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February 2006 Some classes of multivariate infinitely divisible distributions admitting stochastic integral representations
Ole E. Barndorff-Nielsen, Makoto Maejima, Ken-Iti Sato
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Bernoulli 12(1): 1-33 (February 2006).

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.

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Ole E. Barndorff-Nielsen. Makoto Maejima. Ken-Iti Sato. "Some classes of multivariate infinitely divisible distributions admitting stochastic integral representations." Bernoulli 12 (1) 1 - 33, February 2006.

Information

Published: February 2006
First available in Project Euclid: 28 February 2006

zbMATH: 1102.60013
MathSciNet: MR2202318

Rights: Copyright © 2006 Bernoulli Society for Mathematical Statistics and Probability

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Vol.12 • No. 1 • February 2006
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