2018 | Buch

# Stochastic Models for Time Series

verfasst von: Prof. Paul Doukhan

Verlag: Springer International Publishing

Buchreihe : Mathématiques et Applications

2018 | Buch

verfasst von: Prof. Paul Doukhan

Verlag: Springer International Publishing

Buchreihe : Mathématiques et Applications

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. It goes on to address nonlinearity from polynomial or chaotic models for which explicit expansions are available, then turns to Markov and non-Markov linear models and discusses Bernoulli shifts time series models. Finally, the volume focuses on the limit theory, starting with the ergodic theorem, which is seen as the first step for statistics of time series. It defines the distributional range to obtain generic tools for limit theory under long or short-range dependences (LRD/SRD) and explains examples of LRD behaviours. More general techniques (central limit theorems) are described under SRD; mixing and weak dependence are also reviewed. In closing, it describes moment techniques together with their relations to cumulant sums as well as an application to kernel type estimation.The appendix reviews basic probability theory facts and discusses useful laws stemming from the Gaussian laws as well as the basic principles of probability, and is completed by R-scripts used for the figures. Richly illustrated with examples and simulations, the book is recommended for advanced master courses for mathematicians just entering the field of time series, and statisticians who want more mathematical insights into the background of non-linear time series.

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Abstract

The present chapter deals with the standard notion of stochastic independence. This is a crucial concept, since this monograph aims to understand how to weaken it, in order to define asymptotic independence. We discuss in detail the limits of this idea through various examples and counter-examples.

Abstract

This chapter describes a simple Gaussian limit theory; namely we restate simple central limit theorems together with applications and moment/exponential inequalities for partial sums behaving asymptotically as Gaussian random variables. A relevant reference for the whole chapter is Petrov (Limit theorems of probability theory. Sequences of independent random variables, Oxford University Press, Oxford, 1975), results without a precise reference should be found in this reference, and the others are in Hall and Heyde (Martingale limit theory and its application, Academic Press, London, 1980). Topics related to empirical processes are covered by van der Vaart and Wellner (Weak convergence and empirical processes, Springer, New York, 1998) and Rosenblatt (Stochastic curve estimation. NSF-CBMS regional conference series in probability and statistics, 1991).

Abstract

Many statistical procedures are derived from probabilistic inequalities and results; such procedures may need more precise bounds as this is proved in the present chapter for the independent case. Basic notations are those from Appendix B.1. Developments may be found in van der Vaart (Asymptotic statistics, Cambridge University Press, Cambridge, 1998) and those related with functional estimation may be found in the monograph (Rosenblatt, Stochastic curve estimation, NSF-CBMS regional conference series in probability and statistics, vol 3, 1991). We begin the chapter with applications of the moment inequalities in Lemma 2.2.1 which are useful for empirical procedures. Then we describe empirical estimators, contrast estimators and non-parametric estimators. The developments do not reflect the relative interest of the topics but are rather considered with respect to possible developments under dependence conditions hereafter.

Abstract

Some bases for the theory of time series are given below. The chapter deals with the widely used assumption of stationarity which yields a simpler theory for time series. This concept is widely considered in Rosenblatt (Stationary processes and random fields, Birkhäuser, Boston, 1985) and in Brockwell and Davis (Time series: theory and methods, 2nd edn., Springer series in statistics, Springer, New York, 1991). The latter reference is more involved with linear time series.

Abstract

We consider stationary sequences generated through independent identically distributed \((\xi _n)_{n\in \mathbb {Z}}\). A reference is Brockwell and Davis (Time series: theory and methods. Springer, New York, 1991). Such models are natural in signal theory since they appear through linear filtering of a white noise. The usual setting is that \((\xi _n)_{n\in \mathbb {Z}}\) is only a \(\mathbb {L}^2 \)-stationary white noise sequence and not an independent identically distributed sequence.

Abstract

This chapter aims at describing stationary sequences generated from independent identically distributed samples \((\xi _n)_{n\in \mathbb {Z}}\). Most of the material in this chapter is specific to this monograph so that we do not provide a global reference. However Rosenblatt (Stationary processes and random fields. Birkhäuser, Boston, 1985) performs an excellent approach to modelling. Generalized linear models are presented in Kedem and Fokianos (Regression models for time series analysis. Wiley, Hoboken, 2002). The Markov case has drawn much attention, see Duflo (Random iterative models. Springer, New-York, 1996), and for example Douc et al. (Nonlinear time series: theory, methods, and applications with R examples. CRC Press, Chapman & Hall, Boca Raton, 2015) for the estimation of such Markov models. Many statistical models will be proved in this way. The organization follows the order from natural extensions of linearity to more general settings. From linear processes it is natural to build polynomial models or their limits. Then we consider more general Bernoulli shift models to define recurrence equations besides the standard Markov setting.

