, ν, γ) of µ = α (µ) can be calculated from (A, ν, γ) of µ by A = Γ(1 + 2/α) A, ∞ ν(B) = α ν(u−1 B)αuα−1 e−u du, d ( B∈ \ {0}), (2.1) ∞ γ = Γ(1 + 1/α) γ + α αuα e−u du x + |ux|2 d − 1 + |x|2 ν(d x). (2.2) (µ) (µ) If additionally µ ∈ I( d ) is such that {X t } has bounded variation with drift γ0 , then also {X t } is of bounded variation with drift γ0 = Γ(1 + 1/α)γ0 . (2.3) (iii) The mapping α : I( d ) → I( d ) is one-to-one. (i v) Let µn ∈ I( d ), n = 1, 2, . . . If µn converges weakly to some µ ∈ I( d ) as n → ∞, then α (µn ) converges weakly to α (µ) as n → ∞. Conversely, if α (µn ) converges weakly to some distribution µ as n → ∞, then µ = α (µ) for some µ ∈ I( d ) and µn converges weakly to µ as n → ∞. In particular, the range α (I( d )) is closed under weak convergence. (v) For any µ ∈ I( d ) we also have α (µ) = log 1/α 1− t (µ) dXt = lim s↓0 s αt (µ) (log t −1 )1/α−1 X t dt , 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) = (log t −1 )1/α 1(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, we have ∞ (log t −1 )2/α d t A= u2/α e−u du A= A = Γ(1 + 2/α) A, ∞ ν((log t −1 )−1/α B)d t = ν(B) = 1124 α ν(u−1 B) αuα−1 e−u du, (log t −1 )1/α γ= γ+ x d ∞ =γ ∞ v 1/α −v e dv + 1 + |(log t −1 )1/α x|2 α αuα e−u du + |ux|2 d ν(d x) d t + |x|2 x − − ν(d x). + |x|2 The additional part follows immediately from Theorem 3.15 in Sato [15]. d (iii) By (i), we have for each z ∈ , ∞ C α (µ) (z) = Cµ (z(log t −1 1/α ) Hence we conclude that for each u > and z ∈ u ∞ C α (µ) −1/α (u Cµ (z v 1/α )e−v d v. )dt = z) = u Cµ v d , ∞ 1/α z u e −v Cµ (w 1/α z)e−uw d w. dv = Hence we see that for each z ∈ d , the function (0, ∞) → , u → u−1 C α (µ) (u−1/α z) is the Laplace transform of (0, ∞) → , w → Cµ (w 1/α z). Hence for each fixed z ∈ d , Cµ (w 1/α z) is determined by α (µ) for almost every w ∈ (0, ∞), and by continuity for every w > 0. In particular for w = 1, we see that Cµ (z) is determined by α (µ) for every z ∈ d . (iv) Apart from minor adjustments, the proof is the same as that of Proposition 2.4 (v) in BarndorffNielsen 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 (µ) (log t −1 )1/α d X t = −X s(µ) (log s−1 )1/α + s s αt (µ) (log t −1 )1/α−1 X t d t a.s. (µ) But by Proposition 47.11 in Sato [11], applied to each component of X t separately, it holds (µ) (µ) lim sup t↓0 t −1/2 |X t | = a.s. for each > 0, which shows lims↓0 X s (log s−1 )1/α = 0, the almost sure convergence of the integral at 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 CX (z), (X ) is symmetric if and only if CX (z) = C−X (z). Let X and X have distributions µ and α (µ), respectively. Then CX (z) = CX (z(− log t)1/α )d t and C−X (z) = C−X (z(− log t)1/α )d t. Hence, if CX = C−X , then CX = C−X . Conversely, if CX = C−X , then CX = C−X by the one-to-one property of α . Since = Υ and E1 ( d ) = B( the case of general α > 0. d ), the following is an extension of the fact E1 ( Theorem 2.3. For α > 0, Eα ( d )= α (I( 1125 d )). d )= (I( d )) to Proof (i) (Proof for that Eα ( (log t −1 1/α ) (µ) dXt d ) ⊃ for some µ ∈ I( α (I( d d )).) Let µ ∈ α (I( d )). Then µ = ), and hence ∞ α ν(u−1 B)uα−1 e−u du, ν(B) := νµ (B) = α where ν is the Lévy measure of µ and νξ 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 νξ of ν satisfies that, for B ∈ ((0, ∞)), ∞ νξ (B) = α ∞ =α 1B (xu)νξ (d x) ∞ νξ (d x) ∞ ∞ α uα−1 e−u du α 1B ( y)( y/x)α−1 e−( y/x) x −1 d y 1B ( y) y α−1 gξ ( y α )d y, =: where ∞ gξ (r) = αx −α −r/x α e ∞ e−ruQ ξ (du), νξ (d x) = with the measure Q ξ being defined by ∞ 1B (x −α )x −α νξ (d x), Q ξ (B) = α ((0, ∞)). B∈ We conclude that gξ (·) is completely monotone. Thus, νξ (d y) = y α−1 gξ ( y α )d y for some completely monotone function gξ . This concludes that µ ∈ Eα ( (ii) (Proof for that Eα ( d )⊂ α (I( d )).) Let µ ∈ Eα ( ∞ ν(B) = λ(dξ) d d ). ) with Lévy measure ν of the form 1B (rξ)r α−1 gξ (r α )d r, B∈ ( d \ {0}), S where gξ (r) is completely monotone in r and measurable in ξ. For each ξ, there exists a Borel measure Q ξ on [0, ∞) such that gξ (r) = [0,∞) e−r t Q ξ (d t) and Q ξ (B) is measurable in ξ for each B ∈ ([0, ∞)) (see the proof of Lemma 3.3 in Sato [10]). For ν to be a Lévy measure, it is necessary and sufficient that ∞ ∞> r α+1 gξ (r α ) d r + λ(dξ) S = S S α r α+1 d r λ(dξ) λ(dξ) [0,∞) e−r t Q ξ (d t) 1126 r α−1 gξ (r α ) d r ∞ λ(dξ) + S λ(dξ)α−1 = [0,∞) S t −1−2/αQ ξ (d t) e−r t Q ξ (d t) [0,∞) t 2/α −u u e du λ(dξ)α−1 + α r α−1 d r [0,∞) S t −1 e−t Q ξ (d t), 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}) = (which