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Showing papers on "Asymptotology published in 2000"



Book
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


Journal ArticleDOI
TL;DR: The main results are in Section 4 (see Theorem 21) and the illustrative example in the last section is given.

71 citations


Journal ArticleDOI
TL;DR: In this article, a formal asymptotic solution of the Hamiltonian boundary-value problem with delay in the state variables is proposed and the justification of this solution is done.
Abstract: The Hamiltonian boundary-value problem, associated with a singularly-perturbed linear-quadratic optimal control problem with delay in the state variables, is considered. A formal asymptotic solution of this boundary-value problem is constructed by application of the boundary function method. The justification of this asymptotic solution is done. The asymptotic solution of the Hamiltonian boundary-value problem is constructed and justified assuming boundary-layer stabilizability and detectability.

48 citations


Book ChapterDOI
01 Jan 2000
TL;DR: The adjective “asymptotic” was added to the term “Geometric Analysis” to be more specific, and it will first explain the subject of this talk.
Abstract: The term “Geometric Analysis” is a recent one but it has quickly become fashionable and is used too often and for very different mathematics. So, we added the adjective “asymptotic” to be more specific, and we will first explain the subject of this talk.

41 citations



Journal ArticleDOI
TL;DR: In this article, a diffusing particle in one dimension is subject to a time-dependent drift or potential field, and a reflecting barrier constrains the particle's position to the half-line X ≥ 0.
Abstract: We consider a diffusing particle in one dimension that is subject to a time-dependent drift or potential field. A reflecting barrier constrains the particle's position to the half-line X ≥ 0. Such models arise naturally in the study of queues with time-dependent arrival rates, as well as in advection-diffusion problems of mathematical physics. We solve for the probability distribution of the particle as a function of space and time. Then we do a detailed study of the asymptotic properties of the solution, for various ranges of space and time. We also relate our asymptotic results to those obtained by probabilistic approaches, such as central limit theorems and large deviations. We consider drifts that are either piecewise constant or linear functions of time.

27 citations


BookDOI
01 Jan 2000

25 citations


Journal ArticleDOI
TL;DR: In this article, a reduction method is used to prove the existence and uniqueness of strong solutions to stochastic Kolmogorov-Petrovskii-Piskunov (KPP) equations, where the initial condition may be anticipating.
Abstract: A reduction method is used to prove the existence and uniqueness of strong solutions to stochastic Kolmogorov–Petrovskii–Piskunov (KPP) equations, where the initial condition may be anticipating. The asymptotic behaviour of the solution for large time and space and the random travelling waves are then studied under two different basic assumptions.

25 citations


Journal ArticleDOI
TL;DR: In this article, uniform asymptotic expansion of the Schwartz kernels of the Bremmer coupling series solution of the wave equation in inhomogeneous media is derived in the high frequency limit, taking into account critical scattering angle phenomena.
Abstract: The Bremmer coupling series solution of the wave equation, in generally inhomogeneous media, requires the introduction of pseudodifferential operators. In this paper, in two dimensions, uniform asymptotic expansions of the Schwartz kernels of these operators are derived. Also, we derive a uniform asymptotic expansion of the one-way propagator appearing in the series. We focus on designing closed-form representations, valid in the high-frequency limit, taking into account critical scattering-angle phenomena. Our expansion is not limited by propagation angle. In principle, the uniform asymptotic expansion of a kernel follows by matching its asymptotic behaviors away from and near its diagonal. The Bremmer series solver consists of three steps: directional decomposition into up- and downgoing waves, one-way propagation, and interaction of the counter-propagating constituents. Each of these steps is represented here by a kernel for which a uniform asymptotic expansion is found. The associated algorithm provid...

18 citations


Journal ArticleDOI
TL;DR: In this article, the spectral mapping theorem was used to derive the spectrum of the strongly continuous semigroup of the forced elongation in the isothermal regime, and the spectral properties of the eigenvalues were determined by solving a characteristic integral equation in the complex plane.

Book
28 Nov 2000
TL;DR: In this article, the authors studied local densities of measures and limit theorems for stochastic processes, as well as the behavior of the likelihood ratio in problems of distinguishing between simple hypotheses for semimartingales.
Abstract: Local densities of measures and limit theorems for stochastic processes Asymptotic distinguishing between simple hypotheses in the scheme of general statistical experiments Asymptotic behavior of the likelihood ratio in problems of distinguishing between simple hypotheses for semimartingales Asymptotic estimation of parameters Asymptotic information-theoretic problems in parameter estimation Bibliographical notes References Index.




