A new family of mappings of infinitely divisible distributions related to the Goldie-Steutel-Bondesson class
Takahiro Aoyama, Tokyo University of Science
Alexander Lindner, Technische Universitat Braunschweig
Makoto Maejima, Keio University
Abstract
Let {Xt(μ),t≥ 0} be a Lévy process on Rd whose distribution at time 1 is a
d-dimensional infinitely distribution μ.
It is known that the set of all infinitely divisible
distributions on Rd, each of which is represented by the law of a stochastic
integral
∫01log(1/t)dXt(μ)
for some infinitely divisible distribution on Rd, 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
∫01(log(1/t))1/αdXt(μ) for general
α> 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.
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