Fractional integrals and extensions of selfdecomposability Ken-iti Sato Abstract After characterizations of the class L of selfdecomposable distributions on Rd are recalled, the classes K p,α and L p,α with two continuous parameters < p < ∞ and −∞ < α < satisfying K1,0 = L1,0 = L are introduced as extensions of the class L They are defined as the classes of distributions of improper stochastic integrals ∫ ∞− (ρ ) (ρ ) f (s)dXs , where f (s) is an appropriate non-random function and Xs is a d L´evy process on R with distribution ρ at time The description of the classes is given by characterization of their L´evy measures, using the notion of monotonicity of order p based on fractional integrals of measures, and in some cases by addition of the condition of zero mean or some weaker conditions that are newly introduced— having weak mean or having weak mean absolutely The class Ln,0 for a positive integer n is the class of n times selfdecomposable distributions Relations among the classes are studied The limiting classes as p → ∞ are analyzed The Thorin class T , the Goldie–Steutel–Bondesson class B, and the class L∞ of completely selfdecomposable distributions, which is the closure (with respect to convolution and weak convergence) of the class S of all stable distributions, appear in this context Some subclasses of the class L∞ also appear The theory of fractional integrals of measures is built Many open questions are mentioned AMS Subject Classification 2000: Primary: 60E07, 60H05 Secondary: 26A33, 60G51, 62E10, 62H05 Key words: Class L, Class L∞ , Completely monotone, Fractional integral, Improper stochastic integral, Infinitely divisible, L´evy measure, L´evy process, L´evyKhintchine triplet, Monotone of order p, Multiply selfdecomposable, Radial decomposition, Selfdecomposable, Spherical decomposition, Weak mean Hachiman-yama 1101-5-103, Tenpaku-ku, Nagoya, 468-0074 Japan, e-mail: ken-iti.sato@ nifty.ne.jp; homepage: http://ksato.jp/ Ken-iti Sato Introduction 1.1 Characterizations of selfdecomposable distributions A distribution µ on Rd is called infinitely divisible if, for each positive integer n, there is a distribution µn such that µ = µn ∗ µn ∗ · · · ∗ µn , n where ∗ denotes convolution The class of infinitely divisible distributions on Rd is denoted by ID = ID(Rd ) Let Cµ (z), z ∈ Rd , denote the cumulant function of µ ∈ ID, that is, the unique complex-valued continuous function on Rd with Cµ (0) = such that the characteristic function µ (z) of µ is expressed as µ (z) = eCµ (z) If µ ∈ ID, then Cµ (z) is expressed as Cµ (z) = − ⟨z, Aµ z⟩ + ∫ Rd (ei⟨z,x⟩ − − i⟨z, x⟩1{|x|≤1} (x))νµ (dx) + i⟨γµ , z⟩ (1.1) Here ⟨z, x⟩ is the canonical inner product of z and x in Rd , |x| = ⟨x, x⟩1/2 , 1{|x|≤1} is the indicator function of the set {|x| ≤ 1}, Aµ is a d × d symmetric nonnegativedefinite matrix, called the Gaussian covariance matrix of µ , νµ is a measure on Rd ∫ satisfying νµ ({0}) = and Rd (|x| ∧ 1)νµ (dx) < ∞, called the L´evy measure of µ , and γµ is an element of Rd The triplet (Aµ , νµ , γµ ) is uniquely determined by µ Conversely, to any triplet (A, ν , γ ) there corresponds a unique µ ∈ ID such that A = Aµ , ν = νµ , and γ = γµ Throughout this article Aµ , νµ , and γµ are used in this sense A distribution µ on Rd is called selfdecomposable if, for any b > 1, there is a distribution µb such that µ (z) = µ (b−1 z)µb (z), z ∈ Rd (1.2) Let L = L(Rd ) denote the class of selfdecomposable distributions on Rd It is characterized in the following four ways (a) A distribution µ on Rd is selfdecomposable if and only if µ ∈ ID and its L´evy measure νµ has a radial (or polar) decomposition νµ (B) = ∫ S λ (d ξ ) ∫ ∞ 1B (rξ )r−1 kξ (r)dr (1.3) for Borel sets B in Rd , where λ is a finite measure on the unit sphere S = {ξ ∈ Rd : |ξ | = 1} (if d = 1, then S is a two-point set {1, −1}) and kξ (r) is a nonnegative function measurable in ξ and decreasing and right-continuous in r (See Proposition 3.1 for exact formulation of radial decomposition.) Fractional integrals and extensions of selfdecomposability (b) Let {Zk : k = 1, 2, } be independent random variables on Rd and Yn = n ∑k=1 Zk Suppose that there are bn > and γn ∈ Rd for n = 1, 2, such that the law of bnYn + γn weakly converges to a distribution µ as n → ∞ and that {bn Zk : k = 1, , n; n = 1, 2, } is a null array (that is, for any ε > 0, max1≤k≤n P(|bn Zk | > ε ) → as n → ∞) Then