Linear Processes in Function Spaces: Theory and Applications

Front Cover
Springer Science & Business Media, Aug 1, 2000 - Mathematics - 283 pages
1 Review
The 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 pro- cesses 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.
  

What people are saying - Write a review

We haven't found any reviews in the usual places.

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
References
269
Index
277
Copyright

Common terms and phrases

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

All Book Search results »

About the author (2000)

Denis Bosq is a Professor at the Laboratory of Theoretical and Applied Statistics, University of Pierre & Marie Curie - Paris 6. He has over 100 published papers, 5 books, and is chief editor of the journal 'Statistical Inference for Stochastic Processes' as well as associate editor for the 'Journal of Non-Parametric Statistics'. He is a well-known specialist in the field of non-parametric statistical inference.

Bibliographic information