Jan Rosiński

Spectral representations of infinitely divisible processes

SummaryThe spectral representations for arbitrary discrete parameter infinitely divisible processes as well as for (centered) continuous parameter infinitely divisible processes, which are separable in probability, are obtained. The main tools used for the proofs are (i) a “polar-factorization” of an arbitrary Lévy measure on a separable Hilbert space, and (ii) the Wiener-type stochastic integrals of non-random functions relative to arbitrary “infinitely divisible noise”.

Series Representations of Lévy Processes from the Perspective of Point Processes

Several methods of generating series representations of a Levy process are presented under a unified approach and a new rejection method is introduced in this context. The connection of such representations with the Levy-Ito integral representation is precisely established. Four series representations of a gamma process are given as illustrations of these methods.

On A Class of Infinitely Divisible Processes Represented as Mixtures of Gaussian Processes

Variance mixtures of the normal distribution with infinitely divisible mixing measures and a class G of stochastic processes, which naturally arises from such distributions, are studied.

Gaussian approximation of multivariate Levy processes with applications to simulation of tempered and operator stable processes

Problem of simulation of multivariate Levy processes is investigated. The method based on shot noise series expansions of such processes combined with Gaussian approximation of the remainder is established in full generality. Formulas that can be used for simulation of tempered stable, operator stable and other multivariate processes are obtained.

On path properties of certain infinitely divisible processes

Let {X(t): t [set membership, variant] T} be a stochastic process equal in distribution to {[integral operator]sf(t, s)[Lambda](ds): t [set membership, variant] T}, where [Lambda]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 [set membership, variant] S, are inherited by the sample paths of X, provided X has no Gaussian component. The analogous statement for Gaussian processes is false. As a main tool, LePage-type series representation is fully developed for symmetric stochastic integral processes and this may be of independent interest.

Simple conditions for mixing of infinitely divisible processes

Let (Xt)t[epsilon]T be a real-valued, stationary, infinitely divisible stochastic process. We show that (Xt)t[epsilon]T is mixing if and only if Eei(Xt - X0) --> EeiX02, provided the Levy measure of X0 has no atoms in 2[pi]Z. We also show that if (Xt)t[epsilon]T is given by a stochastic integral with respect to an infinitely divisible measure then the mixing of (Xt)t[epsilon]T is equivalent to the essential disjointness of the supports of the representing functions.

The Equivalence of Ergodicity and Weak Mixing for Infinitely Divisible Processes

The equivalence of ergodicity and weak mixing for general infinitely divisible processes is proven. Using this result and [9], simple conditions for ergodicity of infinitely divisible processes are derived. The notion of codifference for infinitely divisible processes is investigated, it plays the crucial role in the proofs but it may be also of independent interest.

Asymptotic independence of multiple Wiener-Itô integrals and the resulting limit laws

We characterize the asymptotic independence between blocks consisting of multiple Wiener-It\^{o} 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-It\^{o} integrals, that involve Gaussian and non Gaussian limits. We give applications to the study of the asymptotic behavior of functions of short and long-range dependent stationary Gaussian time series and establish the asymptotic independence for discrete non-Gaussian chaoses.

THE CLASS OF T Y P E G DISTRIBUTIONS ON Rd AND RELATED SUBCLASSES OF INFINITELY DIVISIBLE DISTRIBUTIONS

Classes of infinitely divisible distributions obtained by iteration of Gaussian randomization of Levy measures are introduced and studied. Their relation to Urbanik-Sato nested classes of selfdecomposable distributions is also established.

Spectral representation and structure of self-similar processes

In this paper we establish a spectral representation of any symmetric stable self-similar process in terms of multiplicative flows and cocycles. Applying the Lamperti transformation we obtain a unique decomposition of a symmetric stable self-similar process into three independent parts: mixed fractional motion, harmonizable and evanescent.