### Abstract

Let (X^{(μ)}_{t} ≥ 0) be a Levy process on ℝ^{d} whose distribution at time 1 is a d-dimensional infinitely distribution μ. It is known that the set of all infinitely divisible distributions on ℝ^{d}, each of which is represented by the law of a stochastic integral ∫^{1}_{0} log 1/t dX^{(μ)}_{t} for some infinitely divisible distribution on 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 ∫^{1}_{0} (log 1/t)^{1/α} dX^{(μ)}_{t} 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.

Original language | English |
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Pages (from-to) | 1119-1142 |

Number of pages | 24 |

Journal | Electronic Journal of Probability |

Volume | 15 |

DOIs | |

Publication status | Published - Jan 1 2010 |

Externally published | Yes |

### Keywords

- Compound Poisson process
- Infinitely divisible distribution
- Limit of the ranges of the iterated mappings
- Stochastic integral mapping
- The Goldie-Steutel-Bondesson class

### ASJC Scopus subject areas

- Statistics and Probability
- Statistics, Probability and Uncertainty

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## Cite this

*Electronic Journal of Probability*,

*15*, 1119-1142. https://doi.org/10.1214/EJP.v15-791