Relations Between the $s$-Selfdecomposable and Selfdecomposable Measures

Factorization property of generalized $s$-selfdecomposable measures and class $L^F$ distributions

“…Following Jurek-Vervaat (1983) or Jurek (1985) we recall the following Definition 2. For a measure ν ∈ ID log and a Lévy process Y ν let us define…”

“…Finally, let us note that the random integral representations for classes U f β (Corollary 1(a)) and L f (Corollary 3) provide more examples for the conjectured "meta-theorem" in The Conjecture on www.math.uni.wroc.p/∼zjjurek or see Jurek (1985) and (1988).…”

“…were characterized in terms of distributions of some random integrals; cf. Jurek (1985Jurek ( , 1988 , Jurek and Vervaat (1983), Jurek and Mason (1993), Jurek and Yor (2004), Iksanov, Jurek and Schreiber (2004) and recently Aoyama and Maejima (2007). More precisely, for each of those classes one integrates a fixed deterministic function with respect to a class of Lévy processes, with possibly a time scale change.…”

“…The terminology has its origin in the fact that distributions from the class U 1 ≡ U were called s-selfdecomposable distribution (the "s-", stands here for the shrinking operations that were used originally in the definition of U); cf. Jurek (1985), (1988) and references therein.…”

Remarks on the Selfdecomposability and New Examples

“…where in both cases Y is a Levy process such that F(l) = v. More about the class U one can find in Jurek (1985). [Let us add here that 1 and J are isomorphisms between the corresponding topological convolution semigroups; Theorems 2.6 and 3.6 in Jurek (1985)].…”