Fourier transforms of measures from the classes [beta]' -2 < [beta] <= -1

Subclasses [beta](E), -2 < [beta] <= -1, of the Lévy class L of self-decomposable measures on a Banach space E are examined. They are closed convolution subsemigroups of the semigroup of infinitely divisible probability distributions on E, defined as limit distributions of some prescribed schemes of summation. Each element of [beta](E) belongs to the domain of normal attraction of a stable measure with exponent -[beta]. Their Fourier transforms are characterized. The symmetric measures in [beta](E) are shown to decompose uniquely into the convolution product of a symmetric stable measure with exponent -[beta] and the probability distribution of a random integral of the form [integral operator](0,1) t dY(t[beta]), where Y is a Lévy process with paths in the Skorohod space DE[0, [infinity]) and Y(1) has finite (-[beta])-moment. Topological and algebraic properties of the random-integral mapping [beta]: (Y(1)) --> [[integral operator](0,1) t dY(t[beta])] are investigated when E is a Hilbert space. As an application of the fact that [beta] is a continuous isomorphism, generators for [beta] are found as the images of compound Poisson distributions. Finally, the connection between the distributions [beta] and thermodynamic limits in the Ising model with zero external field is pointed out.