µ ∈ L Conversely, any µ ∈ L is obtained in this way (ρ ) (c) Given ρ ∈ ID, let {Xt : t ≥ 0} be a L´evy process on Rd (that is, a stochastic process continuous in probability, starting at 0, with time-homogeneous independent increments, with cadlag paths) having distribution ρ at time If ∫ ∞− −s (ρ ) ∫ is de|x|>1 log |x|ρ (dx) < ∞, then the improper stochastic integral e dXs finable and its distribution (∫ ∞− ) (ρ ) µ =L e−s dXs (1.4) is selfdecomposable Here L (Y ) denotes the distribution (law) of a random element Y Conversely, any µ ∈ L is obtained in this way On the other hand, if ∫ ∞− −s (ρ ) ∫ is not definable (See Section 3.4 for |x|>1 log |x|ρ (dx) = ∞, then e dXs improper stochastic integrals.) To see that µ of (1.4) is selfdecomposable, notice that ∫ ∞− (ρ ) e−s dXs ∫ log b = (ρ ) e−s dXs ∫ ∞− + log b (ρ ) e−s dXs = I1 + I2 , I1 and I2 are independent, and ∫ ∞− I2 = (ρ ) (ρ ) e− log b−s dXlog b+s = b−1 (ρ ) ∫ ∞− (ρ ) e−s dYs , where {Ys } is identical in law with {Xs }, and hence µ satisfies (1.2) (d) Let {Yt : t ≥ 0} be an additive process on Rd , that is, a stochastic process continuous in probability with independent increments, with cadlag paths, and with Y0 = If, for some H > 0, it is H-selfsimilar (that is, for any a > 0, the two processes {Yat : t ≥ 0} and {aH Yt : t ≥ 0} have an identical law), then the distribution µ of Y1 is in L Conversely, for any µ ∈ L and H > 0, there is a process {Yt : t ≥ 0} satisfying these conditions and L (Y1 ) = µ Historically, selfdecomposable distributions were introduced by L´evy [18] in 1936 and written in his 1937 book [19] under the name “lois-limites”, to characterize the limit distributions in (b) L´evy wrote in [18, 19] that this characterization problem had been posed by Khintchine, and Khintchine’s book [16] in 1938 called these distributions “of class L” The book [9] of Gnedenko and Kolmogorov uses the same naming Lo`eve’s book [20] uses the name “selfdecomposable” The property (c) gives a characterization of the stationary distribution of an Ornstein–Uhlenbeck type process (sometimes called an Ornstein–Uhlenbeck process driven by a L´evy process) {Vt : t ≥ 0} defined by Ken-iti Sato Vt = e−t V0 + ∫ t (ρ ) es−t dXs , (ρ ) where V0 and {Xt : t ≥ 0} are independent The stationary Ornstein–Uhlenbeck type process and the selfsimilar process in the property (d) are connected via the so-called Lamperti transformation (see [11], [26]) For historical facts concerning (c) see [33], pp 54–55 The proofs of (a)–(d) and many examples of selfdecomposable distributions are found in Sato’s book [39] The main purpose of the present article is to give two families of subclasses of ID, with two continuous parameters, related to L, using improper stochastic integrals and extending the characterization (c) of L 1.2 Nested classes of multiply selfdecomposable distributions If µ ∈ L, then, for any b > 1, the distribution µb in (1.2) is infinitely divisible and uniquely determined by µ and b If µ ∈ L and µb ∈ L for all b > 1, then µ is called twice selfdecomposable Let n be a positive integer ≥ A distribution µ is called n times selfdecomposable, if µ ∈ L and if µb is n − times selfdecomposable Let L1,0 = L1,0 (Rd ) = L(Rd ) and let Ln,0 = Ln,0 (Rd ) be the class of n times selfdecomposable distributions on Rd Then we have ID ⊃ L = L1,0 ⊃ L2,0 ⊃ L3,0 ⊃ · · · (1.5) These classes and the class L∞ (Rd ) in Section 1.4 were introduced by Urbanik [52, 53] and studied by Sato [37] and others (In [37, 52, 53] the class Ln,0 is written as Ln−1 , but this notation is inconvenient in this article.) An n times selfdecomposable distribution is characterized by the property that µ ∈ ID with L´evy measure νµ having radial decomposition (1.3) in (a) with kξ (r) = hξ (log r) for some function hξ (y) monotone of order n for each ξ (see Section 1.5 and Proposition 2.11 for the monotonicity of order n) In the propeerty (b), µ ∈ Ln,0 is characterized by the property that L (Zk ) ∈ Ln−1,0 for k = 1, 2, In (c), µ ∈ Ln,0 is characterized by ρ ∈ Ln−1,0 in (1.4) A direct generalization of (1.4) using exp(−s1/n ) or, equivalently, exp(−(n! s)1/n ) in place of e−s is also possible In (d), µ ∈ Ln,0 if and only if, for any H, the corresponding process {Yt : t ≥ 0} satisfies L (Yt −Ys ) ∈ Ln−1,0 for < s < t The proofs are given in [12, 25, 33, 37] Fractional integrals and extensions of selfdecomposability 1.3 