we shall assume without comment from now on) and ∞ t −1Q ξ (d t) < ∞, λ(dξ) S t −1−2/αQ ξ (d t) < ∞. λ(dξ) (2.4) S In part (i) we have defined Q ξ = U(ρξ ) as the image measure of ρξ under the mapping U : (0, ∞) → (0, ∞), r → r −α , where ρξ has density r → αr −α with respect to νξ . Denoting by V : r → r −1/α , the inverse of U, it follows that ρξ is the image measure of Q ξ under the mapping V . Hence, given Q ξ , we define νξ as having density r → α−1 r α with respect to the image measure V (Q ξ ) of Q ξ under V , i.e. ∞ νξ (B) = α−1 1B (r −1/α )r −1Q ξ (d r), ((0, ∞)). B∈ Define further a measure ν to have spherical component λ = λ and radial parts νξ , i.e. ∞ ν(B) = λ(dξ) 1B (rξ)νξ (d r), B∈ ( d \ {0}). S Then ν is a Lévy measure, since ∞ (r ∧ 1)νξ (d r) λ(dξ) S ∞ = λ(dξ) = r νξ (d r) + ∞ S νξ (d r) 1 −1 −2/α −1 λ(dξ) S λ(dξ) S α r r Q ξ (d r) + α−1 r −1Q ξ (d r), λ(dξ) S 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 Eα ( d ) and its subclasses The first result below shows that the classes Eα ( d ) are increasing as α increases. 1127 Theorem 3.1. For any < α < β, Eα ( Proof. Let < α < β. Then if µ ∈ Eα ( νξ (d r) = r α−1 α d d ) ⊂ Eβ ( d ). ), νξ of µ is gξ (r )d r = r β−1 gξ (r α/β )β r β−α dr = r β−1 gξ (r α/β )β r (β−α)/β β 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 (x α/β ) is completely monotone and then hξ (x) is also completely monotone, and we have νξ (d r) = r β−1 hξ (r β ). Hence µ ∈ Eβ ( d ). In the following, we shall call a class F of distributions in d closed under scaling if for every d valued 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 µs∗ ∈ F for any s > 0, where µs∗ 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 Eα ( d ) if ν = or ν = and νξ in (1.1) satisfies νξ (d r) = r α−1 gξ (r α )d r, r > 0, for some function gξ (r), which is completely monotone in r ∈ (0, ∞) for λ-a.e. ξ, and is measurable in ξ for each r > 0. Denote S+ := {ξ = (ξ1 , . . . , ξd ) ∈ S : ξ1 , . . . , ξd ≥ 0}. (α) (α) Theorem 3.2. Let α > and Y1 and Z1 be compound Poisson distributions on with Lévy measures α α νY (α) (d x) = |x|α−1 e−x d x and ν Z (α) (d x) = x α−1 e−x 1(0,∞) (x) d x, respectively. Then we have the following. (i) The class Eα ( d ) is the smallest class of infinitely divisible distributions on d which is closed under convolution, weak convergence, scaling, taking of powers and contains each of the distributions (α) (Z1 ξ) with ξ ∈ S. Further, Eα ( d ) is c.c.s.s. (ii) The class Eα+ ( d ) = {µ ∈ Eα ( d ) : µ( d \ [0, ∞)d ) = 0} is the smallest class of infinitely divisible distributions on d which is closed under convolution, weak convergence, scaling, taking of powers and (α) contains each of the distributions (Z1 ξ) with ξ ∈ S+ . sym d d (iii) The class Eα ( ) = Eα ( ) ∩ Isym ( d ) is the smallest class of infinitely divisible distributions on d which is closed under convolution, weak convergence, scaling, taking of powers and contains each of (α) the distributions (Y1 ξ) with ξ ∈ S. 1128 Proof. By the definition it is clear that all the classes under consideration are closed under convolution, scaling and taking of powers. The class Eα ( d ) is closed under weak convergence by sym Proposition 2.1 (iv) and Theorem 2.3, and hence so are Eα+ ( d ) and Eα ( d ). Further, it is easy to (α) see that all the given classes contain the specified distributions, since the Lévy measure of (Z1 ξ) for ξ ∈ S has polar decomposition λ = δξ and νξ (d r) = r α−1 gξ (r α ) d r with gξ (r) = e−r , and a (α) similar argument works for (Y1 ξ). Finally, Eα ( 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, (α) weak convergence, scaling, taking of powers and which contains (Z1 ξ) for every ξ ∈ S. As d already shown, this implies F ⊂ Eα ( ). Recall from Theorem 2.3 that α defines a bijection from I( d ) onto Eα ( d ), and let G := α−1 (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, µξ := −1 (A = 0, ν = α δξ , γ) for some γ ∈ (α) d , so that (µ ) {X t ξ } (α) −1 α (Z1 ξ) has generating triplet has bounded variation, and its drift is (Z1 ξ) zero by (2.3) since {X t } has zero drift. This shows that µξ = (N1 ξ) where {Nt } t≥0 is a Poisson process with parameter 1/α, and we have µξ ∈ G by assumption. Since G is closed under convolution and scaling this implies that (n−1 Nn ξ) ∈ G for each n ∈ and hence E(N1 )ξ ∈ G by the strong law of large numbers since G is closed under weak convergence. Since E(N1 ) > and G is closed under taking of powers this shows that δc ∈ G for all c ∈ d . Hence G