Canonical Representations of Convolution Semigroups

Abstract

In the preceding chapter the problem of embedding an infinitely divisible probability measure in a continuous convolution semigroup was discussed in great detail. The next step towards a solution of the central limit theorem is the canonical representation of all continuous convolution semigroups in the sense of a Lévy-Khintchine formula. As in the classical theory this formula will enable us in subsequent chapters to characterize special classes of infinitely divisible measures and to illuminate the rôle of the Gauss measures. A systematic presentation of the theory of canonical representations of continuous convolution semigroups of probability measures on a locally compact group G requires some knowledge of the theory of positive semigroups and their infinitesimal generators or generating functionals.