A new family of mappings of infinitely divisible distributions related to the Goldie-Steutel-Bondesson class

Takahiro Aoyama, Alexander Lindner, Makoto Maejima

Research output: Contribution to journalArticle

6 Citations (Scopus)

Abstract

Let {Xt (μ), t ≥ 0} be a Lévy process on Rd whose distribution at time 1 is a d-dimensional infinitely distribution μ. It is known that the set of all infinitely divisible distributions on Rd, each of which is represented by the law of a stochastic integral, for some infinitely divisible distribution on Rd, 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, 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 languageEnglish
Pages (from-to)1119-1142
Number of pages24
JournalElectronic Journal of Probability
Volume15
Publication statusPublished - 2010
Externally publishedYes

Fingerprint

Infinitely Divisible Distribution
Stochastic Integral
Range of data
Compound Poisson Process
Exponential distribution
Weak Convergence
One Dimension
Convolution
Closed
Family
Class
Integral

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

Cite this

A new family of mappings of infinitely divisible distributions related to the Goldie-Steutel-Bondesson class. / Aoyama, Takahiro; Lindner, Alexander; Maejima, Makoto.

In: Electronic Journal of Probability, Vol. 15, 2010, p. 1119-1142.

Research output: Contribution to journalArticle

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