Limit Theorems for Tapered Toeplitz Quadratic Functionals of Continuous-time Gaussian Stationary Processes

Let { X ( t ), t ∈ ℝ} be a centered real-valued stationary Gaussian process with spectral density f . The paper considers a question concerning asymptotic distribution of tapered Toeplitz type quadratic functional Q T h of the process X ( t ), generated by an integrable even function g and a taper function h . Sufficient conditions in terms of functions f , g and h ensuring central limit theorems for standard normalized quadratic functionals Q T h are obtained, extending the results of Ginovyan and Sahakyan (Probability Theory and Related Fields 138 , 551–579, 2007) to the tapered case and sharpening the results of Ginovyan and Sahakyan (Electronic Journal of Statistics 13 , 255–283, 2019) for the Gaussian case.