contains every infinitely divisible distribution with Gaussian part zero and Lévy measure α−1 δξ with ξ ∈ S. Since G is closed under convolution, scaling and taking of powers it also contains all infinitely n divisible distributions with Gaussian part zero and Lévy measures of the form ν = i=1 δci with n ∈ , ≥ and ci ∈ d \ {0}. Since every finite Borel measure on d is the weak limit of a n sequence of measures of the form i=1 δci , 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 = Eα ( d ) by Theorem 2.3. (α) (ii) and (iii) follow in analogy to the proof of (i), where for (iii) observe that α−1 (Y1 ξ) has characteristic triplet (A = 0, ν = α−1 δξ +α−1 δ−ξ , γ = 0), so that, by an argument similar to the proof of (i), every symmetric compound Poisson distribution is in α−1 (F ) and hence so every symmetric infinitely divisible distribution is. Here F is the smallest class of infinitely divisible distributions on d which is closed under convolution, weak convergence, scaling, taking of powers and contains (α) each of the distributions (Y1 ξ) with ζ ∈ S. Corollary 2.2 and Theorem 2.3 then imply F = sym Eα ( d ). Remark 3.3. In the introduction it was mentioned that B( d ) is the smallest class of distributions on d closed under convolution and weak convergence and containing the distributions of all elementary mixed exponential random variables in d . Theorem 3.2 for α = gives a new interpretation of B( d ), since it is based on a compound Poisson distribution, rather than on an exponential distribution. 1129 Remark 3.4. Once we are given a mapping α , we can construct nested classes of Eα ( d ) by the iteration of the mapping α , which is αm = α ◦ · · · ◦ α (m-times composition). It is easy to see that D( αm ) = I( d ) for any m ∈ . Then we can characterize αm (I( d )) as the smallest class of infinitely divisible distributions which is closed under convolution, weak convergence, scaling and taking of powers and contains αm (N1 ξ) for all ξ ∈ S and N1 being a Poisson distribution with mean 1/α. The same proof of Theorem 3.4 works, but we not go into the details here. Characterization of subclasses of Eα ( d ) by stochastic integrals with respect to some compound Poisson processes For any Lévy process Y = {Yt } 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 ∞ Dom(Y ) = h ∈ L(0,∞) (Y ) : h(t)d Yt is definable , Dom↓ (Y ) = {h ∈ Dom(Y ) : h is a left-continuous and decreasing function such that lim h(t) = 0}. t→∞ Here, following Definition 3.1 of Sato [15], by saying that the (improper stochastic integral) ∞ q h(t)d Yt is definable we mean that p h(t)d Yt converges in probability as p ↓ 0, q → ∞, with the ∞ limit random variable being denoted by h(t)d Yt . The property of h belonging to Dom(Y ) can be characterized in terms of the generating triplet (AY , νY , γY ) of Y and assumptions on h, cf. Sato [15], Theorems 2.6, 3.5 and 3.10. In particular, if AY = 0, then h ∈ Dom(Y ) if and only if h is measurable, ∞ (|h(s)x|2 ∧ 1) νY (d x) < ∞, ds q h(s)γY + h(s)x + |h(s)x|2 d p (4.1) d − 1 + |x|2 νY (d x) ds < ∞ (4.2) for all < p < q < ∞ and q h(s)γY + lim p↓0,q→∞ In this case, given by h(s)x d p ∞ 1 + |h(s)x|2 − 1 + |x|2 νY (d x) ds exists in d . (4.3) h(t) d Yt is infinitely divisible without Gaussian part and its Lévy measure νY,h is ∞ νY,h(B) = 1B (h(s)x) νY (d x), ds B∈ ( d \ {0}). (4.4) d If νY is symmetric and γY = 0, then (4.2) and (4.3) are automatically satisfied, so that h ∈ Dom(Y ) ∞ if and only if (4.1) is satisfied, in which case γY,h in the generating triplet of h(t) d Yt is 0. 1130 Recall that Eα0,ri ( d ) = {µ ∈ Eα ( d ) : µ has no Gaussian part} ∩ Iri ( d ). The next theorem characterizes Eα0,ri ( 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 . (α) Theorem 4.1. Let α > and denote by Y (α) = {Yt } t≥0 a compound Poisson process on ∞ α measure νY (α) (B) = S dξ 1B (rξ)r α−1 e−r d r, equivalently d with Lévy α νY (α) (dξd r) = dξr α−1 e−r d r, ξ ∈ S, r > (4.5) (without drift). Then ∞ Eα0,ri ( d )= (α) : h ∈ Dom(Y (α) ) (4.6) (α) : h ∈ Dom↓ (Y (α) ) . (4.7) h(t)d Yt ∞ = h(t)d Yt The function h ∈ Dom↓ (Y (α) ) in representation (4.7) is uniquely determined by µ ∈ Eα0,ri ( d ). Proof. Let µ ∈ Eα0,ri ( d ). By definition and Remark 1.1, the Lévy measure ν of µ has the polar decomposition (λ, νξ ) given by νξ (d r) = r α−1 g(r α )d r, r > 0, λ(dξ) = dξ, (4.8) and g is independent of ξ and completely monotone. (If µ = δ0 we define gξ = 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 measure Q on [0, ∞) such