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Models, inequalities, and limit theorems for stationary sequences. (English) Zbl 1032.62081

Doukhan, Paul (ed.) et al., Theory and applications of long-range dependence. Boston, MA: Birkhäuser. 43-100 (2003).
Summary: Recently, the notion of dependence in time series has received major attention in the research literature. In statistics, the two types of dependence that are usually considered are weak and strong dependence. Even though these types of dependence correspond to real phenomena, they have not yet received a satisfactory definition. A major objective of statistics is to build consistent tests by using limit theorems in distribution. Thus, a natural perspective is to classify the statistical model by means of related limit theorems. We restrict our attention to two types of statistics: those based on partial sums and those based on Kolmogorov-Smirnov statistics.
After listing some classes of stationary times series models, which are commonly used in statistics as well as in econometrics and finance, we shall recall some classes of weak dependence conditions and present the dependence properties of the previous models in terms of these dependence conditions.
Limit theorems associated to Donsker and Kolmogorov statistics yield a first classification between weak and strong dependence situations, according to two kinds of considerations. Under weak dependence, the limit theorems have normalization of order \(\sqrt n\), and the limiting processes are rather irregular (typically, Brownian motion), whereas strong dependence involves higher order normalizations and very regular limiting processes (such as products of deterministic functions and random variables). This classification is a first step towards a global time series analysis. A second step would be to generate statistics suitable to test the type of dependence in a stationary time series.
For the entire collection see [Zbl 1005.00017].

MSC:

62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
60F05 Central limit and other weak theorems
60G18 Self-similar stochastic processes
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