Limit theorems for Toeplitz quadratic functionals of continuous-time stationary processes

Let X(t), t is an element of R, be a centered real-valued stationary Gaussian process with spectral density f (lambda). 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(lambda). Sufficient conditions in terms of f (lambda) and g(lambda) ensuring central limit theorems for standard normalized quadratic functionals Q(T) are obtained, extending the results of Fox and Taqqu (Prob. Theory Relat. Fields 74: 213-240, 1987), Avram (Prob. Theory Relat. Fields 79:37-45, 1988), Giraitis and Surgailis (Prob. Theory Relat. Fields 86: 87-104, 1990), Ginovian and Sahakian (Theory Prob. Appl. 49:612-628, 2004) for discrete time processes.