Abstract
Let X(t), \(t\in\mathbb{R}\), be a centered real-valued stationary Gaussian process with spectral density f(λ). 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(λ). Sufficient conditions in terms of f(λ) and g(λ) 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.
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Ginovyan, M.S., Sahakyan, A.A. Limit theorems for Toeplitz quadratic functionals of continuous-time stationary processes. Probab. Theory Relat. Fields 138, 551–579 (2007). https://doi.org/10.1007/s00440-006-0037-y
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DOI: https://doi.org/10.1007/s00440-006-0037-y