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A New Family of Mappings of Infinitely Divisible Distributions Related to the Goldie-Steutel-Bondesson Class

Dublin Core		PKP Metadata Items	Metadata for this Document
1.	Title	Title of document	A New Family of Mappings of Infinitely Divisible Distributions Related to the Goldie-Steutel-Bondesson Class
2.	Creator	Author's name, affiliation, country	Takahiro Aoyama; Tokyo University of Science; Japan
2.	Creator	Author's name, affiliation, country	Alexander Lindner; Technische Universitat Braunschweig; Germany
2.	Creator	Author's name, affiliation, country	Makoto Maejima; Keio University; Japan
3.	Subject	Discipline(s)
3.	Subject	Keyword(s)	infinitely divisible distribution; the Goldie-Steutel-Bondesson class; stochastic integral mapping; compound Poisson process; limit of the ranges of the iterated mappings
3.	Subject	Subject classification	60E07
4.	Description	Abstract	Let $\{X_t^\mu,t\geq0\}$ be a Lévy process on $\mathbb{R}^d$ whose distribution at time $1$ is a $d$-dimensional infinitely distribution $\mu$. It is known that the set of all infinitely divisible distributions on $\mathbb{R}^d$, each of which is represented by the law of a stochastic integral $\int_0^1\!\log(1/t)\,dX_t^\mu$ for some infinitely divisible distribution on $\mathbb{R}^d$, 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 $\int_0^1\!(\log(1/t))^{1/\alpha}\,dX_t^\mu$ 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.
5.	Publisher	Organizing agency, location
6.	Contributor	Sponsor(s)
7.	Date	(YYYY-MM-DD)	2010-07-07
8.	Type	Status & genre	Peer-reviewed Article
8.	Type	Type
9.	Format	File format	PDF
10.	Identifier	Uniform Resource Identifier	http://ejp.ejpecp.org/article/view/791
10.	Identifier	Digital Object Identifier	10.1214/EJP.v15-791
11.	Source	Journal/conference title; vol., no. (year)	Electronic Journal of Probability; Vol 15
12.	Language	English=en
14.	Coverage	Geo-spatial location, chronological period, research sample (gender, age, etc.)
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