## Electronic Journal of Probability

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

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

Takahiro Aoyama, Alexander Lindner, and Makoto Maejima

#### 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