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- Jan Rosinski
- 2004

A tempered stable Lévy process combines both the α–stable and Gaussian trends. In a short time frame it is close to an α–stable process while in a long time frame it approximates a Brownian motion. In this paper we consider a general and robust class of multivariate tempered stable distributions and establish their identifiable parametrization. We prove… (More)

- Jan Rosinski
- 2007

Several methods of generating series representations of a Lévy process are presented under a unified approach and a new rejection method is introduced in this context. The connection of such representations with the Lévy–Itô integral representation is precisely established. Four series representations of a gamma process are given as illustrations of these… (More)

A generalized shot noise in Banach spaces is defined as the a.s. limit of certain centered sums of dependent random vectors; and, a necessary and sufficient condition for its existence is given, As an immediate application, the LePage-type series representations of infinitely divisible random vectors are 'V obtained. AMS (1980) Subject Classifications:… (More)

- Serge Cohen, Jan Rosinski
- 2006

The problem of simulation of multivariate Lévy processes is investigated. A method based on generalized shot noise series representations of Lévy processes combined with Gaussian approximation of the remainder is established in full generality. This method is applied to multivariate stable and tempered stable processes and formulas for their approximate… (More)

Sufficient conditions for boundedness and continuity are obtained for stochastically continuous infinitely divisible processes, without Gaussian component, {Y (t), t ∈ T}, where T is a compact metric space or pseudo-metric space. Such processes have a version given by Y (t) = X(t)+ b(t), t ∈ T where b is a deterministic drift function and

In this paper, we consider certain σ-finite measures which can be interpreted as the output of a linear filter. We assume that these measures have regularly varying tails and study whether the input to the linear filter must have regularly varying tails as well. This turns out to be related to the presence of a particular cancellation property in σ-finite… (More)

- Jan Rosinski, Kazimierz Urbanik
- 2006

Minimal integral representations are defined for general stochastic processes and completely characterized for stable processes (symmetric and nonsymmetric). In the stable case, minimal representations are described by rigid subsets of the L-spaces which are investigated here in detail. Exploiting this relationship, various tests for the minimality of… (More)

Let {X(t): teT} be a stochastic process equal in distribution to {fsf(ts)A(ds): teT}, where A is a symmetric independently scattered random measure and f is a suitable deterministic function. It is shown that various * properties of the sections f(*.s). s 6 S. are inherited by the sample paths of 7' X, provided X has no Gaussian component. The analogous… (More)

We characterize the asymptotic independence between blocks consisting of multiple Wiener-Itô integrals. As a consequence of this characterization, we derive the celebrated fourth moment theorem of Nualart and Peccati, its multidimensional extension, and other related results on the multivariate convergence of multiple Wiener-Itô integrals, that involve… (More)

Abstract. A new class of type G selfdecomposable distributions is introduced and characterized in terms of Lévy integrals. In dimension one, this class is a strict subclass of selfdecomposable variance mixtures of normal distributions. The relation to several other known classes of infinitely divisible distributions is established.