Posted Content
TL;DR: The method of asymptotic expansions is used to build an approximation scheme relevant to celestial mechanics in relativistic theories of gravitation in this article, where a scalar theory is considered, both as a simple example and for its own sake.
Abstract: The method of asymptotic expansions is used to build an approximation scheme relevant to celestial mechanics in relativistic theories of gravitation A scalar theory is considered, both as a simple example and for its own sake This theory is summarized, then the relevant boundary problem is seen to be the full initial-value problem It is shown that, with any given system of gravitating bodies, one may associate a one-parameter family of similar systems, the parameter measuring the gravitational field-strength After a specific change of units, the derivation of asymptotic expansions becomes straightforward Two hypotheses could be made as to which time variable has to be used in the expansion The first one leads to an "asymptotic" post-Newtonian approximation (PNA) with instantaneous propagation, differing from the standard PNA in that, in the asymptotic PNA, all fields are expanded The second hypothese could lead to an "asymptotic" post-Minkowskian approximation (PMA) allowing to describe propagation effects, but it is not compatible with the Newtonian limit It is shown that the standard PNA is not compatible with the application of the usual method of asymptotic expansions as envisaged here

Book ChapterDOI
01 Jan 2000
TL;DR: This chapter is devoted to providing a modern asymptotic estimation and testing theory for those various stochastic process models, such as, nonlinear time series models, diffusion processes, point processes, and nonergodic processes.
Abstract: In classical time series analysis the asymptotic estimation and testing theory was developed for linear processes, which include the AR, MA, and ARMA models. However, in the last twenty years a lot of more complicated stochastic process models have been introduced, such as, nonlinear time series models, diffusion processes, point processes, and nonergodic processes. This chapter is devoted to providing a modern asymptotic estimation and testing theory for those various stochastic process models. The approach is mainly based on the LAN results given in the previous chapter. More concretely, in Section 3.1 we discuss the asymptotic estimation and testing theory for non-Gaussian vector linear processes in view of LAN. The results are very general and grasp a lot of other works dealing with AR, MA, and ARMA models as special cases. Section 3.2 reviews some elements of nonlinear time series models and the asymptotic estimation theory based on the conditional least squares estimator and maximum likelihood estimator (MLE). We address the problem of statistical model selection in general fashion. Also the asymptotic theory for nonergodic models is mentioned. Recently much attention has been paid to continuous time processes (especially diffusion processes), which appear in finance. Hence, in Section 3.3 we describe the foundation of stochastic integrals and diffusion processes. Then the LAN-based asymptotic theory of estimation for them is studied.

Journal ArticleDOI
TL;DR: In this article, the authors study the complicated asymptotic character of a simple first-order differential equation, which involves a term with a low exponent of the dependent variable.
Abstract: We study the surprisingly complicated asymptotic character of a simple first-order differential equation, which involves a term with a low exponent of the dependent variable. While numerical solutions and straightforward asymptotic expansions indicate a clearly defined boundary layer type transition, we find that the correct asymptotic structure involves a hidden' boundary layer, and that a straightforward approach cannot discern this.

Journal ArticleDOI
TL;DR: The asymptotic behaviour of the solution to the coupled system of transport equations and their diffusion approximations introduced in [5] and the rate of the asymPTotic decay are studied.
Abstract: In this paper we study the asymptotic behaviour of the solution to the coupled system of transport equations and their diffusion approximations introduced in [5]. We also give the rate of the asymptotic decay. This analysis is based in an essential way on the methods we introduced in [3,4].

Journal ArticleDOI
TL;DR: In this article, an overview of recent development in asymptotic analysis of fields in multi-structures is presented. But the analysis of time-dependent fields in 1D-3D multi-Structures is not discussed.
Abstract: This paper contains an overview of recent development in asymptotic analysis of fields in multi-structures. We begin with simple examples of scalar dynamic problems in two dimensions, and then present analysis of time-dependent fields in 1D-3D multi-structures. The asymptotic results, presented here, are based on the method of compound asymptotic expansions. Copyright (C) 2000 John Wiley & Sons, Ltd.


23 Mar 2000
TL;DR: In this paper, in two dimensions, uniform asymptotic expansions of the Schwartz kernels of these operators are derived and a uniform ascyptotic expansion of the one-way propagator appearing in the series is derived.
Abstract: The Bremmer coupling series solution of the wave equation, in generally inhomogeneous media, requires the introduction of pseudodifferential operators. In this paper, in two dimensions, uniform asymptotic expansions of the Schwartz kernels of these operators are derived. Also, we derive a uniform asymptotic expansion of the one-way propagator appearing in the series. We focus on designing closed-form representations, valid in the high-frequency limit, taking into account critical scattering-angle phenomena. Our expansion is not limited by propagation angle. In principle, the uniform asymptotic expansion of a kernel follows by matching its asymptotic behaviors away from and near its diagonal. The Bremmer series solver consists of three steps: directional decomposition into up- and downgoing waves, one-way propagation, and interaction of the counter-propagating constituents. Each of these steps is represented here by a kernel for which a uniform asymptotic expansion is found. The associated algorithm provid...

