Probability Surveys

The trace problem for Toeplitz matrices and operators and its impact in probability

Mamikon S. Ginovyan, Artur A. Sahakyan, and Murad S. Taqqu

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Abstract

The trace approximation problem for Toeplitz matrices and its applications to stationary processes dates back to the classic book by Grenander and Szegö, 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.

Article information

Source
Probab. Surveys, Volume 11 (2014), 393-440.

Dates
First available in Project Euclid: 2 December 2014

Permanent link to this document
https://projecteuclid.org/euclid.ps/1417528609

Digital Object Identifier
doi:10.1214/13-PS217

Mathematical Reviews number (MathSciNet)
MR3290440

Zentralblatt MATH identifier
1348.60054

Subjects
Primary: 60G10: Stationary processes 62G20: Asymptotic properties
Secondary: 47B35: Toeplitz operators, Hankel operators, Wiener-Hopf operators [See also 45P05, 47G10 for other integral operators; see also 32A25, 32M15] 15B05: Toeplitz, Cauchy, and related matrices

Keywords
Stationary process spectral density long-memory central limit theorem Toeplitz operator trace approximation singularity

Citation

Ginovyan, Mamikon S.; Sahakyan, Artur A.; Taqqu, Murad S. The trace problem for Toeplitz matrices and operators and its impact in probability. Probab. Surveys 11 (2014), 393--440. doi:10.1214/13-PS217. https://projecteuclid.org/euclid.ps/1417528609


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