that g( y) = [0,∞) e− y t Q(d t). By (2.4), since νξ satisfies ∞ (r ∧ 1)νξ (d r) < ∞, we see that ∞ Q({0}) = 0, t −1 Q(d t) < ∞ t −1−2/α Q(d t) < ∞. and (4.9) Observe that under this condition, we have for each r > 0, ∞ νξ ([r, ∞)) = y α−1 ∞ α ∞ −1 g( y )d y = (αt) Q(d t) r ∞ α αt y α−1 e− y t d y r α (αt)−1 e−r t Q(d t). = 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 ∞ ∞ ds α |h(s)r|2 ∧ r α−1 e−r d r < ∞, 1131 (4.10) ∞ (α) in which case h(t) d Yt is infinitely divisible with the generating triplet (AY,h = 0, νY,h, γY,h = 0) and the Lévy measure νY,h is rotationally invariant. Suppose B = C × [r, ∞), where C ∈ (S) and r > 0. Then by (4.4) and (4.5), ∞ ∞ νY,h(B) = νY,h(C × [r, ∞)) = 1C (ξ)dξ ds ∞ = α−1 |C| e−r α (4.11) r/|h(s)| S /|h(s)|α α x α−1 e−x d x ds 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 ∞ ∞ α e−r t −1 e−r t Q(d t) = α /|h(s)|α for every r > 0. ds (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), let Q satisfy (4.9), and denote t −1 Q(d t), F (x) := x ∈ [0, ∞), (0,x] and by F ← (t) = inf{ y ≥ : F ( y) ≥ t}, t ∈ [0, ∞), its left-continuous inverse, with the usual convention inf = +∞. Now define h = hQ : (0, ∞) → [0, ∞), t → (F ← (t))−1/α . Then h is left-continuous, decreasing, and satisfies lim t→∞ h(t) = 0. Denote Lebesgue measure on (0, ∞) by m1 , and consider the function T : (0, ∞) → (0, ∞], s → 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 −1 Q |(0,∞) (d t). (4.14) Hence it follows that for every r > 0, e−r α /h(s)α e−r m1 (ds) = (0,∞) α T (s) m1 (ds) (0,∞)∩{s:T (s)=∞} α e−r t (T (m1 ))(d t), = (0,∞) yielding (4.12). To show (4.10), namely that h ∈ Dom(Y (α) ), observe that ∞ ∞ ds α |h(s)r|2 ∧ r α−1 e−r d r 1132 (4.15) ∞ = r α+1 −r α e ∞ h(s) ds + dr α r α+1 e−r d r = {s:T (s)≥r α } ∞ α r α+1 e−r d r = {t≥r α } α r α−1 e−r d r ds {s:h(s)≤1/r} ∞ ∞ 1/h(s) ∞ −1 −T (s) T (s)−2/α ds + α e ds ∞ t −1−2/αQ(d t) + α−1 e−t t −1 Q(d t) by (4.14). The second of these terms is clearly finite by (4.9). To estimate the first, observe that ∞ ∞ α r α+1 e−r d r t −1−2/α Q(d t) rα ∞ ≤ ∞ α r α+1 e−r d r r α+1 d r t −1−2/α Q(d t) + r α+1 d r + t −1−2/α Q(d t) ∞ t −1−2/α Q(d t), rα and the first two summands are finite by (4.9), while the last summand is equal to t 1/α t −1−2/α Q(d t) r α+1 t 1+2/α t −1−2/α Q(d t) −1 d r = (α + 2) 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 (m1 ). Define the measure ∞ (α) Q on [0, ∞) by Q({0}) = and equality (4.14). Since h(t) d Yt is automatically infinitely divisible with Lévy measure νY,h given by (4.11), we have as in the proof of (a) for every C ∈ (S) and r > 0, |C| e −r α t ∞ −1 (αt) −1 Q(d t) = α e−r |C| (0,∞) α /h(s)α ds = νY,h(C × [r, ∞)). 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 ∞ (α) the proof of (2.4) that (4.9) must hold. Hence we have seen that ( h(t)d Yt ) ∈ Eα0,ri ( d ) for nonnegative h ∈ Dom(Y (α) ). For general h ∈ Dom(Y (α) ), write h = h+ − h− with h+ := h ∨ and h− := (−h)∨0. Then h+ , h− ∈ Dom(Y (α) ) by (4.10), and Equation (4.4) and the discussion following ∞ (α) it show that h(t)d Yt has no Gaussian part, gamma part and satisfies νY,h = νY,h+ + νY,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). Finally, to show uniqueness of h ∈ Dom↓ (Y (α) ) in the representation (4.7), let h1 , h2 ∈ Dom↓ (Y (α) ) such that ∞ ∞ (α) h1 (t)d Yt (α) = h2 (t)d Yt 1133 . 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 ∞ e −r α /|h1 (s)|α ∞ e−r ds = α /|h2 (s)|α ds < ∞ for all r > 0, which using the argument of (4.15) can be written as α α e−r t (T1 (m1 ))(d t) = e−r t (T2 (m1 ))(d t) < ∞, (0,∞) r > 0. (4.16) (0,∞) Observe that T1 and T2 are 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 every b ∈ (0, ∞) that m1 ({s ∈ (0, ∞) : T1 (s) ≤ b}) = m1 ({s ∈ (0, ∞) : T2 (s) ≤ b}) < ∞. Since T1 and T2 are left-continuous and increasing, this clearly implies T1 = T2 and hence h1 = h2 , completing the proof of the uniqueness assertion in representation (4.7). Next, we assume d = and we ask whether every distribution in Eα0 ( ) := {µ ∈ Eα ( ) : µ has no Gaussian part} can be represented as a stochastic integral with respect to the compound Poisson process Z (α) havα ing Lévy measure ν Z (α) (d x) = x α−1 e−x 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 ( ) which correspond to Lévy processes of bounded variation, but that not every distribution in Eα0 ( ) can be represented in this way. However, every distribution in Eα0 ( ) 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 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 < p < q < ∞ there are real constants τ p,q such that q h(t)d Yt − τ p,q converges in probability as p ↓ 0, q → ∞. We write Domes (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 ∈ Domes (Y ) we denote the class of distributions arising as q possible limits p h(t)d Yt − τ p,q as p ↓ 0, q → ∞ by Φh,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 Domes (Y ) can be expressed in terms of the characteristic triplet (AY , νY , γY ) of Y . In particular, if AY = 0, then a function h on (0, ∞) is in Domes (Y ) if and only if h is measurable and (4.1) and (4.2) hold, and in that case Φh,es (Y ) consists of all infinitely divisible distributions µ with characteristic triplet (AY,h = 0, νY,h, γ), where νY,h is given by (4.4) and γ ∈ is arbitrary (cf. [15], Theorems 3.6 and 3.11). p Recall Eα+ ( ) = {µ ∈ Eα ( Eα+,0 ( ) : µ((−∞, 0)) = 0} and denote ) := {µ ∈ Eα+ ( (µ) ) : {X t } has zero drift}, 1134 EαBV ( ) := {µ ∈ Eα ( EαBV,0 ( ) := {µ ∈ EαBV ( Eα0,sym ( ) := Eα0 ( (µ) ) : {X t } is of bounded variation}, (µ) ) : {X t } has zero drift}, ) ∩ Isym ( ) = Eα0,ri ( ). We then have: (α) Theorem 4.2. Let α > and denote by Z (α) = {Z t } t≥0 a compound Poisson process on with Lévy α measure ν Z (α) (d x) = x α−1 e−x 1(0,∞) (x) d x (without drift). Then it holds: (i) The class of distributions arising as limits of essential improper stochastic integrals with respect to Z (α) is Eα0 ( ) : Eα0 ( Φh,es (Z (α) ). )= (4.17) h∈Domes (Z (α) ) (ii) Distributions in EαBV,0 ( ) and Eα+,0 ( (0, ∞) with respect to Z (α) . More precisely Eα+,0 ( ) EαBV,0 ( ) can be expressed as improper stochastic integrals over ∞ = (α) : h ∈ Dom(Z (α) ), h ≥ , (4.18) (α) : h ∈ Dom(Z (α) ) such that (4.19) h(t)d Z t ∞ )= h(t)d Z t ∞ (|h(s)x| ∧ 1)ν Z (α) (d x) < ∞ . ds In particular, Eα+ ( ) ∞ = (α) h(t)d Z t : h ∈ Dom(Z (α) ), h ≥ 0, b ∈ [0, ∞) . +b (4.20) (iii) Not every distribution in Eα0 ( ) can be represented as an improper stochastic integral over (0, ∞) with respect to Z (α) plus some constant. It holds ∞ EαBV ( ) ∪ Eα0,sym ( ) (α) h(t)d Z t +b : b ∈ , h ∈ Dom(Z (α) ) Eα0 ( ). (4.21) Proof. (i) Let h ∈ Domes (Z (α) ) and µ ∈ Φh,es (Z (α) ) and write h = h+ − h− with h+ and h− being the positive and negative parts of h, respectively. Then µ is infinitely divisible without Gaussian part and by (4.4) its Lévy measure ν Z,h satisfies ∞ −1 e−r ν Z,h,1 ([r, ∞)) := ν Z,h([r, ∞)) = α α /h+ (s)α ds, ∞ ν Z,h,−1 ([r, ∞)) := ν Z,h((−∞, −r]) = α−1 e−r α /h− (s)α ds for every r > 0. Define the mappings T1 , T−1 : (0, ∞) → (0, ∞] by T1 (s) = (h+ (s))−α and T−1 (s) = (h− (s))−α and the measures Q and Q −1 on [0, ∞) by Q ξ ({0}) = and (Tξ (m1 ))|(0,∞) (d t) = t −1Q ξ |(0,∞) (d t), 1135 ξ ∈ {−1, 1}. Then as in the proof of Theorem 4.1, α (0,∞) e−r t (αt)−1Q ξ (d t) = ν Z,h,ξ ([r, ∞)), r > 0, ξ ∈ {−1, 1}, and Q and Q −1 satisfy (4.9) and we conclude that ν Z,h,ξ (d r) = r α−1 gξ (r α )d r for completely monotone functions g1 and g−1 , so that Φh,es (Z (α) ) ⊂ Eα0 ( ), giving the inclusion “⊃” in equation (4.17). Now let µ ∈ Eα0 ( [0, ∞) by ) with Lévy measure ν, and define the Lévy measures ν1 and ν−1 supported on ν1 (B) := ν(B), Then ν−1 (B) := ν(−B), ∞ α (αt)−1 e−r t Q ξ (d t), νξ ([r, ∞)) = B∈ r > 0, ((0, ∞)). (4.22) ξ ∈ {−1, 1}, (4.23) for some Borel measures Q and Q −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) with ν Z (α) in place of νY ) and (4.12) hold. Since h1 , 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 of µ1 ∈ Φh1 ,es (Z (α) ) and µ−1 ∈ Φh−1 ,es (Z (α) ) are given by ν1 and ν−1 , respectively. Now define the function h : (0, ∞) → by  h1 (t − n), t ∈ (2n, 2n + 1], n ∈ {1, 2, . . .},   −h (t − n − 1), t ∈ (2n + 1, 2n + 2], n ∈ {1, 2, . . .}, −1 h(t) = (4.24) −k−1 −k −k −k−1  h (t − ), t ∈ (2 , + ], k ∈ {0, 1, 2, . . .},  −h−1 (t − 2−k ), t ∈ (2−k + 2−k−1 , 2−k+1 ], k ∈ {0, 1, 2, . . .