Bernoulli

Some classes of multivariate infinitely divisible distributions admitting stochastic integral representations

Ole E. Barndorff-Nielsen, Makoto Maejima, and Ken-Iti Sato

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Abstract

The class of distributions on generated by convolutions of Γ-distributions and the class generated by convolutions of mixtures of exponential distributions are generalized to higher dimensions and denoted by T ( d) and B ( d) . From the Lévy process { X t ( μ)} on d with distribution μ at t=1, Υ(μ) is defined as the distribution of the stochastic integral 0 1 log(1/t)dX t ( μ) . This mapping is a generalization of the mapping Υ introduced by Barndorff-Nielsen and Thorbjørnsen in one dimension. It is proved that ϒ (ID( d))=B( d) and ϒ (L( d))=T( d) , where ID ( d) and L ( d) are the classes of infinitely divisible distributions and of self-decomposable distributions on d , respectively. The relations with the mapping Φ from μ to the distribution at each time of the stationary process of Ornstein-Uhlenbeck type with background driving Lévy process { X t ( μ)} are studied. Developments of these results in the context of the nested sequence L m ( d) , m =0,1,, , are presented. Other applications and examples are given.

Article information

Source
Bernoulli, Volume 12, Number 1 (2006), 1-33.

Dates
First available in Project Euclid: 28 February 2006

Permanent link to this document
https://projecteuclid.org/euclid.bj/1141136646

Mathematical Reviews number (MathSciNet)
MR2202318

Zentralblatt MATH identifier
1102.60013

Keywords
Goldie-Steutel-Bondesson class infinite divisibility Lévy measure Lévy process self-decomposability stochastic integral Thorin class

Citation

Barndorff-Nielsen, Ole E.; Maejima, Makoto; Sato, Ken-Iti. Some classes of multivariate infinitely divisible distributions admitting stochastic integral representations. Bernoulli 12 (2006), no. 1, 1--33. https://projecteuclid.org/euclid.bj/1141136646


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