The random integral representation hypothesis revisited : new classes of s-selfdecomposable laws

For $\,0<\alpha\le \infty$, new subclasses $\,\mathcal{U}^{<\alpha>}$ of the class $\,\mathcal{U}$, of s-selfdecomposable probability measures, are studied. They are described by random integrals, by their characteristic functions and their L\'evy spectral measures. Also their relations with the classical L\'evy class $L$ of selfdecomposable distributions are investigated.

Limit distribution theory belongs to the core of probability and mathematical statistics. Often limit laws are described by analytical tools such as Fourier or Laplace transforms, but a more stochastic approach (e.g., like stochastic integration, stopping times, random functionals etc.), seems more natural for probability questions. Some illustrations of this paradigm are given in the last paragraph of this note. In a similar spirit, in Jurek (1985) on page 607 (and later repeated in Jurek (1988) on page 474), the following hypothesis was formulated: Each class of limit distributions, derived from sequences of independent random variables, is the image of some subset of ID (the infinitely divisible probability measures) by some mapping defined as a random integral.
Random integral representations, when they can be established, would provide descriptions of limiting laws via stochastic methods, i.e., as the probability distributions of the random integrals of form (a,b] h(t) dY (r(t)) = h(r −1 (s)) dY (s), (1) where h and r are deterministic functions, h : (a, b] → R, r : (a, b] → (0, ∞) and Y (s), 0 ≤ s < ∞, is a stochastic process with independent and stationary increments and cadlag (right continuous with left hand limits) paths; in short, we refer to Y as a Lévy process. In this note we provide new examples of classes of limit distributions for which the above hypothesis holds true. The main results here are Propositions 3, 4 and 5, and Corollaries 5, 6 and 7.

Introduction and notation.
Let E denotes a real separable Banach space, E ′ its conjugate space, < ·, · > the usual pairing between E and E ′ , and ||.|| the norm on E. The σfield of all Borel subsets of E is denoted by B, while B 0 denotes Borel subsets of E \ {0}. By P(E) we denote the (topological) semigroup of all Borel probability measures on E, with convolution " * " and the weak topology, in which convergence is denoted by "⇒". Similarly, by ID(E) we denote the topological convolution semigroup of all infinitely divisible probability measures, i.e., µ ∈ ID(E) iff ∀(natural k ≥ 2) ∃(µ k ∈ P(E)) µ = µ * k k .
Recall also here that ID(E) is a closed topological subsemigroup of P(E).
Finally on a Banach space E we define the transforms T r , for r > 0, as follows: T r x := rx, x ∈ E, and define L(ξ) as the probability distribution of an E-valued random variable ξ. A probability measure µ ∈ P(E) is said to be s-selfdecomposable on E, and we will write µ ∈ U(E), if there exists a sequence ρ n ∈ ID(E) such that ν n := T 1 n (ρ 1 * ρ 2 * ... * ρ n ) * 1/n ⇒ µ, as n → ∞. ( Since we begin with infinitely divisible measures ρ n we do not include the shifts δ xn in (1), and do not assume that the triangle system {T 1 n ρ * 1/n j : 1 ≤ j ≤ n; n ≥ 1} is uniformly infinitesimal, as is usually done in the general limiting distribution theory. Also let us note that our definition (2) is, in fact, the result of Theorem 2.5 in Jurek (1985). There s-selfdecomposability was defined in many different but equivalent forms. Finally, s-selfdecomposable distributions appeared in the context of an approximation of processes by their discretization; cf. Jacod, Jakubowski and Mémin (2001).
Originally the s-selfdecomposable distributions were introduced as limit distributions for sums of shrunken random variables in Jurek (1981). The 's' stands here for shrinking operation defined as follows: Also see the announcement in Jurek (1977). On the real line similar distributions, but not related to s-operation, were studied in O'Connor (1979).
In the present paper we will repeat the scheme (2) successively and will assume that ρ k are chosen from a previously obtained class of limit laws. Such an approach, for another scheme of limiting procedure was introduced by K. Urbanik (1973) and then continued by K. Sato, A. Kumar and B. M. Schreiber, N. Thu, with the most general setting, up to now, described in Jurek (1983), where there is also a list of related references.
For easy reference we collect below some of the known characterizations of the class U(E) of s-selfdecomposable probability measures and indicate only the main steps in the corresponding proofs. PROPOSITION 1. The following statements are equivalent: (iii) there exists a unique Lévy process Y such that µ = L( (0,1) t dY (t)) .