Continuous-parameter extension of multiple selfdecomposability In 1980s Nguyen Van Thu [49, 50, 51] defined a continuous-parameter extension of Ln,0 , replacing the positive integer n by a real number p > He introduced fractional times multiple selfdecomposability and used fractional integrals and fractional difference quotients On one hand he extended the definition of n times selfdecomposability based on (1.2) to fractional times selfdecomposability in the form of infinite products On the other hand he extended essentially the formula (1.4) in the characterization (c), considering its L´evy measure Directly using improper stochastic integrals with respect to L´evy processes, we will define and study the decreasing classes L p,0 for p > 0, which generalize the nested classes Ln,0 for n = 1, 2, Thus the results of Thu will be reformulated as a special case in a family L p,α with two continuous parameters < p < ∞ and −∞ < α < The definition of L p,α will be given in Section 1.6 1.4 Stable distributions and class L∞ Let µ be a distribution on Rd Let < α ≤ We say that µ is strictly α -stable if µ ∈ ID and, for any t > 0, µ (z)t = µ (t 1/α z), z ∈ Rd We say that µ is α -stable if µ ∈ ID and, for any t > 0, there is γt ∈ Rd such that µ (z)t = µ (t 1/α z) exp(i⟨γt , z⟩), z ∈ Rd (When µ is a δ -distribution, this terminology is not the same as in Sato [39].) Let Sα0 = S0α (Rd ) and Sα = Sα (Rd ) be the class of strictly α -stable distributions on Rd and the class of α -stable distributions on Rd , respectively Let S = S(Rd ) ∪ d be the class of stable distributions on R That is, S = 0 of a function f (s) on R in a suitable class is given by Fractional integrals and extensions of selfdecomposability ∫ ∞ cp r (s − r) p−1 f (s)ds, which is the interpolation (1 ≤ p < ∞) and extrapolation (0 < p ≤ 1) of the n times integration ∫ ∞ ∫ ∞ r dsn sn dsn−1 · · · ∫ ∞ s2 f (s1 )ds1 = (n − 1)! ∫ ∞ r (s − r)n−1 f (s)ds However, we need to use fractional integrals of measures Our definition is as follows Let R+ = [0, ∞), R◦+ = (0, ∞) and B(E) for the class of Borel sets in a space E A measure σ is said to be locally finite on R [resp R◦+ ] if σ ([a, b]) < ∞ for all a, b with −∞ < a < b < ∞ [resp < a < b < ∞] Let p > For a measure σ on R [resp R◦+ ], let σ (E) = c p ∫ ∫ dr E (r,∞) (s − r) p−1 σ (ds), E ∈ B(R) [resp B(R◦+ )] (1.9) Let D(I p ) [resp D(I+p )] be the class of locally finite measures σ on R [resp R◦+ ] such that σ is a locally finite measure on R [resp R◦+ ] Define I p σ (E) = σ (E), E ∈ B(R) [resp I+p σ (E) = σ (E), E ∈ B(R◦+ )] for σ ∈ D(I p ) [resp D(I+p )] Thus I p and I+p are mappings from measures to measures on R and R◦+ , respectively D(I p ) and D(I+p ) are their domains We call a [0, ∞]-valued function f (r) on R [resp R◦+ ] monotone of order p on R [resp R◦+ ] if ∫ f (r) = c p (r,∞) (s − r) p−1 σ (ds) (1.10) with some σ ∈ D(I p ) [resp D(I+p )] As will be shown in Example 2.17, functions monotone of order p ∈ (0, 1) have, in general, quite different properties from functions monotone of order p ∈ [1, ∞) We call f (r) completely monotone on R [resp R◦+ ] if it is monotone of order p on R [resp R◦+ ] for all p > This definition of complete monotonicity differs from the usual one in that positive constant functions are not completely monotone Typical completely monotone functions on R and R◦+ are e−r and r−α (α > 0), respectively The properties of fractional integrals of functions are studied in M Riesz [32], Ross (ed.) [35], Samko, Kilbas, and Marichev [36], Kamimura [15], and others Williamson [56] studied fractional integrals of measures on R◦+ for p ≥ and introduced the concept of p-times monotonicity But we not assume any knowledge of them In Sections 2.1–2.3 we build the theory of the fractional integral mappings I p and p I+ for p ∈ (0, ∞) from the point of view that they are mappings from measures to measures A basic relation is the semigroup property I q I p = I p+q and I+q I+p = I+p+q Ken-iti Sato An important property that both I p and I+p are one-to-one is proved The relation between the theories on R and R◦+ is not extension and restriction We need both theories, as will be mentioned at the end of Section 6.2 1.6 Classes K p,α and L p,α generated by stochastic integral mappings The formula (1.4) gives a mapping Φ from ρ ∈ ID(Rd ) to µ ∈ ID(Rd ) Thus (∫ ∞− ) (ρ ) Φρ = L e−s dXs (1.11) The domain of Φ is the class of ρ for