In this paper, we survey some recent results on parametric and nonparametric statistical estimation about the spectrum of stationary models with tapered data, as well as a question concerning robustness of inferences carried out on a linear stationary process contaminated by a small trend. We also discuss some questions concerning tapered Toeplitz matrices and operators, central limit theorems for tapered Toeplitz-type quadratic functionals, and tapered Fejer-type singular integrals. These are the main tools for obtaining the corres ponding results, and also are of interest in themselves. The processes considered will be discrete-time and continuous-time Gaussian, linear or Levy-driven linear processes with memory.
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