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 Probability Surveys > Vol. 11 (2014) open journal systems 


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

Mamikon S. Ginovyan, Boston University
Artur A. Sahakyan, Yerevan State University
Murad S. Taqqu, Boston University


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, antipersistent and short memory.

AMS 2000 subject classifications: Primary 60G10, 62G20; secondary 47B35, 15B05.

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

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Ginovyan, Mamikon S., Sahakyan, Artur A., Taqqu, Murad S., The trace problem for Toeplitz matrices and operators and its impact in probability, Probability Surveys, 11, (2014), 393-440 (electronic). DOI: 10.1214/13-PS217.

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