## Linear Processes in Function Spaces: Theory and ApplicationsThe main subject of this book is the estimation and forecasting of continuous time processes. It leads to a development of the theory of linear processes in function spaces. The necessary mathematical tools are presented in Chapters 1 and 2. Chapters 3 to 6 deal with autoregressive processes in Hilbert and Banach spaces. Chapter 7 is devoted to general linear processes and Chapter 8 with statistical prediction. Implementation and numerical applications appear in Chapter 9. The book assumes a knowledge of classical probability theory and statistics. Denis Bosq is Professor of Statistics at the University of Paris 6 (Pierre et Marie Curie). He is Chief-Editor of Statistical Inference for Stochastic Processes and of Annales de l'ISUP, and Associate Editor of the Journal of Nonparametric Statistics. He is an elected member of the International Statistical Institute, and he has published about 100 papers or works on nonparametric statistics and five books including Nonparametric Statistics for Stochastic Processes: Estimation and Prediction, Second Edition (Springer, 1998). |

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### Contents

Stochastic Processes and Random Variables in Function Spaces | 15 |

12 Random functions | 21 |

13 Expectation and conditional expectation in Banach spaces | 27 |

14 Covariance operators and characteristic functionals in Banach spaces | 30 |

15 Random variables and operators in Hilbert spaces | 33 |

16 Linear prediction in Hilbert spaces | 38 |

NOTES | 42 |

Sequences of Random Variables in Banach Spaces | 43 |

65 Estimation of autocovariance | 164 |

66 The case of C01 | 168 |

67 Some applications to real continuoustime processes | 175 |

NOTES | 180 |

General Linear Processes in Function Spaces | 181 |

71 Existence and first properties of linear processes | 182 |

72 Invertibility of linear processes | 184 |

applications | 188 |

22 Convergence of Brandom variables | 44 |

23 Limit theorems for iid sequences of Brandom variables | 47 |

24 Sequences of dependent random variables in Banach spaces | 54 |

25 Derivation of exponential bounds | 66 |

NOTES | 70 |

Autoregressive Hilbertian Processes of Order 1 | 71 |

32 The ARH1 model | 73 |

33 Basic properties of ARH1 processes | 79 |

34 ARH1 processes with symmetric compact autocorrelation operator | 82 |

35 Limit theorems for ARH1 processes | 86 |

NOTES | 94 |

Estimation of Autocovariance Operators for ARH1 Processes | 95 |

42 Estimation of the eigenelements of C | 102 |

43 Estimation of the crosscovariance operators | 112 |

44 Limits in distribution | 118 |

NOTES | 125 |

Autoregressive Hilbertian Processes of Order p | 127 |

52 Second order moments of ARHp | 133 |

53 Limit theorems for ARHp processes | 136 |

54 Estimation of autocovariance of an ARHp | 140 |

55 Estimation of the autoregression order | 143 |

NOTES | 145 |

Autoregressive Processes in Banach Spaces | 147 |

62 Autoregressive representation of some real continuoustime processes | 150 |

63 Limit theorems | 153 |

64 Weak Banach autoregressive processes | 161 |

74 Limit theorems for LPB and LPH | 191 |

75 Derivation of invertibility | 195 |

NOTES | 202 |

Estimation of Autocorrelation Operator and Prediction | 203 |

81 Estimation of p if H is finite dimensional | 204 |

82 Estimation of p in a special case | 211 |

83 The general situation | 218 |

84 Estimation of autocorrelation operator in C01 | 222 |

85 Statistical prediction | 226 |

86 Derivation of strong consistency | 229 |

NOTES | 236 |

Implementation of Functional Autoregressive Predictors and Numerical Applications | 237 |

92 Choosing and estimating a model | 240 |

93 Statistical methods of prediction | 243 |

94 Some numerical applications | 247 |

NOTES | 251 |

Simulation and prediction of ARH1 processes | 252 |

Appendix | 263 |

Random variables | 264 |

Function spaces | 265 |

Basic function spaces | 266 |

Conditional expectation | 267 |

269 | |

277 | |

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### Common terms and phrases

ARB(l ARH(l ARH(p associated asymptotic autocovariance autoregressive processes autoregressive representation B-random B-valued random Banach space Borel-Cantelli Lemma Bosq bounded linear operator Chapter Cn-C consider continuous-time processes convergence COROLLARY covariance operator cross-covariance operators decomposition defined denotes eigenelements eigenvalues entails estimator Example exists fc=i follows function spaces hence Hilbert space Hilbert-Schmidt operators Hilbertian holds implies inequality integrable invertible large numbers law of large Lemma Limit theorems linear processes martingale martingale difference measurable norm Note obtain orthonormal basis p(Xn pn(Xn prediction predictor process of order proof of Theorem properties random variables real random variable sample paths satisfies scalar product Section separable Banach space sequence standard stationary process stationary solution Statistics stochastic process strictly stationary strong mixing strong white noise subspace Suppose v'jn Wiener process yields

### Popular passages

Page v - If you can look into the seeds of time, And say, which grain will grow, and which will not, Speak then to me, who neither beg, nor fear, Your favours, nor your hate.

Page v - Ainsi l'abbé Blanès n'avait pas communiqué sa science assez difficile à Fabrice; mais, à son insu, il lui avait inoculé une confiance illimitée dans les signes qui peuvent prédire l'avenir.

Page 7 - Let H be a separable Hilbert space with norm [| . || and scalar product Stationary process and white noise in such a space are easily defined by using cross-covariance operators instead of covariances.

### References to this book

Nonparametric Functional Data Analysis: Theory and Practice Frédéric Ferraty,Philippe Vieu Limited preview - 2006 |

Handbook of Econometrics James J. Heckman,Zvi Griliches,Edward E. Leamer,Michael D. Intriligator No preview available - 2007 |