Limit Theorems for Random Fields with Singular Spectrum

1999. Auflage 1999
ISBN: 978-0-7923-5635-6
Verlag: Springer

Buch, Englisch, Band 465, 406 Seiten, Format (B × H): 156 mm x 234 mm, Gewicht: 762 g

Reihe: Mathematics and Its Applications

This book is devoted to an investigation of the basic problems of the the­ ory of random fields which are characterized by certain singular properties (e. g., unboundedness, or vanishing) of their spectral densities. These ran­ dom fields are called, the random fields with singular spectrum, long-memory fields, random fields with long-range dependence, fields with slowly decaying correlations or strongly dependent random fields by various authors. This phenomenon has been observed empirically by many scientists long before suitable mathematical models were known. The methods and results differ significantly from the theory of weakly dependent random fields. The first chapter presents basic concepts of the spectral theory of random fields, some examples of random processes and fields with singular spectrum, Tauberian and Abelian theorems for the covariance function of singular ran­ dom fields. In the second chapter limit theorems for non-linear functionals of random fields with singular spectrum are proved. Chapter 3 summarizes some limit theorems for geometric functionals of random fields with long-range dependence. Limit distributions of the solutions of Burgers equation with random data via parabolic and hyperbolic rescaling are presented in chapter 4. And chapter 5 presents some problems of statistical analysis of random fields with singular spectrum. I would like to thank the editor, Michiel Hazewinkel, for his support. I am grateful to the following students and colleagues: 1. Deriev, A. Olenko, K. Rybasov, L. Sakhno, M. Sharapov, A. Sikorskii, M. Silac-BenSic. I would also like to thank V.Anh, O. Barndorff-Nielsen,Yu. Belyaev, P.

Zielgruppe

Research

Autoren/Hrsg.

Leonenko, Nicolai

Fachgebiete

• Technische Wissenschaften Maschinenbau | Werkstoffkunde Technische Mechanik | Werkstoffkunde Strömungslehre
• Mathematik | Informatik Mathematik Stochastik Mathematische Statistik
• Interdisziplinäres Wissenschaften Wissenschaften: Forschung und Information Datenanalyse, Datenverarbeitung
• Mathematik | Informatik Mathematik Stochastik Stochastische Prozesse
• Naturwissenschaften Physik Mechanik Kontinuumsmechanik, Strömungslehre
• Mathematik | Informatik Mathematik Stochastik Elementare Stochastik
• Mathematik | Informatik Mathematik Stochastik Wahrscheinlichkeitsrechnung
• Mathematik | Informatik Mathematik Numerik und Wissenschaftliches Rechnen Angewandte Mathematik, Mathematische Modelle

Inhaltsverzeichnis

1 Second-Order Analysis of Random Fields.- 1.1 Basic Concepts and Notation.- 1.2 Elements of Spectral Theory of Random Fields.- 1.3 Models of Random Processes and Fields with Singular Spectrum.- 1.4 Tauberian and Abelian Theorems for Correlation Function of Homogeneous Isotropic Random Fields.- 2 Limit Theorems for Non-Linear Transformations of Random Fields.- 2.1 Some Properties of Gaussian and X-Squared Random Fields.- 2.2 Reduction Theorems for the Local Functionals of Random Fields with Slowly Decaying Correlations.- 2.3 Multiple Stochastic Integrals.- 2.4 Non-Central Limit Theorems for Local Functionals of Random Fields.- 3 Asymptotic Distributions of Geometric Functionals of Random Fields.- 3.1 Limit Distributions for Characteristics of the Excess above a Level for Gaussian Fields.- 3.2 Limiting Distributions for the Excess Over a Radial Surface of X-Squared Random Fields.- 3.3 Spherical Measures of Excess over of Moving Level.- 3.4 Sojourns of Multi-Dimensional Gaussian Fields with Dependent Components.- 3.5 Asymptotic Normality of Random ‘Area of Surface’ of Planar Gaussian Field.- 3.6 Asymptotics for Occupation Densities of Gaussian and X-Squared Random Fields.- 4 Limit Theorems For Solutions of The Burgers’ Equation with Random Data.- 4.1 Physical Motivation and Recent History.- 4.2 Hopf-Cole Solution.- 4.3 Parabolic Asymptotics for Weakly Dependent Random Data: the Gaussian Scenario.- 4.4 Parabolic Limits for Strongly Dependent Random Initial Conditions: the Gaussian Scenario.- 4.5 Parabolic Limits for Strongly Dependent Random Data: the Non-Gaussian Scenario.- 4.6 Exact Parabolic Asymptotics for Singular Burgers’ Equation.- 4.7 Hyperbolic Asymptotics for Rescaled Solutions of Burgers’ Equation.- 5 Statistical Problems for Random Fields withSingular Spectrum.- 5.1 Estimation of Mathematical Expectation.- 5.2 Estimation of the Covariance Function.- 5.3 Efficient Estimation of Regression Coefficients of a Random Fields Observed on the Sphere.- 5.4 Estimation in the Frequency Domain.- Comments.