Topic
Asymptotology
About: Asymptotology is a research topic. Over the lifetime, 1319 publications have been published within this topic receiving 35831 citations.
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01 Jan 2000
TL;DR: In this article, the authors present a model for estimating and testing Stochastic Processes based on local asymptotic normality, which is used for estimating long-term memory (LM) processes.
Abstract: 1 Elements of Stochastic Processes.- 1.1 Introduction.- 1.2 Stochastic Processes.- 1.3 Limit Theorems.- Problems.- 2 Local Asymptotic Normality for Stochastic Processes.- 2.1 General Results for Local Asymptotic Normality.- 2.2 Local Asymptotic Normality for Linear Processes.- Problems.- 3 Asymptotic Theory of Estimation and Testing for Stochastic Processes.- 3.1 Asymptotic Theory of Estimation and Testing for Linear Processes.- 3.1.1 Asymptotic Theory Based on a Gaussian Likelihood.- 3.1.2 Asymptotic Theory of Estimation and Testing Based on LAN Results.- 3.2 Asymptotic Theory for Nonlinear Stochastic Models.- 3.2.1 Nonlinear Models.- 3.2.2 Probability Structure of Nonlinear Models.- 3.2.3 Statistical Testing and Estimation Theory for Nonlinear Models.- 3.2.4 Asymptotic Theory Based on the LAN Property.- 3.2.5 Model Selection Problems.- 3.2.6 Nonergodic Models.- 3.3 Asymptotic Theory for Continuous Time Processes.- 3.3.1 Stochastic Integrals and Diffusion Processes.- 3.3.2 Asymptotic Theory for Diffusion Processes.- 3.3.3 Diffusion Processes and Autoregressions with Roots.- Near Unity.- 3.3.4 Continuous Time ARMA Processes.- 3.3.5 Asymptotic Theory for Point Processes.- Problems.- 4 Higher Order Asymptotic Theory for Stochastic Processes.- 4.1 Introduction to Higher Order Asymptotic Theory.- 4.2 Valid Asymptotic Expansions.- 4.3 Higher Order Asymptotic Estimation Theory for Discrete Time Processes in View of Statistical Differential Geometry.- 4.4 Higher Order Asymptotic Theory for Continuous Time Processes.- 4.5 Higher Order Asymptotic Theory for Testing Problems.- 4.6 Higher Order Asymptotic Theory for Normalizing Transformations.- 4.7 Generalization of LeCam's Third Lemma and Higher Order Asymptotics of Iterative Methods.- Problems.- 5 Asymptotic Theory for Long-Memory Processes.- 5.1 Some Elements of Long-Memory Processes.- 5.2 Limit Theorems for Fundamental Statistics.- 5.3 Estimation and Testing Theory for Long-Memory Processes.- 5.4 Regression Models with Long-Memory Disturbances.- 5.5 Semiparametric Analysis and the LAN Approach.- Problems.- 6 Statistical Analysis Based on Functionals of Spectra.- 6.1 Estimation of Nonlinear Functionals of Spectra.- 6.2 Application to Parameter Estimation for Stationary Processes.- 6.3 Asymptotically Efficient Nonparametric Estimation of Functionals of Spectra in Gaussian Stationary Processes.- 6.4 Robustness in the Frequency Domain Approach.- 6.4.1 Robustness to Small Trends of Linear Functionals of a Periodogram.- 6.4.2 Peak-Insensitive Spectrum Estimation.- 6.5 Numerical Examples.- Problems.- 7 Discriminant Analysis for Stationary Time Series.- 7.1 Basic Formulation.- 7.2 Standard Methods for Gaussian Stationary Processes.- 7.2.1 Time Domain Methods.- 7.2.2 Frequency Domain Methods.- 7.2.3 Admissible Linear Procedure: Case of Unequal Mean Vectors and Covariance Matrices.- 7.3 Discriminant Analysis for Non-Gaussian Linear Processes.- 7.4 Nonparametric Approach for Discriminant Analysis.- 7.5 Parametric Approach for Discriminant Analysis.- 7.6 Derivation of Spectral Expressions to Divergence Measures Between Gaussian Stationary Processes.- 7.7 Miscellany.- Problems.- 8 Large Deviation Theory and Saddlepoint Approximation for Stochastic Processes.- 8.1 Large Deviation Theorem 538 8.2 Asymptotic Efficiency for Gaussian Stationary Processes:Large Deviation Approach.- 8.2.1 Asymptotic Theory of Neyman-Pearson Tests.- 8.2.2 Bahadur Efficiency of Estimator.- 8.2.3 Stochastic Comparison of Tests.- 8.3 Large Deviation Results for an Ornstein-Uhlenbeck Process.- 8.4 Saddlepoint Approximations for Stochastic Processes.- Problems.- A.1 Mathematics.- A.2 Probability.- A.3 Statistics.
