Stability and self‐decomposability of semi‐group valued random variables

B. G. Hansen

Statistica Neerlandica, 1996, vol. 50, issue 2, 295-305

Abstract: A random variable X on IR+ is said to be self‐decomposable, dif for all c∈ (0, 1) there exists a random variable Xc on IR+ such that X=dcX+Xc. It is said to be stable if it is self‐decomposable and Xc=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 semi‐groups (S, +) with S having an identical involution, by using characteristic functions. The generalised definitions involve semi‐groups of scaling operators T. There 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 semi‐groups T in the characterisation of stability and self‐decomposability is stressed.

Date: 1996
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