which the improper stochastic integral in (1.11) is definable For functions f (s) in a suitable class, we are interested in the mapping Φ f from ρ ∈ ID to µ ∈ ID defined by (∫ ∞− ) (ρ ) (1.12) µ = Φf ρ = L f (s)dXs The domain D(Φ f ) is the class of ρ for which the improper stochastic integral in (1.12) is definable The range is defined by R(Φ f ) = {Φ f ρ : ρ ∈ D(Φ f )} Let us consider three families of functions For < p < ∞ and −∞ < α < ∞ let ∫ g¯ p,α (t) = c p t ∫ j p,α (t) = c p ∫ ∞ gα (t) = (1 − u) p−1 u−α −1 du, < t ≤ 1, (− log u) p−1 u−α −1 du, t u−α −1 e−u du, t < t ≤ 1, < t < ∞, (1.13) (1.14) (1.15) and a¯ p,α = g¯ p,α (0+), b p,α = j p,α (0+), aα = gα (0+) If α < 0, then a¯ p,α = Γ−α /Γp−α , b p,α = (−α )−p , and aα = Γ−α If α ≥ 0, then a¯ p,α = b p,α = aα = ∞ Let t = f¯p,α (s), l p,α (s), and fα (s) be the inverse functions of s = g¯ p,α (t), j p,α (t), and gα (t), respectively When α < 0, extend f¯p,α (s) for s ≥ a¯ p,α , l p,α (s) for s ≥ b p,α , and fα (s) for s ≥ aα to be zero Define Φ¯ p,α = Φ f¯p,α , Λ p,α = Φl p,α , Ψα = Φ fα Sato [42] studied the mapping Ψα and the mapping Φβ ,α = Φ fβ ,α , −∞ < β < α < ∞, for the inverse function fβ ,α (s) of the function gβ ,α (t) defined by ∫ gβ ,α (t) = cα −β t (1 − u)α −β −1 u−α −1 du, < t ≤ Fractional integrals and extensions of selfdecomposability To make parametrization more convenient, we use Φ¯ p,α = Φα −p,α For Φ¯ p,α , Λ p,α , and Ψα , the domains will be characterized In the analysis of the domains, asymptotic behaviors of f¯p,α (s), l p,α (s), and fα (s) for s → ∞ are essential The behaviors of f¯p,α (s) and fα (s) are similar, but the behavior of l p,α (s) is different from them If α ≥ 2, then D(Φ¯ p,α ) = D(Λ p,α ) = D(Ψα ) = {δ0 } So we will only consider −∞ < α < Define K p,α = K p,α (Rd ) = R(Φ¯ p,α ), (1.16) L p,α = L p,α (Rd ) = R(Λ p,α ) (1.17) It is clear that g¯1,α (t) = j1,α (t), and hence Φ¯ 1,α = Λ1,α , K1,α = L1,α for −∞ < α < (1.18) Since g¯1,0 (t) = j1,0 (t) = − logt, < t ≤ 1, and f¯1,0 (s) = l1,0 (s) = e−s , s ≥ 0, we have Φ¯ 1,0 = Λ1,0 = Φ , K1,0 = L1,0 = L (1.19) So K p,α and L p,α give extensions, with two continuous parameters, of the class L of selfdecomposable distributions Since l p,0 (s) = exp(−(Γp+1 s)1/p ), s ≥ 0, the class L p,0 coincides with the class of n times selfdecomposable distributions if p is an integer n The following are some of the new results in this article For any α and p with −∞ < α < and p > 0, any µ ∈ K p,α has L´evy measure νµ having a radial decomposition ∫ ∫ νµ (B) = S λ (d ξ ) ∞ 1B (rξ )r−α −1 kξ (r)dr (1.20) with kξ (r) measurable in (ξ , r) and monotone of order p on R◦+ in r, and any µ ∈ L p,α has L´evy measure νµ having a radial decomposition νµ (B) = ∫ S λ (d ξ ) ∫ ∞ 1B (rξ )r−α −1 hξ (log r)dr (1.21) with hξ (y) measurable in (ξ , y) and monotone of order p on R in y If −∞ < α < 1, then this property of νµ characterizes K p,α and L p,α If 1∫ < α < 2, then this property of νµ combined with the property of mean (that is, Rd |x|µ (dx) < ∞ and ∫ Rd x µ (dx) = 0) characterizes K p,α and L p,α We will introduce the notion of weak mean of infinitely divisible distributions in Section 3.3 If α = 1, then the property above of νµ and the property of weak mean characterize K p,1 ; the case of L p,1 is still open For each fixed α , the classes K p,α and L p,α are strictly decreasing as p increases and at the limit there appear connections with R(Ψα ) and with the class L∞ of completely selfdecomposable distributions Namely, define K∞,α = ∩ 00 e e d L∞, α = L∞,α (R ) = ∩ p>0 E , E ∈ B((0, 2)), introduced in Section 1.4 These are described by L∞ and L∞ 82 Ken-iti Sato e Theorem 7.11 Descriptions of L∞, α for all α and L∞,α for α ̸= are as follows e L∞, α = L∞ e L∞, α L∞, α L∞,α for −∞ < α ≤ 0, (7.14) for < α < 2, (7.15) for −∞ < α < 1, (7.16) (α ,2) = L∞ e = L∞, α (α ,2) ∫ = { µ ∈ L∞ : Rd xµ (dx) = 0} for < α < (7.17) ∩ e e e Proof (7.14) and (7.15): First, notice that L∞, α = n=1,2, Ln,α Let µ ∈ L∞,α Use Theorem 6.9 Then, for each n = 1, 2, , νµ has a radial decomposition (λ (n) (d ξ ), u−α −1 hξ (log u)du), where hξ (y) is measurable in (ξ , y) and, for λ (n) (n) (n) (n) a e ξ , hξ (y) is monotone of order n on R It follows from Proposition 3.1 that we (n) can choose λ (n) = λ and hξ = hξ independently of n Thus, for λ -a e ξ , hξ (y) is completely monotone on R We choose a modification of hξ (y) completely monotone