Stationary Sequences and Random Fields

ISBN-13: 9780817632649 / Angielski / Miękka / 1985 / 258 str.

Rosenblatt; Murray Rosenblatt

Opis

This book has a dual purpose. One of these is to present material which selec- tively will be appropriate for a quarter or semester course in time series analysis and which will cover both the finite parameter and spectral approach. The second object is the presentation of topics of current research interest and some open questions. I mention these now. In particular, there is a discussion in Chapter III of the types of limit theorems that will imply asymptotic nor- mality for covariance estimates and smoothings of the periodogram. This dis- cussion allows one to get results on the asymptotic distribution of finite para- meter estimates that are broader than those usually given in the literature in Chapter IV. A derivation of the asymptotic distribution for spectral (second order) estimates is given under an assumption of strong mixing in Chapter V. A discussion of higher order cumulant spectra and their large sample properties under appropriate moment conditions follows in Chapter VI. Probability density, conditional probability density and regression estimates are considered in Chapter VII under conditions of short range dependence. Chapter VIII deals with a number of topics. At first estimates for the structure function of a large class of non-Gaussian linear processes are constructed. One can determine much more about this structure or transfer function in the non-Gaussian case than one can for Gaussian processes. In particular, one can determine almost all the phase information.

Spis treści

I Stationary Processes.- 1. General Discussion.- 2. Positive Definite Functions.- 3. Fourier Representation of a Weakly Stationary Process.- Problems.- Notes.- II Prediction and Moments.- 1. Prediction.- 2. Moments and Cumulants.- 3. Autoregressive and Moving Average Processes.- 4. Non-Gaussian Linear Processes.- 5. The Kalman-Bucy Filter.- Problems.- Notes.- III Quadratic Forms, Limit Theorems and Mixing Conditions.- 1. Introduction.- 2. Quadratic Forms.- 3. A Limit Theorem.- 4. Summability of Cumulants.- 5. Long-range Dependence.- 6. Strong Mixing and Random Fields.- Problems.- Notes.- IV Estimation of Parameters of Finite Parameter Models.- 1. Maximum Likelihood Estimates.- 2. The Newton-Raphson Procedure and Gaussian ARMA Schemes.- 3. Asymptotic Properties of Some Finite Parameter Estimates.- 4. Sample Computations Using Monte Carlo Simulation.- 5. Estimating the Order of a Model.- 6. Finite Parameter Stationary Random Fields.- Problems.- V Spectral Density Estimates.- 1. The Periodogram.- 2. Bias and Variance of Spectral Density Estimates.- 3. Asymptotic Distribution of Spectral Density Estimates.- 4. Prewhitening and Tapering.- 5. Spectral Density Estimates Using Blocks.- 6. A Lower Bound for the Precision of Spectral Density Estimates.- 7. Turbulence and the Kolmogorov Spectrum.- 8. Spectral Density Estimates for Random Fields.- Problems.- Notes.- VI Cumulant Spectral Estimates.- 1. Introduction.- 2. The Discrete Fourier Transform and Fast Fourier Transform.- 3. Vector-Valued Processes.- 4. Smoothed Periodograms.- 5. Aliasing and Discretely Sampled Time Series.- Notes.- VII Density and Regression Estimates.- 1. Introduction. The Case of Independent Observations.- 2. Density and Regression Estimates for Stationary Sequences.- Notes.- VIII Non-Gaussian Linear Processes.- 1. Estimates of Phase, Coefficients, and Deconvolution for Non-Gaussian.- Linear Processes.- 2. Random Fields.- 3. Non-Gaussian Linear Random Fields.- Notes.- 1. Monotone Functions and Measures.- 2. Hilbert Space.- 3. Banach Space.- 4. Banach Algebras and Homomorphisms.- Postscript.- Author Index.