The Structure of Self-similar Stable Processes with Stationary Increments

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0102410Taqqu The focus of this research is on a special class of stochastic processes with the following three properties: stationary increments, self-similar and stable non-Gaussian probability laws (stable sssi processes, in short). Unlike the Gaussian case, there are infinitely many different stable sssi processes. This overwhelming variety may be regarded as a fundamental problem. One now has to understand how these processes are different or what it is that they have in common. However, non-Gaussianity also brings to the picture new tools that were unavailable in the Gaussian case. It has been known for quite some time now that non-Gaussian stable processes having some invariance property, like self-similarity or stationarity of the increments, can be associated with nonsingular flows. It is then based on some properties of these flows that one can describe the structure of the corresponding stable processes. The focus will be at first on an important subclass of stable sssi processes called self-similar mixed moving averages. The connection of self-similar mixed moving averages to nonsingular flows allows one to decompose them into separate, independent processes and then explore each part in the decomposition separately. The purpose of this research is to better understand a class of random processes that have characteristics that one encounters in many areas of applications. Examples of such processes include the limit of the so-called renewal reward processes applied in telecommunications and "random wavelet expansions" introduced in probabilistic modeling of images. These processes are fractal-like. They display scale invariance and tend to take often extreme values that deviate greatly from the mean. Their mathematical structure is complex. The goal of this research is to develop tools that can be used to analyze that structure