Abstract

The notion of association, or positive correlation, was naturally introduced in two different fields: reliability (Esary, Proschan, Walkup in Annal Math Stat 38:1466–1474, 1967) and statistical physics (Fortuin, Kasteleyn, Ginibre in Commun Math Phys 22–2:89–103, 1971) to model a tendency that the coordinates of a vector valued random variable admit such behaviours. We refer the reader to Newman (Inequalities in Statistics and Probability, Elsevier, Amsterdam, pp. 127–140, 1984) for more details. This notion deserves much attention since it provides a class of random variables for which independence and orthogonality coincide. Another case for which this feature holds is the Gaussian case, see Chap. 5. The notion of independence is more related to \(\sigma \)-algebras but in those two cases it is related to the geometric notion of orthogonality. Those remarks are of interest for modelling dependence as this is the aim of Chap. 9.

Abstract

We propose an overview of the notions of dependence in this chapter, good references are Doukhan et al. (Theory and applications of long-range dependence. Birkhaüser, Boston, 2002b) for long-range dependence, and Doukhan (Mixing: properties and examples. Lecture notes in statistics. Springer, New York, 1994) and Dedecker et al. (Weak dependence: with examples and applications. Lecture notes in statistics. Springer, New York, 2007) for weak-dependence.

Abstract

Long-range dependent (LRD) phenomena were first exhibited by Hurst for hydrology purposes. This phenomenon occurs from the superposition of independent sources, e.g. confluent rivers provide this behaviour (see Fig. 4.2). Such aggregation procedures provide this new phenomenon. Hurst (Trans Am Soc Civ Eng 116:770–799, 1951) originally determined the optimum dam sizing for the Nile river’s volatile rain and drought conditions observed over a long period of time. LRD is characterized by slow decorrelation properties and the behaviour of partial sums’s variances. This phenomenon is discussed above, see Sects. 4.3 and 4.4. Asymptotic properties of instantaneous functions of Gaussian processes are provided in Remark 5.2.4. Infinite moving averages models with LRD properties are provided in Sects. 6.4 and 6.6. We refer the reader to Doukhan et al. (Theory and applications of long-range dependence. Birkhaüser, Boston, 2002b) for much more.

Abstract

This chapter introduces some simple ideas. We investigate conditions on time series such that the standard limit theorems obtained for independent identically distributed sequences still hold. After a general introduction to weak-dependence conditions an example states the fact that the most classical strong-mixing condition from (Rosenblatt (1956). Proc Natl Acad Sci U S A 42:43–47.) Rosenblatt (1956) may fail to work, see (Andrews (1984). J Appl Probab 21:930–934.) Andrews (1984). When dealing with any weak-dependence condition (including strong mixing), additional decay rates and moment conditions necessary to ensure CLTs. Decay rates will be essential to derive asymptotic results. Coupling arguments as proposed in Sect. 7.4.2 are widely used for this. Finally to make clearer the need for decay rates, we explain how CLTs may be proved under such assumptions. The monograph (Dedecker, Doukhan, Lang, León, Louhichi, Prieur (2007). Lect Notes Stat 190.) (Dedecker et al. 2007) is used as the reference for weak-dependence; in this monograph we developed more formal results together with their proofs. We refer a reader to this monograph for more rigorous results.

Abstract

This chapter is devoted to moment methods. The use of moments relies on their importance in deriving asymptotic of several estimators, based on moments and limit distributions. Cumulants are linked with spectral or multispectral estimation which are main tools of time series analysis. Such functions do not characterize the dependence of non-linear processes; indeed we have already examples of orthogonal and non-independent sequences. This motivates the introduction of higher order characteristics. A multispectral density is defined over \(\mathbb {C}^{p-1}\) by

$$g(\lambda )=\sum _{k=-\infty }^\infty \mathrm {Cov}\,(X_0,X_k)e^{-ik\lambda }.$$

$$g(\lambda _2,\ldots ,\lambda _p)=\sum _{k_2=-\infty }^\infty \!\!\cdots \!\!\sum _{k_p=-\infty }^\infty \kappa (X_0,X_{k_2},\ldots , X_{k_p}) e^{-i(k_2\lambda _2+\cdots +k_p\lambda _p)}.$$