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A random variable X on R(+) is said to be self-decomposable, if for all c (0, 1) there exists a random variable X(c) on R(+) such that X=(d) cX + X(c). It is said to be stable if it is self dccomposable and X(c)=(d) (1-c)X', where X and X' are identically and independently distributed. The notions of stability and self decomposability for infinitely divisible random variables are generalised to abelian semigroups (S,+) with S having an identical involution by using characteristic functions. The generalised definitions involve semigroups of scaling operators T. These operators can be interpreted in a slightly different context as generalised continuous time branching processes (with immigration). The underlying importance of the generator of the semigroups T in the characterisation of stability and self decomposability is stressed.

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Authors: HANSEN B, JRC Ispra (IT)
Bibliographic Reference: Article : Statistica Neerlandica (1995)
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