On the trace approximations of products of Toeplitz matrices
TL;DR: In this paper, error orders for integral limit approximations to the traces of products of Toeplitz matrices generated by integrable real symmetric functions defined on the unit circle are established.
About: This article is published in Statistics & Probability Letters.The article was published on 2013-03-01 and is currently open access. It has received 9 citations till now. The article focuses on the topics: Toeplitz matrix & Levinson recursion.
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TL;DR: The trace approximation problem for Toeplitz matrices and its applications to stationary processes dates back to the classic book by Grenander and Szego as discussed by the authors, which describes applications to discrete-and continuous-time stationary processes.
Abstract: The trace approximation problem for Toeplitz matrices and its applications to stationary processes dates back to the classic book by Grenander and Szego, Toeplitz forms and their applications (University of California Press, Berkeley, 1958). It has then been extensively studied in the literature. In this paper we provide a survey and unified treatment of the trace approximation problem both for Toeplitz matrices and for operators and describe applications to discrete- and continuous-time stationary processes. The trace approximation problem serves indeed as a tool to study many probabilistic and statistical topics for stationary models. These include central and non-central limit theorems and large deviations of Toeplitz type random quadratic functionals, parametric and nonparametric estimation, prediction of the future value based on the observed past of the process, hypotheses testing about the spectrum, etc. We review and summarize the known results concerning the trace approximation problem, prove some new results, and provide a number of applications to discrete- and continuous-time stationary time series models with various types of memory structures, such as long memory, anti-persistent and short memory.
9 citations
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TL;DR: This paper provides a survey and unified treatment of the trace approximation problem both for Toeplitz matrices and for operators and describes applications to discrete- and continuous-time stationary processes.
Abstract: The trace approximation problem for Toeplitz matrices and its applications to stationary processes dates back to the classic book by Grenander and Szeg\"o, "Toeplitz forms and their applications". It has then been extensively studied in the literature.
In this paper we provide a survey and unified treatment of the trace approximation problem both for Toeplitz matrices and for operators and describe applications to discrete- and continuous-time stationary processes.
The trace approximation problem serves indeed as a tool to study many probabilistic and statistical topics for stationary models. These include central and non-central limit theorems and large deviations of Toeplitz type random quadratic functionals, parametric and nonparametric estimation, prediction of the future value based on the observed past of the process, etc.
We review and summarize the known results concerning the trace approximation problem, prove some new results, and provide a number of applications to discrete- and continuous-time stationary time series models with various types of memory structures, such as long memory, anti-persistent and short memory.
9 citations
TL;DR: In this paper , the authors investigated error orders for integral limit approximations to traces of products of Toeplitz matrices generated by integrable functions on [ − π, π ] having some singularities at the origin.
Abstract: We investigate error orders for integral limit approximations to traces of products of Toeplitz matrices generated by integrable functions on [ − π, π ] having some singularities at the origin. Even though a sharp error order of the above approximation is derived in Theorem 2 of [7], its proof contains an inaccuracy as pointed out by [4]. In the present paper, we reinvestigate the claim given in Theorem 2 of [7] and give an alternative proof of their claim.
1 citations
TL;DR: The analysis of the trace of a product of two matrices in the case where one of them is the inverse of a given positive definite matrix while the other is nonnegative definite is studied in this article.
Abstract: This short note is devoted to the analysis of the trace of a product of two matrices in the case where one of them is the inverse of a given positive definite matrix while the other is nonnegative definite. In particular, a relation between the trace of A –1 H and the values of diagonal elements of the original matrix A is analysed. MSC 2010: 15A09, 15A42, 15A63
Cites background from "On the trace approximations of prod..."
...They also arise naturally in the applications of mathematical statistics, especially in regression analysis, [1-3] or the analysis of discrete-time stationary processes [4]....
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Peer Review•
TL;DR: A survey of recent results on central and non-central limit theorems for quadratic functionals of stationary processes can be found in this paper , where the underlying processes are Gaussian, linear or Lévy-driven linear processes with memory, and are defined either in discrete or continuous time.
Abstract: This is a survey of recent results on central and non-central limit theorems for quadratic functionals of stationary processes. The underlying processes are Gaussian, linear or Lévy-driven linear processes with memory, and are defined either in discrete or continuous time. We focus on limit theorems for Toeplitz and tapered Toeplitz type quadratic functionals of stationary processes with applications in parametric and nonparametric statistical estimation theory. We discuss questions concerning Toeplitz matrices and operators, Fejér-type singular integrals, and Lévy-Itô-type and Stratonovich-type multiple stochastic integrals. These are the main tools for obtaining limit theorems. MSC2020 subject classifications: Primary 60F05, 60G10, 60G15; secondary 62F12, 62G05.
