Convergence of Probability Measures. By P. Billingsley. Chichester, Sussex, Wiley, 1968. xii, 253 p. 9 1/4“. 117s.

Convergence of Probability Measures

Stable non-Gaussian random processes , by G. Samorodnitsky and M. S. Taqqu. Pp. 632. £49.50. 1994. ISBN 0-412-05171-0 (Chapman and Hall).

Preface to the revised edition Remarks on notation 1. Basic examples 2. Characterization and existence 3. Stable processes and their extensions 4. The Levy-Ito decomposition of sample functions 5. Distributional properties of Levy processes 6. Subordination and density transformation 7. Recurrence and transience 8. Potential theory for Levy processes 9. Wiener-Hopf factorizations 10. More distributional properties Supplement Solutions to exercises References and author index Subject index.

Lévy processes and infinitely divisible distributions

Most of the signal processing that we will study in this course involves local operations on a signal, namely transforming the signal by applying linear combinations of values in the neighborhood of each sample point. You are familiar with such operations from Calculus, namely, taking derivatives and you are also familiar with this from optics namely blurring a signal. We will be looking at sampled signals only. Let's start with a few basic examples. Local difference Suppose we have a 1D image and we take the local difference of intensities, DI(x) = 1 2 (I(x + 1) − I(x − 1)) which give a discrete approximation to a partial derivative. (We compute this for each x in the image.) What is the effect of such a transformation? One key idea is that such a derivative would be useful for marking positions where the intensity changes. Such a change is called an edge. It is important to detect edges in images because they often mark locations at which object properties change. These can include changes in illumination along a surface due to a shadow boundary, or a material (pigment) change, or a change in depth as when one object ends and another begins. The computational problem of finding intensity edges in images is called edge detection. We could look for positions at which DI(x) has a large negative or positive value. Large positive values indicate an edge that goes from low to high intensity, and large negative values indicate an edge that goes from high to low intensity. Example Suppose the image consists of a single (slightly sloped) edge:

http://ieeexplore.ieee.org/iel5/5/31307/01456462.pdf

Linear systems

/pdf/introduction-to-the-modern-theory-of-dynamical-systems-1eced70mum.pdf

Introduction to the Modern Theory of Dynamical Systems

We use a discrete-time analysis, giving necessary and sufficient conditions for the almost-sure convergence of ARCH(1) and GARCH(1,1) discrete-time models, to suggest an extension of the ARCH and GARCH concepts to continuous-time processes. Our ‘COGARCH’ (continuous-time GARCH) model, based on a single background driving Levy process, is different from, though related to, other continuous-time stochastic volatility models that have been proposed. The model generalises the essential features of discrete-time GARCH processes, and is amenable to further analysis, possessing useful Markovian and stationarity properties.

/pdf/a-continuous-time-garch-process-driven-by-a-levy-process-lq2q3jawc8.pdf

A continuous-time GARCH process driven by a Lévy process: stationarity and second-order behaviour

Exponential functionals of Levy processes appear as stationary distributions of generalized Ornstein–Uhlenbeck (GOU) processes. In this paper we obtain the infinitesimal generator of the GOU process and show that it is a Feller process. Further, we use these results to investigate properties of the mapping $$\Phi $$
 , which maps two independent Levy processes to their corresponding exponential functional, where one of the processes is assumed to be fixed. We show that in many cases this mapping is injective, and give the inverse mapping in terms of (Levy) characteristics. Also, continuity of $$\Phi $$
 is treated, and some results on its range are obtained.

/pdf/on-exponential-functionals-of-levy-processes-39q6adwiul.pdf

On Exponential Functionals of Lévy Processes

This paper collects some of the well known probabilistic properties of GARCH(p, q) processes. In particular, we address the question of strictly and of weakly stationary solutions. We further investigate moment conditions as well as the strong mixing property of GARCH processes. Some distribu- tional properties such as the tail behaviour and continuity properties of the stationary distribution are also included.

/pdf/stationarity-mixing-distributional-properties-and-moments-of-4gk0k9unv8.pdf

Stationarity, Mixing, Distributional Properties and Moments of GARCH(p, q)-Processes

https://www.uni-ulm.de/fileadmin/website_uni_ulm/mawi.inst.050/people/lindner/lindner-maller1.pdf

Lévy integrals and the stationarity of generalised Ornstein-Uhlenbeck processes

We show that the conjectured generalization of the Bourgain-Tzafriri restricted-invertibility theorem is equivalent to the conjecture of Feichtinger, stating that every bounded frame can be written as a finite union of Riesz basic sequences We prove that any bounded frame can at least be written as a finite union of linearly independent sequences We further show that the two conjectures are implied by the paving conjecture Finally, we show that Weyl-Heisenberg frames over rational lattices are finite unions of Riesz basic sequences

/pdf/frames-and-the-feichtinger-conjecture-v32fwnhd8u.pdf

Alexander Lindner

Papers

A continuous-time GARCH process driven by a Lévy process: stationarity and second-order behaviour

On Exponential Functionals of Lévy Processes

Stationarity, Mixing, Distributional Properties and Moments of GARCH(p, q)-Processes

Lévy integrals and the stationarity of generalised Ornstein-Uhlenbeck processes

Frames and the Feichtinger conjecture