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2010 A New Family of Mappings of Infinitely Divisible Distributions Related to the Goldie-Steutel-Bondesson Class
Takahiro Aoyama, Alexander Lindner, Makoto Maejima
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Electron. J. Probab. 15: 1119-1142 (2010). DOI: 10.1214/EJP.v15-791

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.

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Takahiro Aoyama. Alexander Lindner. Makoto Maejima. "A New Family of Mappings of Infinitely Divisible Distributions Related to the Goldie-Steutel-Bondesson Class." Electron. J. Probab. 15 1119 - 1142, 2010. https://doi.org/10.1214/EJP.v15-791

Information

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

zbMATH: 1225.60026
MathSciNet: MR2659759
Digital Object Identifier: 10.1214/EJP.v15-791

Subjects:
Primary: 60E07

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

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