References
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TL;DR: In this article, Toeplitz forms are used for the trigonometric moment problem and other problems in probability theory, analysis, and statistics, including analytic functions and integral equations.
Abstract: Part I: Toeplitz Forms: Preliminaries Orthogonal polynomials. Algebraic properties Orthogonal polynomials. Limit properties The trigonometric moment problem Eigenvalues of Toeplitz forms Generalizations and analogs of Toeplitz forms Further generalizations Certain matrices and integral equations of the Toeplitz type Part II: Applications of Toeplitz Forms: Applications to analytic functions Applications to probability theory Applications to statistics Appendix: Notes and references Bibliography Index.
2,279 citations
Book•
01 Jan 1971
TL;DR: In this article, the authors propose a method for approximating by Singular Integrals of Periodic Functions using Fourier Transform Transform Transformions of Derivatives (FTDFs).
Abstract: 0 Preliminaries.- 0 Preliminaries.- 0.1 Fundamentals on Lebesgue Integration.- 0.2 Convolutions on the Line Group.- 0.3 Further Sets of Functions and Sequences.- 0.4 Periodic Functions and Their Convolution.- 0.5 Functions of Bounded Variation on the Line Group.- 0.6 The Class BV2?.- 0.7 Normed Linear Spaces, Bounded Linear Operators.- 0.8 Bounded Linear Functional, Riesz Representation Theorems.- 0.9 References.- I Approximation by Singular Integrals.- 1 Singular Integrals of Periodic Functions.- 1.0 Introduction.- 1.1 Norm-Convergence and-Derivatives.- 1.1.1 Norm-Convergence.- 1.1.2 Derivatives.- 1.2 Summation of Fourier Series.- 1.2.1 Definitions.- 1.2.2 Dirichlet and Fejer Kernel.- 1.2.3 Weierstrass Approximation Theorem.- 1.2.4 Summability of Fourier Series.- 1.2.5 Row-Finite ?-Factors.- 1.2.6 Summability of Conjugate Series.- 1.2.7 Fourier-Stieltjes Series.- 1.3 Test Sets for Norm-Convergence.- 1.3.1 Norms of Some Convolution Operators.- 1.3.2 Some Applications of the Theorem of Banaeh-Steinhaus.- 1.3.3 Positive Kernels.- 1.4 Pointwise Convergence.- 1.5 Order of Approximation for Positive Singular Integrals.- 1.5.1 Modulus of Continuity and Lipschitz Classes.- 1.5.2 Direct Approximation Theorems.- 1.5.3 Method of Test Functions.- 1.5.4 Asymptotic Properties.- 1.6 Further Direct Approximation Theorems, Nikolski? Constants.- 1.6.1 Singular Integral of Fejer-Korovkin.- 1.6.2 Further Direct Approximation Theorems.- 1.6.3 Nikolski? Constants.- 1.7 Simple Inverse Approximation Theorems.- 1.8 Notes and Remarks.- 2 Theorems of Jackson and Bernstein for Polynomials of Best Approximation and for Singular Integrals.- 2.0 Introduction.- 2.1 Polynomials of Best Approximation.- 2.2 Theorems of Jackson.- 2.3 Theorems of Bernstein.- 2.4 Various Applications.- 2.5 1.- 4.2.1 The Case p = 2.- 4.2.2 The Case p ? 2.- 4.3 Finite Fourier-Stieltjes Transforms.- 4.3.1 Fundamental Properties.- 4.3.2 Inversion Theory.- 4.3.3 Fourier-Stieltjes Transforms of Derivatives.- 4.4 Notes and Remarks.- 5 Fourier Transforms Associated with the Line Group.- 5.0 Introduction.- 5.1 L1-Theory.- 5.1.1 Fundamental Properties.- 5.1.2 Inversion Theory.- 5.1.3 Fourier Transforms of Derivatives.- 5.1.4 Derivatives of Fourier Transforms, Moments of Positive Functions Peano and Riemann Derivatives.- 5.1.5 Poisson Summation Formula.- 5.2 Lp-Theory, 1 < p ? 2.- 5.2.1 The Case p = 2.- 5.2.2 The Case 1 2.- 5.2.3 Fundamental Properties.- 5.2.4 Summation of the Fourier Inversion Integral.- 5.2.5 Fourier Transforms of Derivatives.- 5.2.6 Theorem of Plancherel.- 5.3 Fourier-Stieltjes Transforms.