}. Then also h ∈ Domes (Z (α) ) and any µ ∈ Φh,es (Z (α) ) has Lévy measure ν, showing the inclusion “⊂” in equation (4.17). (ii) Let h ∈ Dom(Z (α) ). Then ∞ (α) h(t)d Z t ∞ (α) h(t) d Z t ∈ Eα0 ( ) by (i). Further, by Theorem 3.15 in Sato [15], is the distribution at time of a Lévy process of bounded variation if and only if ∞ ds (|h(s)x| ∧ 1)ν Z (α) (d x) < ∞, (4.25) ∞ (α) in which case this Lévy process will have zero drift. Since ( h(t)d Z t ) has trivially support contained in [0, ∞) if h ≥ 0, this gives the inclusion “⊃” in (4.18) and (4.19). Now suppose that µ ∈ EαBV,0 ( ) with Lévy measure ν, define ν1 and ν−1 by (4.22) and choose Borel measures Q and Q −1 such that (4.23) holds. Then it can be shown in complete analogy to the proof ∞ leading to (2.4) that for ξ ∈ {−1, 1}, νξ satisfies (1 ∧ x)νξ (d x) < ∞ if and only if ∞ Q ξ ({0}) = 0, t −1 Q ξ (d t) < ∞ t −1−1/αQ ξ (d t) < ∞. and (4.26) For ξ ∈ {−1, 1} and x ∈ [0, ∞) define Fξ (x) := (0,x] t −1Q ξ (d t), hξ = (Fξ← )−1/α and Tξ = (hξ )−α = Fξ← . Then it follows in complete analogy to the proof of (a) of Theorem 4.1, using (4.26), that (4.12) 1136 and (4.25) hold for hξ and Q ξ . By Theorem 3.15 in Sato [15] this then shows that hξ ∈ Dom(Z (α) ) for ξ ∈ {−1, 1}. Now if µ ∈ Eα+,0 ( ), define h(t) := h1 (t), and for general µ ∈ EαBV,0 , define h(t) by (4.24). In each case h satisfies (4.25), h ∈ Dom(Z (α) ), and µ = inclusions “⊂” in (4.18) and (4.19). 0,sym (iii) Let µ ∈ Eα ( ∞ (α) f (t)d Yt ( ) = Eα0,ri ( ( ∞ (α) h(t)d Z t ), giving the ). By Theorem 4.1 there exists f ∈ Dom↓ (Y (α) ) such that µ = ). Write h1 = h−1 := f and define the function h : (0, ∞) → by (4.24). We claim (α) that h ∈ Dom(Z ). To see this, observe that h clearly satisfies (4.1) with respect to ν Z (α) since f has the corresponding property with respect to νY (α) . Next, since |h(s)x|(1 + |h(s)x|2 )−1 is bounded by 1/2 and ν Z (α) ( ) is finite, it follows that ∞ q h(s)x α + |h(s)x|2 x α−1 e−x d x ds < ∞ ∀ q > 0. (4.27) But since Z (α) has the generating triplet ∞ A Z (α) = 0, ν Z (α) , γ Z (α) = x 1+ α x2 x α−1 e−x d x , (4.27) shows that (4.2) is satisfied for h with respect to ν Z (α) . Finally, by the definition of h, for ∞ q γ Z,h,0,q := h(s)x α + |h(s)x|2 x α−1 e−x d x ds, q > 0, we have γ Z,h,0,q = for q = 2, 4, 6, . . ., and since lim t→∞ h(t) = it follows that limq→∞ γ Z,h,0,q exists and is equal to 0. We conclude that (4.3) is satisfied, so that h ∈ Dom(Z (α) ). By (4.4) we ∞ ∞ (α) (α) clearly have h(t)d Z t = f (t)d Yt = µ. Together with (4.17) and (4.19) and 0 this shows (4.21) apart from the fact that the inclusions are proper. 0,sym To show that the first inclusion in (4.21) is proper, let µ ∈ Eα ( ) \ EαBV ( ). 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 t −1−1/αQ(d t) = ∞. As already shown, there exists h ∈ Dom(Z (α) ) such that ∞ (α) µ = ( h(t)d Z t ). Then h + 1[1,2] ∈ Dom(Z (α) ), and neither symmetric nor of finite variation. ( ∞ To see that the second inclusion in (4.21) is proper, let µ ∈ Eα0 ( supported on [0, ∞) such that ∞ (α) (α) (h(t) + 1[1,2] (t)d Z t ) is clearly ) with Lévy measure ν being x ν(d x) = ∞. Suppose there are b ∈ and h ∈ Dom(Z (α) ) such that µ = ( h(t)d Z t + b). Since ν is supported on [0, ∞), we must have h ≥ Lebesgue almost surely, so that we can suppose that h ≥ everywhere. Then we have from (4.1) and (4.3) that ∞ ∞ (|h(s)x|2 ∧ 1) ν Z (α) (d x) < ∞ ds and ∞ ∞ ds h(s)x + h(s)x 1137 ν Z (α) (d x) < ∞. Together these two equations imply ∞ ∞ (|h(s)x| ∧ 1)ν Z (α) (d x) < ∞, ds so that µ ∈ EαBV ( ) by (4.19), contradicting x ν(d x) = ∞. This completes the proof of (4.21). Proof of Theorem 1.2. This is an immediate consequence of Equation (4.18) since B ( +,0 E1 ( ). The composition of Φ with ∞ (µ) Recall that Φ(µ) = with D(Φ) = Ilog ( e−t d X t tion Φ ◦ α . We start with the following proposition. d Proposition 5.1. Let α > 0, m ∈ {1, 2, . . .} and µ ∈ I( d ). α (µ) ∈ I logm ( Proof. Let ν and ν denote the Lévy measures of µ and that ∞ ν(d x) d d d ∞ (log |x|) ν(d x) = ν(d x) d |x|>1 = 1/|x| m ν(d x) d d ). Then µ ∈ Ilogm ( α (µ), d ) if and only if respectively. By (2.1), we conclude α ϕ(ux)αuα−1 e−u du → [0, ∞]. In particular, we have α (log(u|x|))m αuα−1 e−u du ∞ m h(x)ν(d x), α (log u)n αuα−1 e−u du (log |x|)m−n n n=0 =: ). In this section we study the composi- for every measurable nonnegative function ϕ : m = and its application α ϕ(x) ν(d x) = +) 1/|x| say. d Then it is easy to see that h(x) = o(|x|2 ) as |x| ↓ and that lim|x|→∞ h(x)/(log |x|)m = ∞ α αuα−1 e−u du = 1. Hence, ν(d x) < ∞, giving the claim. |x|>1 (log |x|)m ν(d x) < ∞ if and only if |x|>1 (log |x|)m Theorem 5.2. Let α > and ∞ α u−1 e−u du, nα (x) = x > 0. x Let x = n∗α (t), t > 0, be its inverse function, and define the mapping ∞ α (µ) = (µ) n∗α (t) d X t 1138 , α µ ∈ Ilog ( : Ilog ( d ). d ) → I( d ) by It then holds Φ◦ α = α ◦Φ= α, (5.1) . (5.2) including the equality of the domains. In particular, we have Φ◦ = ◦Φ= Proof. We first note that D( α ) is independent of the value of α and equals Ilog ( Theorem 2.3 of [8], (essentially in Theorem 2.4 (i) of [14].) d ), shown in As mentioned right after Equation (1.5), D(Φ) = Ilog ( d ). Thus it follows from Proposition 5.1 that both Φ ◦ α as well as α ◦ Φ are well defined on Ilog ( d ) and that they have the same domain. Note that ∞ C (z) = α (µ) Cµ (log t −1 )1/α z d t = Cµ (u1/α z)e−u du and ∞ Cµ e−t z d t. CΦ(µ) (z) = Then, if we are allowed to exchange the order of the integrals by Fubini’s theorem, we have ∞ C( α ◦Φ)(µ) ∞ −s (z) = Cµ (s1/α e−t z)d t e ds ∞ = (5.3) ∞ αuα−1 Cµ (uz) ∞ α αt eαt−u e d t du α Cµ (uz)u−1 e−u du = ∞ =− Cµ (uz)d nα (u) ∞ = Cµ (n∗α (t)z)d t, and the same calculation can be carried out for C(Φ◦ α )(µ) (z) = ∞ Cµ (n∗α (t)z)d t. In order to assure the exchange of the order of the integrations by Fubini’s theorem, it is enough to show that ∞ ∞ Cµ (s1/α e−t z) d t < ∞. e−s ds (5.4) This is Equation (4.5) in Barndorff-Nielsen et al. [3] with the replacement of s by s1/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 ∞ C(Φ◦ (z) = C( α )(µ) (z) = α ◦Φ)(µ) Cµ (n∗α (t)z) d t = C 1139 α (µ) (z), z∈ d , and that Φ ◦ α = α ◦Φ= α. Since = , this shows in particular (5.2). It is well known that Φ(Ilog ( d )) = L( d ), the class of selfdecomposable distributions on immediate consequence of Theorem 5.2 is the following. d . An Theorem 5.3. Let α > 0. Then d Φ(Eα ( ) ∩ Ilog ( d )) = d α (L( )) = d α (I log ( )). 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 mapping α . We need some lemmas. In the following, m m+1 = αm ◦ α . α is defined recursively as α Lemma 5.4. Let α > 0. For m = 1, 2, . . ., we have m α ) D( = Ilogm ( d m α ) and = Φm ◦ m α = m α ◦ Φm . Proof. By Proposition 5.1, we have µ ∈ Ilogm ( d ) if and only if α (µ) ∈ Ilogm ( d ). As shown in the proof of Lemma 3.8 in [9], we also have that µ ∈ Ilogm+1 ( d ) if and only if µ ∈ Ilog ( d ) and Φ(µ) ∈ Ilogm ( d ), and thus D(Φm ) = Ilogm ( d ). Since α = Φ ◦ α = α ◦ Φ, we conclude that µ ∈ Ilogm+1 ( d ) if and only if µ ∈ Ilog ( d ) and α (µ) ∈ Ilogm ( d ). (5.5) Now we prove D( αm ) = Ilogm ( d ) inductively. For m = this is known, so assume that D( αm ) = Ilogm ( d ) for some m ≥ 1. If µ ∈ D( αm+1 ), then αm+1 (µ) = αm ( α (µ)) is well-defined. Thus, m d ) by assumption, so that µ ∈ Ilogm+1 ( d ) by (5.5). Conversely, if µ ∈ α (µ) ∈ D( α ) = I logm ( Ilogm+1 ( d ), then µ ∈ Ilog ( d ) and α (µ) ∈ Ilogm ( d ) by (5.5), so that αm ( α (µ)) is well-defined by assumption. This shows D( αm+1 ) = Ilogm+1 ( d ). That αm = Φm ◦ αm = αm ◦ Φm for every m then follows easily from (5.1), Proposition 5.1 and D(Φm ) = Ilogm ( d ). Let S( d ) be the class of all stable distributions on d , and for m = 0, 1, . . . denote L m ( d ) = d m+1 Φm+1 (Ilogm+1 ( d )), L∞ ( d ) = ∩∞ ), Nα,m ( d ) = (Ilogm+1 ( d )) and Nα,∞ ( d ) = m=0 L m ( α d ∩∞ ). Lemma 5.4 implies that Nα,m ( d ) ⊃ Nα,m+1 ( d ), so that the family Nα,m , m = m=0 Nα,m ( 0, 1, . . ., is nested. It is known (cf. Sato [10]) that L∞ ( d ) = S( d ), where the closure is taken under weak convergence and convolution. In order to show that also Nα,∞ ( d ) = S( d ), we need two further lemmas. Lemma 5.5. For α > 0, α maps S( d ) bijectively onto S( α (S( d )) = S( d d ), namely ). This is an immediate consequence of Proposition 2.1 (ii). Lemma 5.6. Let α > 0. For m = 0, 1, . . . , Nα,m ( and S( d ) ⊂ Nα,m ( d ) = d ) is closed under convolution and weak convergence, m+1 (L m ( α 1140 d )) ⊂ L m ( d ). (5.6) Proof. By Lemma 5.4, Nα,m ( hence S( d d m+1 (Ilogm+1 ( α )= ) ⊂ Nα,m ( Nα,m ( d d d m+1 α )) = ( ◦ Φm+1 )(Ilogm+1 ( ) by Lemma 5.5 and the fact that S( ) = (Φm+1 ◦ m+1 )(Ilogm+1 ( α d d )) = m+1 (L m ( α ) ⊂ Lm( d ). Further, )) ⊂ Φm+1 (Ilogm+1 ( d d d )) = L m ( d )), ). Next observe that α and hence αm+1 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 Nα,m ( d ) is closed under convolution and weak convergence, too. We can now characterize Nα,∞ ( gence: d ) as the closure of S( d ) under convolution and weak conver- Theorem 5.7. Let α > 0. It holds L∞ ( d d ) = Nα,∞ ( ) = S( d ). )) = S( d ). In particular, lim m m→∞ (Ilogm ( d Proof. By (5.6) we have S( But since each Nα,m ( section Nα,∞ ( d ) = d d) = L∞ ( d ) ⊃ Nα,∞ ( d ) ⊃ S( d ). ) is closed under convolution and weak convergence, so must be the interd ), and together with = the assertions follow. ∞ m=0 Nα,m ( 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. References [1] Aoyama, T.: Nested subclasses of the class of type G selfdecomposable distributions on Probab. Math. Statist. 29, 135–154 (2009) MR2553004 d . [2] Aoyama T., Maejima M. and Rosi´ nski J.: A subclass of type G selfdecomposable distributions. J. Theor. Probab. 