177 citations
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10 Mar 2012
TL;DR: In this article, the analysis of the solutions of a system of Ordinary Differential Equations with turning points is studied. But the authors focus on the complex case and do not consider the real case.
Abstract: 1. The Analytic Theory of Differential Equations.- 1. Analyticity of the Solutions of a System of Ordinary Differential Equations.- 2. Regular Singular Points.- 3. Irregular Singular Points.- 2. Second-Order Equations on the Real Line.- 1. Transformations of Second-Order Equations.- 2. WKB-Bounds.- 3. Asymptotic Behaviour of Solutions of a Second-Order Equation for Large Values of the Parameter.- 4. Systems of Two Equations Containing a Large Parameter.- 5. Systems of Equations Close to Diagonal Form.- 6. Asymptotic Behaviour of the Solutions for Large Values of the Argument.- 7. Dual Asymptotic Behaviour.- 8. Counterexamples.- 9. Roots of Constant Multiplicity.- 10. Problems on Eigenvalues.- 11. A Problem on Scattering.- 3. Second-Order Equations in the Complex Plane.- 1. Stokes Lines and the Domains Bounded by them.- 2. WKB-Bounds in the Complex Plane.- 3. Equations with Polynomial Coefficients. Asymptotic Behaviour of a Solution in the Large.- 4. Equations with Entire or Meromorphic Coefficients.- 5. Asymptotic Behaviour of the Eigenvalues of the Operator -d2 / dx2 + ?2q(x). Self-Adjoint Problems.- 6. Asymptotic Behaviour of the Discrete Spectrum of the Operator -y? + ?2q(x)y. Non-Self-Adjoint Problems.- 7. The Eigenvalue Problem with Regular Singular Points.- 8. Quasiclassical Approximation in Scattering Problems.- 9. Sturm-Liouville Equations with Periodic Potential.- 4. Second-Order Equations with Turning Points.- 1. Simple Turning Points. The Real Case.- 2. A Simple Turning Point. The Complex Case.- 3. Some Standard Equations.- 4. Multiple and Fractional Turning Points.- 5. The Fusion of a Turning Point and Regular Singular Point.- 6. Multiple Turning Points. The Complex Case.- 7. Two Close Turning Points.- 8. Fusion of Several Turning Points.- 5. nth-Order Equations and Systems.- 1. Equations and Systems on a Finite Interval.- 2. Systems of Equations on a Finite Interval.- 3. Equations on an Infinite Interval.- 4. Systems of Equations on an Infinite Interval.- 5. Equations and Systems in the Complex Plane.- 6. Turning Points.- 7. A Problem on Scattering, Adiabatic Invariants and a Problem on Eigenvalues.- 8. Examples.- References.
173 citations
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28 Oct 1999
TL;DR: In this paper, a boundary value problem for the Laplacian in a multi-structure is introduced. But the boundary value is not a boundary-value problem for 3D-1D multi-structures.
Abstract: 1. Introduction to compound asymptotic expansions 2. A boundary value problem for the Laplacian in a multi-structure 3. Auxiliary facts from mathematical elasticity 4. Elastic multi-structure 5. Non-degenerate elastic multi-structure 6. Spectral analysis for 3D-1D multi-structures Bibliographical remarks Bibliography Index
161 citations
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TL;DR: In this article, the authors extended Aronson's and Weinberger's concept of asymptotic speed to the nonlinear integral equation t J g(u(t-s,x+y))k(s9\\y\\)dsdy, which is a spatial version of Kermack's and McKendrick's epidemic model.
Abstract: Recently Aronson [1] extended the concept of asymptotic speed which he and Weinberger [3], [4] had developed for nonlinear diffusion problems in population genetics, combustion and nerve propagation, to an epidemic model proposed by Kendall [11], [12] in 1957 (1965). In this model (which is a spatial version of Kermack's and McKendrick's epidemic model [13]) the aflfected individuals become immediately infectious and are removed at a constant rate. The model does not take into account that with most infectious diseases the affected individuals underlie an incubation period, before they become infective, and that they remain infective for a fixed period only. These features cannot be described by the equation considered by Aronson [1] which contains a derivative with respect to time and an integral with respect to space. It is therefore desirable to extend Aronson's and Weinberger's concept of asymptotic speed to the nonlinear integral equation t J g(u(t-s,x+y))k(s9\\y\\)dsdy
159 citations