Sketch of proofs. Characterizations (i) and (ii) are equivalent by Theorem 2.5 and Corollary 2.3 in Jurek (1985). Equivalence of (ii) and (iii) follows from Theorem 1.1 and Theorem 1.2(a) in Jurek (1988), where one needs to take the constat β = 1 and the linear operator Q = I.
For our purposes we define random integrals by the formal formula of integration by parts: where the later integral is defined as a limit of the appropriate Rieman-Stieltjes partial sums. This "limited" approach to integration is sufficient for our purposes; cf. Jurek and Vervaat (1983) or Jurek and Mason (1993), Section 3.6. On the other hand, since Lévy processes are semi-martingales, the integrals (1) or the above, can be defined as the stochastic integrals as well.
COROLLARY 1. The class U of s-selfdecomposable probability measures is closed topological convolution subsemigroup of ID. Moreover, it also is closed under the convolution powers (i.e, for t > 0 and µ we have that µ ∈ U if and only if µ * t ∈ U) and the dilations T d , for d ∈ R ( i.e., µ ∈ U if and only if T d µ ∈ U).
Proof. Both algebraic properties follow from (ii) in Proposition 1 and the following identities (T d (ν * ρ)) * t = T d ν * t * T d ρ * t , for t > 0, d ∈ R, and ν, ρ ∈ ID. To show that U is closed in weak convergence topology we use again the factorization (ii) together with Theorem 1.
where Y ρ (·) is a Lévy process (i.e., a process with independent and stationary increments, starting from zero and with cadlag paths) such that L(Y ρ (1)) = ρ. We refer to Y (·) as the background driving Lévy process (in short, the BDLP) for the s-selfdecomposable measure J (ρ).
REMARK 1. The random integral mapping J is an isomorphism between the closed topological semigroups ID(E) and U(E); cf. Jurek (1985), Theorem 2.6.
Finally, letμ be the characteristic function (the Fourier transform) of a measure µ. Then for random integrals (1) we infer that L( when h is a deterministic function, r is an increasing (or monotone) time change in (0, ∞) and Y ρ (.) a Lévy process; cf. Lemma 1.2 in Jurek and Vervaat (1983) or Lemma 1.1 in Jurek (1985) or simply approximate the right-hand integral by Rieman-Stieltjes partial sums.
Our results are given in the generality of a Banach space E, however, below in many formulas we will skip the dependence on E.
2. m-times s-selfdecomposable probability measures. Let us put U <1> := U(E) and for m ≥ 2, let U <m> denotes the class of limiting measures in (2), when ρ k ∈ U <m−1> , for k = 1, 2, ... . As a convention we assume that U <0> := ID. Our first characterization is proved along the lines of the proofs of Theorem 1.1 and 1.2 in Jurek (1988), however one needs not to confuse the classes U β introduced there, with those of U <m> investigated here. Needed changes in arguments are explained as they are deemed.
PROPOSITION 2. For m = 1, 2, ..., the following are equivalent descriptions of m-times s-selfdecomposable probability measures: (iii) There exists a unique (in distribution) Lévy process Y ρ such that Proof. For m = 1, the above is just the Proposition 1. Now suppose that the proposition is proved for m. If µ ∈ U <m+1> then, by the definition (formula (1)), ρ k ∈ U <m> , for k = 1, 2, ... . For given 0 < c < 1, let us choose natural numbers m n such that 1 ≤ m n ≤ n and m n /n → c, as n → ∞. From (2) we have By Theorems 1.2 and 2.1 in Parthasarathy (1967), the second convolution factor in (5) converges, say to µ c , which must be in U <m> by Corollary 1. Thus we get the factorization (ii) for m + 1, i.e., (i) implies (ii).
If (ii) holds we have a family C := {µ c : 0 ≤ c ≤ 1} ⊂ U <m> , where µ 1 = δ 0 and µ 0 = µ, from which we construct sequence (ρ k ) as follows Using the factorization (ii) for c = (k − 1)/k, then applying to both sides the dilation T k and then raising to the (convolution) power k, gives the equality Hence which completes the proof that (ii) implies (i).
Since we have we infer that (iii) implies (ii). To prove the converse that (ii) implies (iii) we proceed as in Jurek (1988), page 482 (formula (3.1)) till page 484, taking β = 1 and Q = I (identity operator). Thus we construct process Z(t) with independent increments and cadlag paths such that L(Z(t)) = µ e −t ∈ U <m> . Because of Corollary 1 we conclude that has increments with probability distributions in U <m> . All in all we have proved (iii).