- 5.3.1 Fundamental Properties.- 5.3.2 Inversion Theory.- 5.3.3 Fourier-Stieltjes Transforms of Derivatives.- 5.4 Notes and Remarks.- 6 Representation Theorems.- 6.0 Introduction.- 6.1 Necessary and Sufficient Conditions.- 6.1.1 Representation of Sequences as Finite Fourier or Fourier-Stieltjes Transforms.- 6.1.2 Representation of Functions as Fourier or Fourier-Stieltjes Transforms.- 6.2 Theorems of Bochner.- 6.3 Sufficient Conditions.- 6.3.1 Quasi-Convexity.- 6.3.2 Representation as L1/2? Transform.- 6.3.3 Representation as L1-Transform.- 6.3.4 A Reduction Theorem.- 6.4 Applications to Singular Integrals.- 6.4.1 General Singular Integral of Weierstrass.- 6.4.2 Typical Means.- 6.5 Multipliers.- 6.5.1 Multipliers of Classes of Periodic Functions.- 6.5.2 Multipliers on LP.- 6.6 Notes and Remarks.- 7 Fourier Transform Methods and Second-Order Partial Differential Equations.- 7.0 Introduction.- 7.1 Finite Fourier Transform Method.- 7.1.1 Solution of Heat Conduction Problems.- 7.1.2 Dirichlet's and Neumann's Problem for the Unit Disc.- 7.1.3 Vibrating String Problems.- 7.2 Fourier Transform Method in L1.- 7.2.1 Diffusion on an Infinite Rod.- 7.2.2 Dirichlet's Problem for the Half-Plane.- 7.2.3 Motion of an Infinite String.- 7.3 Notes and Remarks.- III Hilbert Transforms.- 8 Hilbert Transforms on the Real Line.- 8.0 Introduction.- 8.1 Existence of the Transform.- 8.1.1 Existence Almost Everywhere.- 8.1.2 Existence in L2-Norm.- 8.1.3 Existence in Lp-Norm, 1 ?.- 8.2 Hilbert Formulae, Conjugates of Singular Integrals, Iterated Hilbert Transforms.- 8.2.1 Hilbert Formulae.- 8.2.2 Conjugates of Singular Integrals: 1 ?.- 8.2.3 Conjugates of Singular Integrals: p = 1.- 8.2.4 Iterated Hilbert Transforms.- 8.3 Fourier Transforms of Hilbert Transforms.- 8.3.1 Signum Rule.- 8.3.2 Summation of Allied Integrals.- 8.3.3 Fourier.- 8.3.4 Norm-Convergence of the Fourier Inversion Integral.- 8.4 Notes and Remarks.- 9 Hilbert Transforms of Periodic Functions.- 9.0 Introduction.- 9.1 Existence and Basic Properties.- 9.1.1 Existence.- 9.1.2 Hilbert Formulae.- 9.2 Conjugates of Singular Integrals.- 9.2.1 The Case 1 ?.- 9.2.2 Convergence in C2? and L1/2?.- 9.3 Fourier Transforms of Hilbert Transforms.- 9.3.1 Conjugate Fourier Series.- 9.3.2 Fourier Transforms of Derivatives of Conjugate Functions, the Classes (W~)xr2?'(V~)rx2?.- 9.3.3 Norm-Convergence of Fourier Series.- 9.4 Notes and Remarks.- IV Characterization of Certain Function Classes 355.- 10 Characterization in the Integral Case.- 10.0 Introduction.- 10.1 Generalized Derivatives, Characterization of the Classes Wrx2?.- 10.1.1 Riemann Derivatives in X2?-Norm.- 10.1.2 Strong Peano Derivatives.- 10.1.3 Strong and Weak Derivatives, Weak Generalized Derivatives.- 10.2 Characterization of the Classes Vr2?.- 10.3 Characterization of the Classes (V~)rx2?.- 10.4 Relative Completion.- 10.5 Generalized Derivatives in Lp-Norm and Characterizations for 1 ? p ?2.- 10.6 Generalized Derivatives in X(R)-Norm and Characterizations of the Classes Wrx(R) and Vrx(R).- 10.7 Notes and Remarks.- 11 Characterization in the Fractional Case.- 11.0 Introduction.- 11.1 Integrals of Fractional Order.- 11.1.1 Integral of Riemann-Liouville.- 11.1.2 Integral of M. Riesz.- 11.2 Characterizations of the Classes W[LP |?|?], V[LP |?|?], 1 ? p ? 2.- 11.2.1 Derivatives of Fractional Order.- 11.2.2 Strong Riesz Derivatives of Higher Order, the Classes V[LP |?|? ].- 11.3 The Operators R?{?} on Lp 1 ? p ? 2.- 11.3.1 Characterizations.- 11.3.2 Theorems of Bernstein-Titchmarsh and H. Weyl.- 11.4 The Operators R?(?} on 2?.