An Application of Series Representations to Zero - One Laws for Infinitely Divisible Random Vectors

Abstract

In this paper we give an application of series representations of infinitely divisible random vectors to the zero-one laws for measurable subgroups. In Proposition 2 of Section 1 we provide a series representation of Poissonian-type random vectors suitable for this purpose. Then the zero-one dichotomy is a consequence of the observation that either all terms of the series representing an infinitely divisible measure μ are members of a subgroup G or infinitely many of them lie outside of G. In the first case the zero-one law follows from Hewitt-Savage zero-one law and in the second case the whole sum lies outside of G, so μ(G) = 0. The latter assertion is justified by a generalization of a theorem of P. Levy which we prove in Section 2. This provides a transparent probabilistic argument for the zero-one law due to A. Jannsen [2] (Section 3 of this paper). Jannsen’s result, which gives a complete answer to the zero-one dichotomy problem for infinitely divisible measures, relies on special topological and algebraic techniques which are not needed here, due to series representations originated by Ferguson - Klass [1] and LePage [4], and developed by the author [7].