Proof. Part (a) follows from the characterization (ii) in Proposition 2. To prove that U <m> are closed we use Theorem 1.7.1 in Jurek and Mason (1993) or cf. Chapter 2 in Parthasarathy (1967).
Part(b). Since U ⊂ ID, therefore applying successively the random integral mapping J to both sides gives the inclusion U <m+1> ⊂ U <m> . For the second inclusion L k ⊂ U <k> , note that it is true for k = 1, cf. Corollary 4.1 in Jurek (1985). Assume it is true n, i.e., L n ⊂ U <n> and let µ ∈ L n+1 ⊂ L n . Then for any 0 < c < 1 there exits ν c ∈ L n such that because, by the induction assumption, ν c and µ are in L n . Consequently, by (ii), in Proposition 2, µ ∈ U <n+1> and this completes the proof.
Our next aim is to describe m-times s-selfdecomposability in terms parameters of infinitely divisible laws. Recall that each ID distribution µ is uniquely determined by a triple: a shift vector a ∈ E, a Gaussian covariance operator R, and a Lévy spectral measure M; we will write ρ = [a, R, M]. These are the parameters in the Lévy-Khintchine representation of the characteristic functionμ, namely Φ is called the Lévy exponent ofρ (cf. Araujo and Giné (1980), Section 3.6). Furthermore, by the Lévy spectral function of ρ we mean the function where D is a Borel subset of unit sphere S := {x : ||x|| = 1} and r > 0. Note that L M uniquely determines M.
Since the Lévy processes have infinitely divisible increments (from the class ID) and ID is a topologically closed convolution semigroup, and also closed under dilations T a (a multiplication of random variable by a scalar a), therefore the random integrals (a,b] h(t) dY (r(t)) have probability distributions in ID as well . If [a h,r , R h,r , M h,r ] denotes the triple corresponding to the probability distribution of the integral in question, and [a, R, M] denotes the one corresponding to the law of Y (1) then (4) and (7) give the following equation: and finally for the shift vector we have In order to get the second equality in (13) one needs to observe that 1 {t||x||≤1} (x) = 1 {0<t≤||x|| −1 } (t) or to change the order of integration. Thus Now we may characterize the m-times s-selfdecomposable distributions in terms of the triples in their Lévy-Khintchine formula.
Similarly from (8) and (11), and from the change of the order of integration, we get which proves (15). In order to prove the formula for the shift, first note that by (11) and by change of order of integration, we have for m = 1, 2, ... . Note that for m = 0 the above formula gives the second summand in (13). In terms of w m , (19) gives the recurrence relation where a <0> := a. Thus, if the formula for the shifts (16) holds for m, then the above gives that it also holds for m + 1, which completes the proof the proposition.
Let us recall that the functions are called the incomplete gamma functions. Simple calculations shows that Consequently, the formula (16) may be written as Let us introduce rescales of time in the interval (0, 1) as follows Note that τ α is the cumulative probability distribution function of the random variable g α := e −Gα , where G α is the gamma random variable with the probability density (Γ(α)) −1 x α−1 e −x for x > 0, and zero elsewhere. Hence , for s > 0, 0 < c < 1, (24) and (15) can rewritten as Now we can establish the random integral representation for the subclasses U <m> of s-selfdecomposable probability measures.
where g m = e −Gm and G m is the gamma random variable.
Proof. Use Proposition 4 together with the formula (4). Note that there are not restrictions on a shift vector and a Gaussian covariance operator R. Finally, for m=1 this is Theorem 2.9 in Jurek (1985). REMARK 2. Using the series representation of the exponential function and (24) we get Since the previous characterization, of m-times s-selfdecomposability, has only a restriction on the Lévy spectral measure, therefore we have a characterization of U <m> in terms of Lévy spectral functions.
for all sets D and all r > 0, or equivalently for all sets D and all r > 0.
Proof. In view of the Proposition 3 we have that M = G <m> for a unique Lévy spectral measure G, and (15) gives the first part of the corollary. Since the relation (18), in terms Lévy spectral functions, reads for j = 1, 2..., therefore the inductive argument proves the second part of the corollary.
COROLLARY 5. In order that a Lévy spectral measure G to be a Lévy spectral measure of an m-times s-selfdecomposable probability measure, it is necessary and sufficient that its Lévy spectral functions r → L G (D, r) are m-times differentiable, except at countable many points r, and the function L(D, r) = (A m (L G (D, ·)))(r) is a Lévy spectral function.
The operator A m is the m-time composition of the linear differential operator A, which is defined as follows for once differentiable real-valued functions h defined on (0, ∞).
