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

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.