Spectral Theory of Random Fields, 1983, 267 pp., ISBN 0-911575-00-6, $52.00
THIS TITLE IS OUT OF PRINT BUT THE PUBLISHER CAN PREPARE A COPY
First book of its kind presenting a unified treatment
of the spectral theory of Random Fields, including a wealth of concrete
formulas relevant to applications. Treats Likelihood Ratios, Prediction
and Extrapolation, Optimal Regression, Mean-value Estimation. Includes
comprehensive bibliography.
TABLE OF CONTENTS
CHAPTER I. SPECTRAL THEORY OF HOMOGENEOUS
AND ISOTROPIC RANDOM FIELDS ..... 1
Section 1. Homogeneous and Isotropic Random Fields ..... 1
Section 2. Spherical Averages of Homogeneous and Isotropic Random Fields ..... 24
Section 3. Homogeneous and Isotropic Random Fields of the Markov Type ..... 51
Section 4. Homogeneous and Isotropic Random Fields in Hilbert Space ..... 61
Section 5. Isotropic Random Fields on Spheres ..... 71
Section 6. Isotropic Random Fields on Euclidean Spaces ..... 86
Section 7. Strong Law of Large Numbers for Isotropic Random
Fields ..... 101
CHAPTER II. LOCAL BEHAVIOR OF SAMPLE FUNCTIONS
OF RANDOM FIELDS ..... 105
Section 1. On the Modulus of Continuity of Random Fields in Euclidean Spaces ..... 105
Section 2. Local Behavior of Sample Functions of Gaussian
Random Fields ..... 115
CHAPTER III. ON ABSOLUTE CONTINUITY AND SINGULARITY
OF MEASURES CORRESPONDING TO RANDOM FIELDS ..... 131
Section 1. Preliminary Results on Absolute Continuity of Measures over Hilbert Spaces ..... 131
Section 2. On Absolute Continuity of Measures Corresponding to Gaussian Homogeneous Random Fields with Identical Correlation Functions and Different Means ..... 135
Section 3. Absolute Continuity of Measures Corresponding
to Gaussian Homogeneous Zero Mean Random Fields with Different Correlation
Functions ..... 148
CHAPTER IV. SELECTED PROBLEMS CONCERNING STATISTICS
OF RANDOM FIELDS ..... 170
Section 1. Linear Forecasting for Random Fields Observed on a Sphere ..... 170
Section 2. On Extrapolation of a Homogeneous and Isotropic Random Field from Observation on a Countable System of Concentric Spheres ..... 193
Section 3. Optimal Estimates of Regression Coefficients and Mean Values of an Isotropic Random Field Observed on a Sphere ..... 201
Section 4. On the Integral Equations for the Statistics of Homogeneous and/or Isotropic Random Fields ..... 214
A Brief Survey of the Research Literature in the Theory of Random Fields ..... 227
Notes ..... 230
Bibliography ..... 237
Index ..... 258