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

Abstract

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.