On the Central Limit Theorem for Toeplitz Quadratic Forms of Stationary Sequences∗
نویسندگان
چکیده
Abstract. Let X(t), t = 0,±1, . . . , be a real-valued stationary Gaussian sequence with a spectral density function f(λ). The paper considers the question of applicability of the central limit theorem (CLT) for a Toeplitz-type quadratic form Qn in variables X(t), generated by an integrable even function g(λ). Assuming that f(λ) and g(λ) are regularly varying at λ = 0 of orders α and β, respectively, we prove the CLT for the standard normalized quadratic form Qn in a critical case α+ β = 1 2 . We also show that the CLT is not valid under the single condition that the asymptotic variance of Qn is separated from zero and infinity.
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