Linear Processes in Function Spaces: Theory And Applications

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.