21, 14–34 (2008) MR2384471 [3] Barndorff-Nielsen, O.E., Maejima, M. and Sato, K.: Some classes of multivariate infinitely divisible distributions admitting stochastic integral representations. 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Stat. 1, 47–87 (2006) MR2235174 [15] Sato, K.: Transformations of infinitely divisible distributions via improper stochastic integrals. ALEA Lat. Am. J. Probab. Math. Stat. 3, 67–110 (2007) MR2349803 [16] Wolfe, S.J.: On a continuous analogue of the stochastic difference equation X n = ρX n−1 + Bn . Stoch. Proc. Appl. 12, 301–312 (1982) MR0656279 1142 [...]... Maejima, M and Nakahara, G.: A note on new classes of infinitely divisible distributions on d Electr Comm Probab 14, 358–371 (2009) MR2535084 [9] Maejima, M and Sato, K.: The limits of nested subclasses of several classes of infinitely divisible distributions are identical with the closure of the class of stable distributions Probab Theory Relat Fields 145, 119–142 (2009) MR2520123 [10] Sato, K.: Class. .. of Mathematics at Keio University He thanks for their hospitality and support References [1] Aoyama, T.: Nested subclasses of the class of type G selfdecomposable distributions on Probab Math Statist 29, 135–154 (2009) MR2553004 d [2] Aoyama T., Maejima M and Rosi´ ski J.: A subclass of type G selfdecomposable distributions n J Theor Probab 21, 14–34 (2008) MR2384471 [3] Barndorff-Nielsen, O.E., Maejima,... integrals Illinois J Math 50, 825–851 (2006) MR2247848 [14] Sato, K.: Two families of improper stochastic integrals with respect to Lévy processes, ALEA Lat Am J Probab Math Stat 1, 47–87 (2006) MR2235174 [15] Sato, K.: Transformations of infinitely divisible distributions via improper stochastic integrals ALEA Lat Am J Probab Math Stat 3, 67–110 (2007) MR2349803 [16] Wolfe, S.J.: On a continuous analogue...Proof By the definition it is clear that all the classes under consideration are closed under convolution, scaling and taking of powers The class Eα ( d ) is closed under weak convergence by sym + Proposition 2.1 (iv) and Theorem 2.3, and hence so are Eα ( d ) and Eα ( d ) Further, it is easy to (α) see that all the given classes contain the specified distributions, since the Lévy measure of (Z1... L of multivariate distributions and its subclasses J Multivar Anal 10, 207–232 (1980) MR0575925 [11] Sato, K.: Lévy Processes and Infinitely Divisible Distributions Cambridge University Press, Cambridge (1999) MR1739520 [12] Sato, K.: Stochastic integrals in additive processes and application to semi-Lévy processes Osaka J Math 41, 211–236 (2004) MR2040073 [13] Sato, K.: Additive processes and stochastic... for ξ ∈ S has polar decomposition λ = δξ and νξ (d r) = r α−1 gξ (r α ) d r with gξ (r) = e−r , and a (α) similar argument works for (Y1 ξ) Finally, Eα ( 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. .. weak convergence, scaling and m taking of powers and contains α (N1 ξ) for all ξ ∈ S and N1 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 4 Characterization of subclasses of Eα ( d ) by stochastic integrals with respect to some compound Poisson processes For any Lévy process Y = {Yt } t≥0 on d , denote by L(0,∞) (Y ) the class of. .. and Theorem 2.3 then imply F = sym Eα ( d ) Remark 3.3 In the introduction it was mentioned that B( d ) is the smallest class of distributions on d closed under convolution and weak convergence and containing the distributions of all elementary mixed exponential random variables in d Theorem 3.2 for α = 1 gives a new interpretation of B( d ), since it is based on a compound Poisson distribution, rather... p,q may 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 ) of Y In particular, if AY = 0, then a function h on (0, ∞) is in Domes (Y ) if and only if h is measurable and (4.1) and (4.2) hold, and in that case Φh,es (Y ) consists of all infinitely divisible distributions µ with characteristic... Gaussian part zero and Lévy measure α−1 δξ with ξ ∈ S Since G is closed under convolution, scaling and taking of powers it also contains all infinitely n divisible distributions with Gaussian part zero and Lévy measures of the form ν = i=1 ai δci with n ∈ , ai ≥ 0 and ci ∈ d \ {0} Since every finite Borel measure on d is the weak limit of a n sequence of measures of the form i=1 ai δci , it follows from Theorem . 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. 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. 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