on R for all ξ ∈ S Further, we choose λ to be a probability measure For y0 ∈ R, the function hξ (y0 + y), y ∈ R◦+ , is completely monotone on R◦+ and hence ∫ hξ (y0 + y) = (0,∞) e−yβ Γξy0 (d β ), y>0 with a unique measure Γξy0 on (0, ∞) by Bernstein’s theorem (recall that our definition of complete monotonicity involves hξ (y0 + y) → as y → ∞, so that Γξy0 has no mass at 0) In particular, we have Γξ0 for y0 = If y0 < 0, then hξ (y) = hξ (y0 + (y − y0 )) = ∫ (0,∞) e−(y−y0 )β Γξy0 (d β ), y>0 and hence ey0 β Γξy0 (d β ) = Γξ0 (d β ) Thus ∫ hξ (y0 + y) = (0,∞) e−(y0 +y)β Γξ0 (d β ), ∫ Therefore hξ (y) = (0,∞) e−yβ Γξ0 (d β ), y0 < 0, y > y ∈ R We see that {Γξ0 : ξ ∈ S} is a measurable family Indeed, if Γξ0 is a continuous measure for every ξ , then it is proved from the inversion formula (see [55], p 285) ∫ s [ys] (−y)m (d/dy)m (hξ (y)), m! m=0 ∑ y→∞ Γξ0 (d β ) = lim s > 0, where [ys] is the largest integer ≤ ys If not, it is proved by approximating Γξ0 by the convolutions with continuous measures We have Fractional integrals and extensions of selfdecomposability ∫ ∞> |x|≤1 ∫ = S ∫ = S ∫ = S |x|2 νµ (dx) = λ (d ξ ) λ (d ξ ) λ (d ξ ) ∫ ∫ S ∫ u1−α du ∫ λ (d ξ ) (0,∞) ∫ udu (α ,∞) ∫ (α ,∞) Γξ (d β ) 83 ∫ u1−α hξ (log u)du u−β Γξ0 (d β ) u−β Γξ (d β ) ∫ u1−β du, where we define Γξ (E) = Since ∫ (0,∞) 1E (α + β )Γξ0 (d β ), E ∈ B((α , ∞)) ∫ 1−β du = ∞ for β ≥ 2, we obtain Γξ ([2, ∞)) = for λ -a e ξ We have u ∫ |x|≤1 |x|2 νµ (dx) = ∫ ∫ λ (d ξ ) S (α ,2) (2 − β )−1Γξ (d β ) We also have ∫ ∞> |x|>1 ∫ = S ∫ = S ∫ = S ∫ νµ (dx) = λ (d ξ ) λ (d ξ ) λ (d ξ ) ∫ ∞ S u−α −1 du ∫ ∞ u−1 du (α ,2) ∫ ∫ (0,∞) Γξ (d β ) u−α −1 hξ (log u)du u−β Γξ0 (d β ) u−β Γξ (d β ) (α ,2) ∫ ∞ ∫ ∫ ∞ λ (d ξ ) u−β −1 du, ∫ and 1∞ u−β −1 du = ∞ for β ≤ Hence, if α < 0, then Γξ ((α , 0]) = for λ -a e ξ For any α < we have ∫ νµ (dx) = |x|>1 ∫ S λ (d ξ ) ∫ (α ∨0,2) β −1Γξ (d β ) Similarly, it follows from νµ (B) = that νµ (B) = ∫ S ∫ S λ (d ξ ) λ (d ξ ) ∫ ∞ 1B (uξ )u−α −1 hξ (log u)du ∫ (α ∨0,2) Γξ (d β ) ∫ ∞ 1B (uξ )u−β −1 du (7.18) (7.19) The measure λ (d ξ )Γξ (d β ) on S × (α ∨ 0, 2) is written to Γ (d β )λβ (d ξ ), where Γ (d β ) is a measure on (α ∨ 0, 2) satisfying 84 Ken-iti Sato ∫ (α ∨0,2) (β −1 + (2 − β )−1 )Γ (d β ) < ∞ and {λβ : β ∈ (α ∨ 0, 2)} is a measurable family of probability measures on S (α ∨0,2) e Therefore L∞, α ⊂ L∞ (α ∨0,2) Conversely, suppose that µ ∈ L∞ with L´evy measure νµ satisfying (1.6) Then, defining λ (d ξ ) and Γξ (d β ) in the converse direction and letting hξ (y) = ∫ −y(β −α ) Γ (d β ), we see (7.19) and then (7.18) with h (y) completely ξ ξ (α ∨0,2) e e This completes the proof of (7.14) and (7.15) monotone on R Hence µ ∈ L∞, α Assertions (7.16) and (7.17) follow from Theorem 6.12 (i) and (ii), respectively (α ,2) Note that if µ ∈ L∞ with < α < 2, then ∫ |x|>1 |x|νµ (dx) = and hence ∫ Rd ∫ (α ,2) Γ (d β ) ∫ S λβ (d ξ ) ∫ ∞ r−β dr = ∫ (α ,2) (β − 1)−1Γ (d β ) < ∞ |x|µ (dx) < ∞ ⊓ ⊔ Remark 7.12 Open problem: Give the description of the class L∞,1 ⊓ ⊔ Remark 7.13 Open question: Does there exist a function f (s), s ≥ 0, such that L∞, α e e e or L∞,α is equal to R(Φ f ), R (Φ f ), or R (Φ f )? In particular, for L∞ = L∞,0 = L∞,0 , this is a long-standing question ⊓ ⊔ Theorem 7.14 We have e K∞, α e L∞, α for −∞ < α < (7.20) K∞, α L∞, α for α ∈ (−∞, 1) ∪ (1, 2) (7.21) e −α −1 Proof We know that µ ∈ K∞, α if and only if νµ has radial decomposition (λ , u ◦ e kξ (u)du) with kξ (u) completely monotone on R+ On the other hand, µ ∈ L∞,α if and only if νµ has radial decomposition (λ , u−α −1 hξ (log u)du) with hξ (y) completely monotone on R Since the complete monotonicity of hξ (y) on R implies that e e of hξ (log u) on R◦+ , we have K∞, α ⊃ L∞,α To see the strictness of the inclusion, use y the functions h(y) = e−ce and k(u) = h(log u) = e−cu with c > 0; k(u) is completely monotone on R◦+ but h(y) is not completely monotone on R, since h′′ (y) = −h(y)ceu (1 − cey ) < for y close to −∞ Hence (7.20) is true Assertion (7.21) for α ∈ (−∞, 1) is automatic from (7.20) For α ∈ (1, 2), combine (7.20) with the condition of zero mean ⊓ ⊔ When µ ∈ L∞ , let Γµ denote the measure Γ in the representation (1.6) of νµ We give some moment properties of distributions in L∞ Proposition 7.15 Let µ ∈ L∞ Let < α < Fractional integrals and extensions of selfdecomposability 85 ∫ (i) If Γµ ((0, α ]) > 0, then Rd |x|α µ (dx) = ∞ ∫ ∫ (ii) Suppose that Γµ ((0, α ]) = Then, Rd |x|α µ (dx) < ∞ if and only if (α ,2) (β − α )−1Γµ (d β ) < ∞ Proof Since λβ in (1.6) satisfies λβ (S) = 1, we have ∫ |x|>1 |x|α νµ (dx) = ∫ (0,2) Γµ (d β ) ∫ ∞ rα −β −1 dr ∫ Since 1∞ rα −β −1 dr is infinite for β ≤ α and (β − α )−1 for β > α , our assertions follow ⊓ ⊔ Proposition 7.16 (i) Let µ ∈ L∞ and suppose that µ is not Gaussian (that is, νµ ̸= ∫ α Γ Then α 0) Let α0 be the infimum of the support of ∈ [0, 2) and d µ R |x| µ (dx) = ∫ α ∞ for α ∈ (α0 , 2) If α0 > 0, then Rd |x| µ (dx) < ∞ for α ∈ (0, α0 ) ∫ (α ,2) (ii) Let < α < There exists µ ∈ L∞ such that Rd |x|α µ (dx) = ∞ Proof Assertion (i) follows from Proposition 7.15 To see (ii), choose α ′ ∈ (α , 2), ⊓ ⊔ let Γ (d β ) = 1(α ,α ′ ) (β )d β , and use Proposition 7.15 (ii) Remark 7.17 The identity (7.5) expresses the iteration of Λ p,α for α ̸= The iteration of a stochastic integral mapping Φ f generates nested classes of their ranges The description of their intersection is an interesting problem See Maejima and Sato [27] and the references therein ⊓ ⊔ 7.2 L p,α , L0p,α , and Lep,α for α ∈ (−∞, 2) with fixed p Little is known about the one-parameter families {L p,α : α ∈ (−∞, 2)}, {L0p,α : α ∈ (−∞, 2)}, and {Lep,α : α ∈ (−∞, 2)} for fixed p Lemma 7.18 Let n be a positive integer If f (r) is monotone of order n on R, then, for any a > 0, e−ar f (r) is monotone of order n on R Proof This follows from Lemma 5.17, as e−ar is completely monotone on R ⊓ ⊔ Theorem 7.19 Let n be a positive integer Then, for −∞ < α < α ′ < 2, Ln,α Ln,α ′ , Ln, α Ln, α′ , e and Ln, α e Ln, α′ (7.22) e e e Proof Step Let us prove that Ln, α ⊃ Ln,α ′ Let µ ∈ Ln,α ′ Then νµ has radial ′ decomposition (λ (d ξ ), u−α −1 hξ (log u)du) with hξ (y) monotone of order n on R Let ′ hξ♭ (y) = e−(α −α )y hξ (y), ′ Then hξ♭ (y) is monotone of order n on R by the lemma above, and u−α −1 hξ (log u) = e u−α −1 h♭ξ (log u) Hence µ ∈ Ln, α 86 Ken-iti Sato ⊃ L0 If α < 1, then these follow Step Let us prove Ln,α ⊃ Ln,α ′ and Ln, α n,α ′ ∫ from Step Suppose α = and let µ ∈ Ln,α ′ = Ln,α ′ Then, Rd |x|µ (dx) < ∞ and ∫ L ) e 1, νµ ∈ R(Λn,1 Rd x µ (dx) = from Theorem 6.12 (ii) Since µ ∈ Ln,1 from Step ∫ L ) such that ν = Λ L ν We have Thus there is ν ∈ D(Λn,1 µ |x|>1 |x|ν (dx) < ∞ n,1 ∫ from Theorem 6.2 Since |x|>1 |x|νµ (dx) < ∞, we have ∫ ∫ ∞ ds |ln,1 (s)x|>1 |ln,1 (s)x|ν (dx) < ∞ Moreover, γµ = − ∫ |x|>1 xνµ (dx) = − ∫ ∫ ∞ ds |ln,1 (s)x|>1 ln,1 (s)xν (dx) (∫ )−1 ∫ Choose ρ ∈ ID such that νρ = ν , Aρ = 0∞ ln,1 (s)2 ds Aµ , and γρ = − |x|>1 x ν (dx) Then it follows from Proposition 3.18 that ρ ∈ D0 (Λn,1 ) and Λn,1 ρ = µ ⊂ L Similarly, if α > and if µ ∈ L ′ = L0 , then µ ∈ L Hence µ ∈ Ln,1 n,α = n,1 n,α n,α ′ Ln, α Step To show the strictness of the inclusion, let λ be a non-zero finite measure on S and let h(y) = (−y)n−1 1(−∞,0) (y), which is monotone of order n on R (Example 2.17 (a)) Then (λ (d ξ ), u−α −1 h(log u)du) is a radial decomposition of a L´evy mea∫ e sure ν , since 01 u1−α h(log u)du < ∞ Let µ ∈ ID with νµ = ν Then µ ∈ Ln, α but e µ ̸∈ Ln,α ′ , as is seen by an argument similar to the proof of Theorem 5.19 Indeed, we have ′ u−α −1 h(log u) = u−α −1 h♯ (log u) for ′ h♯ (y) = e(α −α )y (−y)n−1 1(−∞,0) (y), which is not monotone of any order on R from Proposition 2.13 (iii) Strictness of the first and second inclusions in (7.22) is obtained from that of the third ⊓ ⊔ Remark 7.20 Open question: Is (7.22) true for p ∈ R◦+ in place of n ? ⊓ ⊔ Acknowledgments The author thanks Makoto Maejima and V´ıctor P´erez-Abreu for their constant encouragement by proposing and exploring many problems related to the subject and for their valuable comments during preparation of this work, and Yohei Ueda for his helpful remarks for improvement of 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alternative approach to multiply self-decomposable probability measures on Banach spaces Probab Theory Relat Fields 72, 35–54 52 U RBANIK , K (1972) Slowly varying sequences of random variables Bull Acad Polonaise Sci S´er Sci Math Astronom Phys 20, 679–682 53 U RBANIK , K (1973) Limit laws for sequences of normed sums satisfying some stability conditions Multivariate Analysis–III (ed Krishnaiah, P R., Academic Press, New York), 225–237 Fractional integrals and extensions of selfdecomposability 89 54 WATANABE , T (2007) Asymptotic estimates of multi-dimensional stable densities and their applications Trans Amer Math