Author | Bosq, Denis. author |
---|---|
Title | Linear Processes in Function Spaces [electronic resource] : Theory and Applications / by Denis Bosq |
Imprint | New York, NY : Springer New York : Imprint: Springer, 2000 |
Connect to | http://dx.doi.org/10.1007/978-1-4612-1154-9 |
Descript | XIV, 286 p. online resource |
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 -- 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) -- 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 -- 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