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The trace problem for Toeplitz matrices and operators and its impact in probability. (English) Zbl 1348.60054

Summary: The trace approximation problem for Toeplitz matrices and its applications to stationary processes dates back to the classic book by U. Grenander and G. Szegö [Toeplitz forms and their applications. Angeles: University of California Press (1958; Zbl 0080.09501)]. 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.

MSC:

60G10 Stationary stochastic processes
47B35 Toeplitz operators, Hankel operators, Wiener-Hopf operators
15B05 Toeplitz, Cauchy, and related matrices
60F05 Central limit and other weak theorems
60F10 Large deviations
62G05 Nonparametric estimation
62G20 Asymptotic properties of nonparametric inference

Citations:

Zbl 0080.09501

Software:

longmemo
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References:

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