Random Integral Representations for Classes of Limit Distributions Similar to Levy Class L ₀. III

Summary

For -1 < β < 0, subclasses, U _β, of the Lévy class L _o of selfdecomposable measures on Banach spaces, are examined. They are defined as limit distributions in certain summation schemes. The main result in Section 1 (Theorem 1.2) shows that each measure μi from Uβ is a convolution of a strictly stable measure with exponent (-β) and a probability distribution of random integral $\int_{(0,1)}tdY(t^{\beta})$, where Y is a Lévy process with finite (-β)th moment. The situation differs essentially from that with positive U _β Theorem 2.2 (in Section 2) shows that the natural mapping

$$\pounds (Y(1))\to \pounds(\int_{(0,1)}tdY(t^{\beta}))$$

is a homeomorphism, which immediately gives so called “generators” for the classU _β In addition, potential applications of measures from Uβ in the Ising model for ferromagnetism are indicated. Finally, Remark 3.B shows that the classes U _β constitute a filtration of the semigroup ID of all infinitely divisible measures.