Proof. If measures M ′ and M are related as in (12) then their corresponding spectral functions (tails) L M ′ and L M satisfy equality Hence L M ′ is at least once differentiable (except on a countable set) and  . If one assume that the formula (12) defines the mapping J on the measure M or its spectral function L M , then A may be viewed as its inverse mapping. Before the next characterization, of the class U <m> distributions, let us recall that a Lévy exponent is just the logarithm of an infinitely divisible Fourier transform; cf. formula (7). Let us note that, if Ψ is the Lévy exponent of ρ and Φ is that of J (ρ), then (3) and (4)  With these equalities and the recursive relation between classes U <m> we have The operator D m is the m-time composition of the following linear differential operator (Dg)(y) := g(y) + d(g(ty))/dt| t=1 , where g : E ′ → C is once differentiable in each direction y ∈ E ′ and t ∈ R.
Note that in a particular case one has (Dg)(y) := g(y) + y dg(y))/dy, when y ∈ E ′ = R and it differs from A in Corollary 5 only by a sign. REMARK 4. If ones defines J on Lévy exponents by (4) then the operator D can be viewed as its inverse, i.e., D = J −1 , on Lévy exponents on a Banach space.

PROPOSITION 5.
A probability measure µ = [a, R, M] is completely sselfdecomposable, i.e., µ ∈ U <∞> := ∞ m=1 U <m> if and only if there exists a unique bi-measure σ(·, ·) on S × (0, 2) such that where A · D := {x ∈ E : x/||x|| ∈ D, ||x|| ∈ A} and for each Borel D ⊂ S, σ(D, ·) is a finite Borel measure on the interval (0, 2) and for each Borel subset A ⊂ (ǫ, ∞) for some ǫ > 0, σ(·, A) is a finite Borel measure on the unit sphere S. Moreover, we have that Proof. If µ = [a, R, M] is completely s-selfdecomposable then by Proposition 3 or Corollary 4, for each m there exists a unique Lévy measure G such that or for all D and r > 0 On the other hand, using (28) the integral i.,e., Σ is the more explicit description of U <∞> . Further, let us recall that

By the formula (27), the last integral is equal
Similar integrability formulas hold for functions g k (x) := log k (1 + ||x||) and Lévy measures M. Recall that the integrability condition of g k appears in the random integral representation for the class L k .

Concluding remarks and two examples.
A). The classes U <m> were introduced by an inductive procedure and thus we have the natural index m. For a positive non-integer α one may proceeds as in Thu (1986) using the fractional calculus. However, we may utilize our random integral approach and define where Y ρ (·) is a Lévy process with L(Y ρ (1)) = ρ. Equivalently, we have cf. (14), (15) and for the shift vector (16) with (21),(22) and (24). Furthermore, for any continuous and bounded f on (0, ∞) and gamma random variables G α and G β we have B). In this subsection we consider only R-valued random variables or Borel measures on the real line. Because of the inclusion L ⊂ U each selfdecomposable distribution is an example of s-selfdecomposable one. On the other hand, by Proposition 3 in Iksanow, Jurek and Schreiber (2002), selfdecomposable distributions of random variables of the form X := ∞ k=1 a k η k , where η k 's are independent identically distributed Laplace (double exponential) random variables and k a 2 k < ∞, have the background driving probability measures ν ∈ U. Furthermore, by Proposition 3 in Jurek (2001) we have that In Jurek (1996) it was noticed that φ S (t) = t/(sinh t) ("S" stands for the hyperbolic 'sine') and φ C (t) := 1/(cosh t) ( "C" stands for the hyperbolic 'cosine') are the characteristic functions of random variables of the above series form X. Using (31) we conclude It might be worthy to mention here that φ S (t) · ψ S (t) is a characteristic function of a conditional Lévy's random area integral ; cf. Lévy (1951) or Yor (1992) and Jurek (2001). Similarly, (φ C (t) · ψ C (t)) 1/2 is a characteristic function of an integral functional of Brownian motion; cf. Wenocur (1986) and Jurek (2001), p. 248. Recently in Jurek and Yor (2002) the probability distributions corresponding to both ψ S and ψ C were expressed in terms of squared Bessel bridges. Also both functions viewed as the Laplace transform in t 2 /2 can be interpreted as the hitting time of 1 by the Bessel process starting from zero; cf. Yor (1997), p. 132. At present we are not aware of any stochastic representation for the analytic expressions in (32). Finally, it seems that the operators A m may be related to some Markov processes.