We give asymptotic bounds for sample paths of discrete time infinitely divisible processes and prove the optimality of such bounds.

Self-similar processes are useful models for natural systems that exhibit scaling. Operator scaling allows a different scale factor in each coordinate. This paper develops practical methods for modeling and simulation. A simulation method is developed for operator scaling Lévy processes, based...

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...

A sufficient condition is given for processes admitting a series expansion with partially dependent components to have nonrandom oscillation. Included are infinitely divisible processes with no Gaussian component. This immediately gives information about the sample paths of such processes, e.g....

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 Lévy 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...

A tempered stable Lévy process combines both the [alpha]-stable and Gaussian trends. In a short time frame it is close to an [alpha]-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...

We derive spectral necessary and sufficient conditions for stationary symmetric stable processes to be metrically transitive and mixing. We then consider some important classes of stationary stable processes: Sub-Gaussian stationary processes and stationary stable processes with a harmonic...