A new family of mappings of infinitely divisible distributions related to the Goldie-Steutel-Bondesson class

作者：Aoyama Takahiro^*; Lindner Alexander; Maejima Makoto

来源：Electronic Journal of Probability, 2010, 15: 1119-1142.

• infinitely divisible distribution
• the Goldie-Steutel-Bondesson class
• stochastic integral mapping
• compound Poisson process
• limit of the ranges of the iterated mappings

Let {X(t)((mu)), t >= 0} be a Levy process on R(d) whose distribution at time 1 is a d-dimensional infinitely distribution mu. It is known that the set of all infinitely divisible distributions on R(d), each of which is represented by the law of a stochastic integral integral(1)(0) log1/tdX(t)((mu)) for some infinitely divisible distribution on R(d), coincides with the Goldie-Steutel-Bondesson class, which, in one dimension, is the smallest class that contains all mixtures of exponential distributions and is closed under convolution and weak convergence. The purpose of this paper is to study the class of infinitely divisible distributions which are represented as the law of integral(1)(0) (log1/t)(1/alpha) dX(t)((mu)) for general alpha > 0. These stochastic integrals define a new family of mappings of infinitely divisible distributions. We first study properties of these mappings and their ranges. Then we characterize some subclasses of the range by stochastic integrals with respect to some compound Poisson processes. Finally, we investigate the limit of the ranges of the iterated mappings.

• 出版日期2010-7-7