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The trace approximation problem for Toeplitz matrices and its applications tostationary processes dates back to the classic book by Grenander and Szeg\"o,"Toeplitz forms and their applications". It has then been extensively studiedin the literature.  In this paper we provide a survey and unified treatment of the traceapproximation problem both for Toeplitz matrices and for operators and describeapplications to discrete- and continuous-time stationary processes.  The trace approximation problem serves indeed as a tool to study manyprobabilistic and statistical topics for stationary models. These includecentral and non-central limit theorems and large deviations of Toeplitz typerandom quadratic functionals, parametric and nonparametric estimation,prediction of the future value based on the observed past of the process, etc.  We review and summarize the known results concerning the trace approximationproblem, prove some new results, and provide a number of applications todiscrete- and continuous-time stationary time series models with various typesof memory structures, such as long memory, anti-persistent and short memory.

The Trace Problem for Toeplitz Matrices and Operators and its Impact in Probability