Linear Processes in Function Spaces

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.