Let X(t), t = 0, ±1,. .. , be a real-valued stationary Gaussian sequence with a spectral density function f (λ). The paper considers the question of applicability of the central limit theorem (CLT) for a Toeplitz-type quadratic form Qn in variables X(t), generated by an integrable even function g(λ). Assuming that f (λ) and g(λ) are regularly varying at λ = 0 of orders α and β, respectively, we prove the CLT for the standard normalized quadratic form Qn in a critical case α + β = 1 2. We also show that the CLT is not valid under the single condition that the asymptotic variance of Qn is separated from zero and infinity.

Let X(t), t ∈ R, be a centered real-valued stationary Gaussian process with spectral density f (λ). The paper considers a question concerning asymptotic distribution of Toeplitz type quadratic functional Q T of the process X(t), generated by an integrable even function g(λ). Sufficient conditions in terms of f (λ) and g(λ) ensuring central limit theorems for standard normalized quadratic functionals Q T are obtained, extending the results of Fox and Taqqu (Prob. Theory Relat.

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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. 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, 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.