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

摘要

Let {X(t), t is an element of 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.