Electronic Journal of Probability

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

Takahiro Aoyama, Alexander Lindner, and Makoto Maejima

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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.

Article information

Source
Electron. J. Probab., Volume 15 (2010), paper no. 35, 1119-1142.

Dates
Accepted: 7 July 2010
First available in Project Euclid: 1 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1464819820

Digital Object Identifier
doi:10.1214/EJP.v15-791

Mathematical Reviews number (MathSciNet)
MR2659759

Zentralblatt MATH identifier
1225.60026

Subjects
Primary: 60E07: Infinitely divisible distributions; stable distributions

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

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

Citation

Aoyama, Takahiro; Lindner, Alexander; Maejima, Makoto. A New Family of Mappings of Infinitely Divisible Distributions Related to the Goldie-Steutel-Bondesson Class. Electron. J. Probab. 15 (2010), paper no. 35, 1119--1142. doi:10.1214/EJP.v15-791. https://projecteuclid.org/euclid.ejp/1464819820


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