Embedding of Infinitely Divisible Probability Measures

Abstract

One of the most prominent subclasses of probability measures on a locally compact group is the class of infinitely divisible probability measures. Its rôle for the solution of the central limit problem is well-known from the classical theory. The most important step on the way to the central limit theorem is the embedding of an infinitely divisible measure into a continuous one-parameter convolution semigroup. Since the embedding theorem does not hold for any general locally compact group, the question arises as to what classes of groups yield the validity of an embedding theorem. Establishing these classes of groups will be the aim of the following chapter. First of all, root compact groups which admit an algebraic version of the embedding theorem are studied. The class of root compact groups enables us to describe the domain of validity for the theorem asserting the closedness of the infinitely divisible probability measures in the semigroup of all probability measures and for the fact that every infinitely divisible measure is submonogeneously embeddable. This particular form of algebraic embedding is also studied without the condition of root compactness. For this purpose it appears necessary to introduce a special class of infinitely divisible measures, the Poisson measures, whose various characterizations are presented in detailed fashion. A deeper analysis of the submonogeneous embedings for general lacally compact Abelian groups has been initiated by considerations of the roots of divisible measures on certain free groups.