Details

Weak Dependence: With Examples and Applications


Weak Dependence: With Examples and Applications


Lecture Notes in Statistics, Band 190

von: Jérome Dedecker, Paul Doukhan, Gabriel Lang, José Rafael Leon, Sana Louhichi, Clémentine Prieur

106,99 €

Verlag: Springer
Format: PDF
Veröffentl.: 29.07.2007
ISBN/EAN: 9780387699523
Sprache: englisch
Anzahl Seiten: 322

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Beschreibungen

Time series and random ?elds are main topics in modern statistical techniques. They are essential for applications where randomness plays an important role. Indeed, physical constraints mean that serious modelling cannot be done - ing only independent sequences. This is a real problem because asymptotic properties are not always known in this case. Thepresentworkisdevotedtoprovidingaframeworkforthecommonlyused time series. In order to validate the main statistics, one needs rigorous limit theorems. In the ?eld of probability theory, asymptotic behavior of sums may or may not be analogous to those of independent sequences. We are involved with this ?rst case in this book. Very sharp results have been proved for mixing processes, a notion int- duced by Murray Rosenblatt [166]. Extensive discussions of this topic may be found in his Dependence in Probability and Statistics (a monograph published by Birkhau ¨ser in 1986) and in Doukhan (1994) [61], and the sharpest results may be found in Rio (2000)[161]. However, a counterexample of a really simple non-mixing process was exhibited by Andrews (1984) [2]. The notion of weak dependence discussed here takes real account of the available models, which are discussed extensively. Our idea is that robustness of the limit theorems with respect to the model should be taken into account. In real applications, nobody may assert, for example, the existence of a density for the inputs in a certain model, while such assumptions are always needed when dealing with mixing concepts.
Weak dependence.- Models.- Tools for non causal cases.- Tools for causal cases.- Applications of strong laws of large numbers.- Central Limit theorem.- Donsker Principles.- Law of the iterated logarithm (LIL).- The Empirical process.- Functional estimation.- Spectral estimation.- Econometric applications and resampling.
<P>This monograph is aimed at developing Doukhan/Louhichi's (1999) idea to measure asymptotic independence of a random process. The authors propose various examples of models fitting such conditions such as stable Markov chains, dynamical systems or more complicated models, nonlinear, non-Markovian, and heteroskedastic models with infinite memory. Most of the commonly used stationary models fit their conditions. The simplicity of the conditions is also their strength.<BR>&nbsp;<BR>The main existing tools for an asymptotic theory are developed under weak dependence. They apply the theory to nonparametric statistics, spectral analysis, econometrics, and resampling. The level of generality makes those techniques quite robust with respect to the model. The limit theorems are sometimes sharp and always simple to apply.<BR>&nbsp;<BR>The theory (with proofs) is developed and the authors propose to fix the notation for future applications. A large number of research papers deals with the present ideas; the authors as well as numerous other investigators participated actively in the development of this theory. Several applications are still needed to develop a method of analysis for (nonlinear) times series and they provide here a strong basis for such studies.<BR>&nbsp;<BR>Jérôme Dedecker (associate professor Paris 6), Gabriel Lang (professor&nbsp;at<BR>Ecole Polytechnique, ENGREF Paris), Sana Louhichi (Paris 11, associate professor at&nbsp;Paris 2), and Clémentine Prieur (associate professor at INSA, Toulouse) are&nbsp;<BR>main contributors for the development of weak dependence. José Rafael León (Polar price, correspondent of the Bernoulli society for Latino-America) is professor at University Central of Venezuela and Paul Doukhan&nbsp; is professor at ENSAE (SAMOS-CES Paris 1 and Cergy Pontoise) and associate editor of <EM>Stochastic Processes and their Applications</EM>. His <EM>Mixing: Properties and Examples</EM> (Springer, 1994) is a main reference for theconcurrent notion of mixing.</P>
Make it simple to read and thus the mathematical level needed is as low as possible Aimed to fix the notions in the area in development May be considered as an introduction to weak dependence Propose models and tools for practitioners hence the sections devoted to examples are really extensive Some of the already developed applications are also quoted for completeness Includes supplementary material: sn.pub/extras
<P>This monograph is aimed at developing Doukhan/Louhichi's (1999) idea to measure asymptotic independence of a random process. The authors propose various examples of models fitting such conditions such as stable Markov chains, dynamical systems or more complicated models, nonlinear, non-Markovian, and heteroskedastic models with infinite memory. Most of the commonly used stationary models fit their conditions. The simplicity of the conditions is also their strength. The main tools for an asymptotic theory are developed under weak dependence. They apply the theory to nonparametric statistics, spectral analysis, econometrics, and resampling. The level of generality makes those techniques quite robust with respect to the model. The limit theorems are sometimes sharp and always simple to apply. The theory (with proofs) is developed and the authors propose to fix the notation for future applications. Several applications are still needed to develop a method of analysis for (nonlinear) times series and they provide here a strong basis for such studies.</P>

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