- 11.5 Integral Representations, Fractional Derivatives of Periodic Functions.- 11.6 Notes and Remarks.- V Saturation Theory.- 12 Saturation for Singular Integrals on X2? and Lp, 1 ? p ? 2 433.- 12.0 Introduction.- 12.1 Saturation for Periodic Singular Integrals, Inverse Theorems.- 12.2 Favard Classes.- 12.2.1 Positive Kernels.- 12.2.2 Uniformly Bounded Multipliers.- 12.2.3 Functional Equations.- 12.3 Saturation in Lp, 1 ? p ? 2.- 12.3.1 Saturation Property.- 12.3.2 Characterizations of Favard Classes: p = 1.- 12.3.3 Characterizations of Favard Classes: 1 < p? 2.- 12.4 Applications to Various Singular Integrals.- 12.4.1 Singular Integral of Fejer.- 12.4.2 Generalized Singular Integral of Picard.- 12.4.3 General Singular Integral of Weierstrass.- 12.4.4 Singular Integral of Bochner-Riesz.- 12.4.5 Riesz Means.- 12.5 Saturation of Higher Order.- 12.5.1 Singular Integrals on the Real Line.- 12.5.2 Periodic Singular Integrals.- 12.6 Notes and Remarks.- 13 Saturation on X(R).- 13.0 Introduction.- 13.1 Saturation of D?(f x t) in X(R), Dual Methods.- 13.2 Applications to Approximation in Lp, 2 ?.- 13.2.1 Differences.- 13.2.2 Singular Integrals Satisfying (12.3.5).- 13.2.3 Strong Riesz Derivatives.- 13.2.4 The Operators R?{?}.- 13.2.5 Riesz and Fejer Means.- 13.3 Comparison Theorems.- 13.3.1 Global Divisibility.- 13.3.2 Local Divisibility.- 13.3.3 Special Comparison Theorems with no Divisibility Hypothesis.- 13.3.4 Applications to Periodic Continuous Functions.- 13.4 Saturation on Banach Spaces.- 13.4.1 Strong Approximation Processes.- 13.4.2 Semi-Groups of Operators.- 13.5 Notes and Remarks.- List of Symbols.- Tables of Fourier and Hilbert Transforms.
967 citations
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..., [3]): Lip(p, γ) = {ψ(λ) ∈ L(T); ωp(ψ; δ) = O(δ ), δ → 0}....
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TL;DR: Asymptotic normality of the maximum likelihood estimator for the parameters of a long range dependent Gaussian process is proved in this paper, where the limit of the Fisher information matrix is derived for such processes which implies efficiency of the estimator.
Abstract: Asymptotic normality of the maximum likelihood estimator for the parameters of a long range dependent Gaussian process is proved. Furthermore, the limit of the Fisher information matrix is derived for such processes which implies efficiency of the estimator and of an approximate maximum likelihood estimator studied by Fox and Taqqu. The results are derived by using asymptotic properties of Toeplitz matrices and an equicontinuity property of quadratic forms.
891 citations
TL;DR: In this paper, a central limit theorem for quadratic forms in strongly dependent linear (or moving average) variables is proved, generalizing the results of Avram [1] and Fox and Taqqu [3] for Gaussian variables.
Abstract: A central limit theorem for quadratic forms in strongly dependent linear (or moving average) variables is proved, generalizing the results of Avram [1] and Fox and Taqqu [3] for Gaussian variables. The theorem is applied to prove asymptotical normality of Whittle's estimate of the parameter of strongly dependent linear sequences.
384 citations
"On the trace approximations of prod..." refers background in this paper
..., [14], [12], [15], [5], [1], [6], [4], [9], [16], [13], [7], [8], and references therein)....
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..., [11], [5], [1], [4], [9], [2], [16], [13], [7]), [8])....
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...Assertion (A2) was proved in [9] (see, also, [8])....
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