Linear Processes in Function Spaces Theory and Applications
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 Hilb...
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| Format: | eBook |
| Language: | English |
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New York, NY
Springer New York
2000, 2000
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| Edition: | 1st ed. 2000 |
| Series: | Lecture Notes in Statistics
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| Subjects: | |
| Online Access: | |
| Collection: | Springer Book Archives -2004 - Collection details see MPG.ReNa |
Table of Contents:
- 5.5. Estimation of the autoregression order
- Notes
- 6. Autoregressive processes in Banach spaces
- 1. Strong autoregressive processes in Banach spaces
- 2. Autoregressive representation of some real continuous-time processes
- 3. Limit theorems
- 4. Weak Banach autoregressive processes
- 5. Estimation of autocovariance
- 6. The case of C[0, 1]
- 7. Some applications to real continuous-time processes
- Notes
- 7. General linear processes in function spaces
- 7.1. Existence and first properties of linear processes
- 7.2. Invertibility of linear processes
- 7.3. Markovian representations of LPH: applications
- 7.4. Limit theorems for LPB and LPH
- 7.5. * Derivation of invertibility
- Notes
- 8. Estimation of autocorrelation operator and prediction
- 8.1. Estimation of p if H is finite dimensional
- 8.2. Estimation of p in a special case
- 8.3. The general situation
- 8.4. Estimation of autocorrelation operator in C[0,1]
- 8.5. Statistical prediction
- 2.2. Convergence of B-random variables
- 2.3. Limit theorems for i.i.d. sequences of B-random variables
- 2.4. Sequences of dependent random variables in Banach spaces
- 2.5. * Derivation of exponential bounds
- Notes
- 3. Autoregressive Hilbertian processes of order one
- 3.1. Stationarity and innovation in Hilbert spaces
- 3.2. The ARH(1) model
- 3.3. Basic properties of ARH(1) processes
- 3.4. ARH(1) processes with symmetric compact autocorrelation operator
- 3.5. Limit theorems for ARH(1) processes
- Notes
- 4. Estimation of autocovariance operators for ARH(1) processes
- 4.1. Estimation of the covariance operator
- 4.2. Estimation of the eigenelements of C
- 4.3. Estimation of the cross-covariance operators
- 4.4. Limits in distribution
- Notes
- 5. Autoregressive Hilbertian processes of order p
- 5.1. The ARH(p) model
- 5.2. Second order moments of ARH(p)
- 5.3. Limit theorems for ARH(p)processes
- 5.4. Estimation of autocovariance of an ARH(p)
- 8.6. * Derivation of strong consistency
- Notes
- 9. Implementation of functional autoregressive predictors and numerical applications
- 9.1. Functional data
- 9.2. Choosing and estimating a model
- 9.3. Statistical methods of prediction
- 9.4. Some numerical applications
- Notes
- Figures
- 1. Measure and probability
- 2. Random variables
- 3. Function spaces
- 4. Basic function spaces
- 5. Conditional expectation
- 6. Stochastic integral
- References
- Synopsis
- 1. The object of study
- 2. Finite-dimensional linear processes
- 3. Random variables in function spaces
- 4. Limit theorems in function spaces
- 5. Autoregressive processes in Hilbert spaces
- 6. Estimation of covariance operators
- 7. Autoregressive processes in Banach spaces and representations of continuous-time processes
- 8. Linear processes in Hilbert spaces and Banach spaces
- 9. Estimation of autocorrelation operator and forecasting
- 10. Applications
- 1. Stochastic processes and random variables in function spaces
- 1.1. Stochastic processes
- 1.2. Random functions
- 1.3. Expectation and conditional expectation in Banach spaces
- 1.4. Covariance operators and characteristic functionals in Banach spaces
- 1.5. Random variables and operators in Hilbert spaces
- 1.6. Linear prediction in Hilbert spaces
- Notes
- 2. Sequences of random variables in Banach spaces
- 2.1. Stochastic processes as sequences of B-valued random variables