The theory of sparse stochastic processes offers a broad class of statistical models to study signals. In this framework, signals are represented as realizations of random processes that are solution of linear stochastic differential equations driven by white L 'evy noises. Among these processes, generalized Poisson processes based on compound-Poisson noises admit an interpretation as random L-splines with random knots and weights. We demonstrate that every generalized L 'evy process-from Gaussian to sparse-can be understood as the limit in law of a sequence of generalized Poisson processes. This enables a new conceptual understanding of sparse processes and suggests simple algorithms for the numerical generation of such objects.
Properties and examples of continuous-time ARMA (CARMA) processes driven by Levy processes are examined. By allowing Levy processes to replace Brownian motion in the definition of a Gaussian CARMA process, we obtain a much richer class of possibly heavy-tailed continuous-time stationary processes with many potential applications in finance, where such heavy tails are frequently observed in practice. If the Levy process has finite second moments, the correlation structure of the CARMA process is the same as that of a corresponding Gaussian CARMA process. In this paper we make use of the properties of general Levy processes to investigate CARMA processes driven by Levy processes W(t) without the restriction to finite second moments. We assume only that W (1) has finite r-th absolute moment for some strictly positive r. The processes so obtained include CARMA processes with marginal symmetric stable distributions.   Introduction.- Asymptotics.- Preliminaries of Levy Processes.- Student-Levy Processes.- Student OU-type Processes.- Student Diffusion Processes.- Miscellanea.- Bessel Functions.- References.- Index.     A retroreflective film comprising a lamination of a first base layer having vacuum met allized hemispherical depressions on one surface thereof and a second layer having substantially hemispherical projections from the surface thereof formed from an optically transparent film wherein the first and second layers are arranged so that the hemispherical depressions of the base layer and the substantially hemispherical projections of the intermediate layer are concentrically arranged and the radius of the hemispherical depressions of the base layer is greater than the radius of the substantially hemispherical projections of the second layer. A laminates construction with matched opposing substantially hemispherical projections on the second layer is preferably provided with a third optically clear overlay film.   1. Introduction 2. Roadmap to the book 3. Mathematical context and background 4. Continuous-domain innovation models 5. Operators and their inverses 6. Splines and wavelets 7. Sparse stochastic processes 8. Sparse representations 9. Infinite divisibility and transform-domain statistics 10. Recovery of sparse signals 11. Wavelet-domain methods 12. Conclusion Appendix A. Singular integrals Appendix B. Positive definiteness Appendix C. Special functions.  It is well documented that natural images are compressible in wavelet bases and tend to exhibit fractal properties. In this paper, we investigate statistical models that mimic these behaviors. We then use our models to make predictions on the statistics of the wavelet coefficients. Following an innovation modeling approach, we identify a general class of finite-variance self-similar sparse random processes. We first prove that spatially dilated versions of self-similar sparse processes are asymptotically Gaussian as the dilation factor increases. Based on this fundamental result, we show that the coarse-scale wavelet coefficients of these processes are also asymptotically Gaussian, provided the wavelet has enough vanishing moments. Moreover, we quantify the degree of Gaussianity by deriving the theoretical evolution of the kurtosis of the wavelet coefficients across scales. Finally, we apply our analysis to one- and two-dimensional signals, including natural images, and show that the wavelet coefficients ... A generalization of fractional Brownian motion (fBm) of parameter H in ]0, 1[ is proposed. More precisely, this work leads to nth-order fBm (n-fBm) of H parameter in ]n-1, n[, where n is any strictly positive integer. They include fBm for the special case n=1. Properties of these new processes are investigated. Their covariance function are given, and it is shown that they are self similar. In addition, their spectral shape is assessed as 1 f sup spl alpha with spl alpha belonging to ]1; + spl infin [, providing a larger framework than classical fBm. Special interest is given to their nth-order stationary increments, which extend fractional Gaussian noises. The covariance function and power spectral densities are calculated. The properties and signal processing tasks such as a Cholesky-type synthesis technique and a maximum likelihood estimation method of the H parameter are presented. The results show that the estimator is efficient (unbiased and reaches the Cramer-Rao lower bound) for a large majority of tested values.