Soc 359, 2851–2879 55 W IDDER , D V (1946) The Laplace Transform Princeton Univ Press, Princeton, NJ 56 W ILLIAMSON , R E (1956) Multiply monotone functions and their Laplace transforms Duke Math J 23, 189–207 57 Z INGER , A A (1965) On a class of limit distributions for normalized sums of independent random variables (in Russian) Teor Verojatnost i Primenen 10, 672–692 Index ϒ -transformation, 42, 57 Goldie-Steutel-Bondesson class, 10 absolutely continuous, 20, 55, 75 additive process, improper stochastic integral, 3, 32 absolutely definable, 32 definable, 32 essentially definable, 33 infinitely divisible distribution, class B, 10 class K∞,α , class K p,α , 9, 46 , 46 class K p, α e , 46 class K p, α class L, 2, class L∞,α , class L∞ , 5, 81 class L p,α , 9, 73 class L0p,α , 73 class Lep,α , 73 class T , 10 class U, 10 completely monotone, completely selfdecomposable, compound Poisson distribution, 55, 75 cumulant function, domain absolute, 33 essential, 33 of Φ Lf , 36 of stochastic integral mapping, 32 elementary Γ -variable, 10 elementary compound Poisson variable, 10 elementary mixed-exponential variable, 10 fractional integral, Gaussian covariance matrix, generalized Γ -convolutions, 11 Jurek class, 10 L´evy measure, L´evy process, L´evy–Khintchine triplet, 26 Lamperti transformation, locally finite, locally square-integrable, 31 location parameter, 26 log-normal distribution, 11 lower semi-continuous, 20 mapping Φ f , 8, 32 measurable family, 21 mixture of exponential distributions, 11 monotone of order p, 7, 12 multiply selfdecomposable, n times selfdecomposable, 4, 73 fractional times selfdecomposable, twice selfdecomposable, nondegenerate, 55 null array, of polar product type, 28 Ornstein–Uhlenbeck process driven by L´evy process, Ornstein–Uhlenbeck type process, Pareto distribution, 11 91 92 polar decomposition, 2, 27 radial decomposition, 2, 27 range absolute, 33 essential, 33 of Φ Lf , 36 of stochastic integral mapping, 33 Riemann–Liouville integral, selfdecomposable, selfsimilar, semigroup property, spherical decomposition, 28 Index stable distribution, 5, 63 strictly, 5, 63 stochastic area, 11 stochastic integral, 31 tempered stable distribution, 59 Thorin class, 10 transformation Φ Lf , 36 triplet, 26 vague convergence, 17, 80 weak mean, 29, 49 have weak mean absolutely, 30, 51 Contents Fractional integrals and extensions of selfdecomposability Ken-iti Sato Introduction 1.1 Characterizations of selfdecomposable distributions 1.2 Nested classes of multiply selfdecomposable distributions 1.3 Continuous-parameter extension of multiple selfdecomposability 1.4 Stable distributions and class L∞ 1.5 Fractional integrals 1.6 Classes K p,α and L p,α generated by stochastic integral mappings 1.7 Remarkable subclasses of ID Fractional integrals and monotonicity of order p > 2.1 Basic properties 2.2 One-to-one property 2.3 More properties and examples Preliminaries in probability theory 3.1 L´evy–Khintchine representation of infinitely divisible distributions 3.2 Radial and spherical decompositions of σ -finite measures on Rd 3.3 Weak mean of infinitely divisible distributions 3.4 Stochastic integral mappings of infinitely divisible distributions 3.5 Transformation of L´evy measures First two-parameter extension K p,α of the class L of selfdecomposable distributions 4.1 Φ f and Φ Lf for f = ϕα L 4.2 Φ¯ p,α and Φ¯ p, α L ¯ 4.3 Range of Φ p,α 2 5 10 11 11 15 18 26 26 27 29 31 36 37 37 40 44 93 94 Contents , and K e 4.4 Classes Kp,α , K p, α p,α One-parameter subfamilies of {K p,α } , and K e for p ∈ (0, ∞) with fixed α 5.1 K p,α , K p, α p,α , and K e for α ∈ (−∞, 2) with fixed p 5.2 K p,α , K p, α p,α Second two-parameter extension L p,α of the class L of selfdecomposable distributions L 6.1 Λ p,α and Λ p, α L 6.2 Range of Λ p,α 6.3 Classes L p,α , L0p,α , and Lep,α 6.4 Relation between K p,α and L p,α One-parameter subfamilies of {L p,α } 7.1 L p,α , L0p,α , and Lep,α for p ∈ (0, ∞) with fixed α 7.2 L p,α , L0p,α , and Lep,α for α ∈ (−∞, 2) with fixed p References 46 56 56 64 66 66 72 73 75 76 76 85 86 Index 91 [...]