Linear Processes in Function Spaces: Theory and Applications

ISBN-13: 9780387950525 / Angielski / Miękka / 2000 / 286 str.

D. Bosq; Denis Bosq

Opis

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. Mathematical tools are presented, as well as autoregressive processes in Hilbert and Banach spaces and general linear processes and statistical prediction. Implementation and numerical applications are also covered. The book assumes knowledge of classical probability theory and statistics.

Spis treści

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.