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.
Highlights
AMS 2000 subject classifications: Primary 60G10, 62G20; secondary 47B35, 15B05
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
The present work is devoted to the problem of approximation of the traces of products of truncated Toeplitz matrices and operators generated by integrable real symmetric functions defined on the unit circle
Summary
We first define the main objects to be studied in this work, namely the truncated Toeplitz matrices and operators, generated by integrable real symmetric functions. Given a number T > 0 and an integrable real symmetric function h(λ) defined on R := (−∞, ∞), the T -truncated Toeplitz operator generated by h(λ), denoted by WT (h), is defined by the following equation (see, e.g., [33, 40, 45]): T [WT (h)u](t) = h(t − s)u(s)ds, u(s) ∈ L2[0, T ],. For a given number T > 0, let AT (hk) denote either the (T × T )-truncated Toeplitz matrix (BT (hk)), or the T -truncated Toeplitz operator (WT (hk)) generated by function hk, defined respectively by (1.1) and (1.2) with hk instead of h. All functions defined on T are assumed to be 2π-periodic and periodically extended to R
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