... is defined for all bounded Borel sets E on R+ and for all ρ ∈ ID, then f is locally square-integrable on R+ ([41]) ⊓ ⊔ Remark 3.24 General treatment (with random integrands in general) of improper stochastic integrals and stochastic integrals up to infinity from the semimartingale point of view is made by Cherny and Shiryaev [6] Stochastic integrals of nonrandom functions with respect to an infinitely... representation Many other choices of the integrand are found in the literature Kwapie´n and Woyczy´nski [17] and Rajput and Rosinski [31] use some form other than in (1.1) and (3.1) Maruyama [29] uses still another form Fractional integrals and extensions of selfdecomposability 27 3.2 Radial and spherical decompositions of σ -finite measures on Rd A measure ν (B), B ∈ B(Rd ), is called σ -finite if there is a Borel... class of L´evy measures of infinitely divisible distributions on Rd The words increase and decrease are used in the non-strict sense In Section 1.5 we defined the mappings I p and I+p for p > 0 and the notion of monotonicity of order p Let us begin with the following remarks (i) If f is monotone of order p > 0 on R, then the restriction of f to R◦+ is monotone of order p on R◦+ (ii) If f is monotone of. .. divisible random measure Λ (B) for B in a σ -ring of subsets of a general parameter space are studied by Rajput and Rosinski [31] The integrability condition suggests that our absolutely definable improper stochastic integral of a nonrandom function with respect to a L´evy process should be identical with the stochastic integral up to infinity of Cherny and Shiryaev [6] and with the stochastic integral of. .. the L´evy–Khintchine triplet of µ Each has its own advantage and disadvantage Weak convergence of a sequence of infinitely divisible distributions can be expressed by the corresponding triplets of the type (Aµ , νµ , γµ♯ ), but cannot by the triplets of the type (Aµ , νµ , γµ ) This is because the integrand in the integral term is continuous with respect to x in (3.1), but not continuous in (1.1) On... definition of Jurek similarly to the proof of Theorem F of [1] See Bodesson [5] and Steutel and van Harn [46] for examples and related classes Especially, many examples in T (R) are known To mention one of them, the distribution of L´evy’s stochastic area of the two-dimensional Brownian motion has density 1/(π cosh x) and belongs to T (R) with L´evy measure dx/(2 |x sinh x|) 2 Fractional integrals and. .. the mapping of Jurek [13] Noting (1.23), we see that T = K∞,0 , (1.30) B = K∞,−1 (1.31) Historically, the class of µ ∈ T (R) on the positive axis was introduced by Thorin [47, 48] in the naming of generalized Γ -convolutions (GGC), to show that Pareto and log-normal distributions are infinitely divisible The class of µ ∈ B(R) on the positive axis was introduced by Bondesson [4] in the naming of generalized... (1.1) On the other hand the formulas derived from (Aµ , νµ , γµ ) are often simpler than those derived from (Aµ , νµ , γµ♯ ) See the book [39] for details In [39] the author uses the symbol γ in the sense of γµ , but in the papers [40]–[44] in the sense of γµ♯ The γµ and γµ♯ are both called the location parameter of µ They depend on the choice of the integrand in the L´evy–Khintchine representation... (Φ f ) if and only if (3.29) and (3.30) are satisfied Proof If we ignore the statements related to γρ , γµt , and γµ and retain those related to γρ♯ , γµ♯t , and γµ♯ , this proposition is proved in Lemma 5.4 and Propositions 5.5– 5.6 of [41], Proposition 2.6 of [42], and Propositions 2.1–2.3 of [43] Let us give remarks concerning the statements related to γρ , γµt , and γµ An important point is that... the stochastic integral of a Λ -integrable function of Rajput and Rosinski [31] In our set-up, improper stochastic integrals in more general cases are studied in [41, 44] ⊓ ⊔ 3.5 Transformation of L´evy measures Let f (s) be a locally square-integrable nonrandom function on R+ Suggested by the equation (3.34), we introduce the transformation Φ Lf in the following way Definition 3.25 For ν ∈ ML = ML (Rd ... stochastic integral up to infinity of Cherny and Shiryaev [6] and with the stochastic integral of a Λ -integrable function of Rajput and Rosinski [31] In our set-up, improper stochastic integrals in. .. semimartingale point of view is made by Cherny and Shiryaev [6] Stochastic integrals of nonrandom functions with respect to an infinitely divisible random measure Λ (B) for B in a σ -ring of subsets of a... square-integrable on R+ ([41]) ⊓ ⊔ Remark 3.24 General treatment (with random integrands in general) of improper stochastic integrals and stochastic integrals up to infinity from the semimartingale