## Bernoulli

### Some classes of multivariate infinitely divisible distributions admitting stochastic integral representations

#### Abstract

The class of distributions on $\mathbb{R}$ 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 $T(\mathbb{R}^d)$ and $B(\mathbb{R}^d)$ . From the Lévy process $\{X_t^{(\mu)}\}$ on $\mathbb{R}^d$ with distribution μ at t=1, Υ(μ) is defined as the distribution of the stochastic integral $\int_0^1 \log(1/t)\d X_t^{(\mu)}$ . This mapping is a generalization of the mapping Υ introduced by Barndorff-Nielsen and Thorbjørnsen in one dimension. It is proved that $\Upsilon(ID(\mathbb{R}^d))=B(\mathbb{R}^d)$ and $\Upsilon(L(\mathbb{R}^d))=T(\mathbb{R}^d)$ , where $ID(\mathbb{R}^d)$ and $L(\mathbb{R}^d)$ are the classes of infinitely divisible distributions and of self-decomposable distributions on $\mathbb{R}^d$ , 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 $\{X_t^{(\mu)}\}$ are studied. Developments of these results in the context of the nested sequence $L_m(\mathbb{R}^d), m=0,1,\ldots,\infty$, $m =0,1,…,∞$ , 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

https://projecteuclid.org/euclid.bj/1141136646

Mathematical Reviews number (MathSciNet)
MR2202318

Zentralblatt MATH identifier
1102.60013

#### 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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