Stochastic Limit Theory

Nowadays it is becoming clearer and clearer that, in the description of natural phenomena, the triadic scheme - microseopie, mesoscopic, macroscopic - is only a rough approximation and that there are many levels of description, probably an infinite hierarchy, in which the specific properties of a given level express some kind of cumulative or collective behaviour of properties or sys­ tems corresponding to the lower levels. One of the most interesting challenges for contemporary natural sciences is the comprehension of the connections among these different levels of description of reality and the deduction of the laws of higher levels in this hierarchy from basic laws corresponding to lower levels. Since these cumulative or collective phenomena are, typically, nonlin­ ear effects, the transition from this general program to concrete scientific achievements requires the developement of techniques which allow physical information to be extracted from nonlinear quantum systems. Explicitly in­ tegrable examples of such systems are rare, and the most interesting physical phenomena are not captured by them. Even in the case of linear systems the fact that an explicit solution is formally available is often useless, since it is impossible to interpret interesting physical phenomena from it.

Authors: Buraczewski, Dariusz, Damek, Ewa, Mikosch, Thomas
Covers fields which are not available in book form and are spread over the literature
Provides an accessible introduction to a complicated stochastic model
A readable overview of one of the most complicated topics on applied probability theory

This book presents essential tools for modelling non-linear time series. The first part of the book describes the main standard tools of probability and statistics that directly apply to the time series context to obtain a wide range of modelling possibilities. Functional estimation and bootstrap are discussed, and stationarity is reviewed. The second part describes a number of tools from Gaussian chaos and proposes a tour of linear time series models.