Logarithm of Probability Measures and Infinite Divisibility of Sub-Stable RandomVectors

摘要

In the paper we discuss infinite divisibility of sub-stable finite dimensionalrandom vectors. Usually one must guess the corresponding Levy measure nu for given a infinitely divisible measure mu, which is rather hard. In some cases however it seems to be possible to get the measure nu as a limit of a sequence of signed measures derived from the measure mu. Obviously the convergence cannot be weak or normed, as is usually done in functional analysis, because a probability measure has variation 1 and this lies on the boundary of the corresponding ball of convergence. In the paper we give some partial results in this area applied to sub-stable random vectors. In view of these results it seems possible that every finite dimensional sub-stable distribution is of the form of generalized Poisson measure for some signed sigma-finite measure on R(sup n). (Copyright (c) 1993 by Faculty of Technical Mathmatics and Informatics, Delft, The Netherlands.)