25 Jun 2000
TL;DR: In this article, a new time-explicit asymptotic method is presented through introducing an auxiliary parameter for the solution of an exact controllability problem of scattering waves.
Abstract: In this paper a new time-explicit asymptotic method is presented through introducing an auxiliary parameter for the solution of an exact controllability problem of scattering waves. Based on an exact control function, the asymptotic iteration is controlled by the auxiliary parameter and the optimal auxiliary parameter is updated during the iteration based on the existing or current iterated solutions of the wave equation. The numerical results show that the new method presented has a significant advantage over the purely asymptotic method in the history of convergence and has the ability to solve the scattering by the multi-bodies.

Journal ArticleDOI
Dan Socolescu1
TL;DR: In this article, it was shown that the asymptotic behavior of a smooth function with bounded Dirichlet integral in an exterior domain is controlled by the first Fourier coefficient.
Abstract: In this Note we show that the asymptotic behaviour of a smooth function with bounded Dirichlet integral in an exterior domain is controlled by the asymptotic behaviour of its first Fourier coefficient. If the function is a velocity solution to the exterior Dirichlet problem for the steady two-dimensional Navier–Stokes equations we can strengthen this result, by proving the pointwise asymptotic convergence of the velocity solution to its asymptotic mean value, i.e., to the asymptotic finite limit of its first Fourier coefficient.


Book ChapterDOI
01 Jan 2000
TL;DR: In this paper, the first approximation of a solution of the system of equations is a solution to the corresponding first-approximation of the problem of finding a singularity, finding parameters determining properties of solutions, and obtaining the asymptotic expansions of solutions.
Abstract: All local and asymptotic first approximations of a polynomial, of a differential polynomial and of a system of such polynomials may be selected algorithmically. Here the first approximation of a solution of the system of equations is a solution of the corresponding first approximation of the system of equations. The power transformations induce linear transformations of vector exponents and commute with the operation of selecting first approximations. In a first approximation of a system of equations they allow to reduce number of parameters and to reduce the presence of some variables to the form of derivatives of their logarithms. If the first approximation is the linear system, then in many cases the system of equations can be transformed into the normal form by means of the formal change of coordinates. The normal form is reduced to the problem of smaller dimension by means of the power transformation. Combining these algorithms, in many problems we can resolve a singularity, find parameters determining properties of solutions and obtain the asymptotic expansions of solutions. Some applications from Mechanics, Celestial Mechanics and Hydrodynamics are indicated.

Journal ArticleDOI
TL;DR: In this paper, the asymptotic stability of a class of nonlinear generalized systems through its slow subsystems under the assumption of regularity is investigated, and one example is given.
Abstract: This paper is mainly used to determine the asymptotic stability of a class of nonlinear generalized systems through its slow subsystems under the assumption of regularity. One example is given.


Journal ArticleDOI
TL;DR: In this paper, the authors considered the Cauchy problem for a perturbed Liouville equation and constructed an asymptotic solution with respect to the perturbation parameter by the two-scale expansion method.
Abstract: We consider the Cauchy problem for a perturbed Liouville equation. An asymptotic solution is constructed with respect to the perturbation parameter by the two-scale expansion method; this construction can be applied over long time intervals. The main result is the definition of a deformation of the leading term of the asymptotic expansion within a slow time scale.

01 Jan 2000
TL;DR: In this article, uniform asymptotic expansions of the Schwartz kernels of the Bremmer coupling series are derived in two dimensions, and a uniform expansion of the one-way propagator appearing in the series is derived.
Abstract: The Bremmer coupling series solution of the wave equation, in generally inho- mogeneous media, requires the introduction of pseudodifferential operators. In this paper, in two dimensions, uniform asymptotic expansions of the Schwartz kernels of these operators are derived. Also, we derive a uniform asymptotic expansion of the one-way propagator appearing in the series. We focus on designing closed-form representations, valid in the high-frequency limit, taking into account critical scattering-angle phenomena. Our expansion is not limited by propagation angle. In principle, the uniform asymptotic expansion of a kernel follows by matching its asymptotic behaviors away from and near its diagonal. The Bremmer series solver consists of three steps: directional decomposition into up- and downgoing waves, one-way propagation, and interaction of the counter- propagating constituents. Each of these steps is represented here by a kernel for which a uniform asymptotic expansion is found. The associated algorithm provides a fundamental improvement of the parabolic-equation and phase-shift/phase-screen style methods applied in ocean acoustics, integrated optics, and exploration seismology.