On the central limit theorem for Toeplitz quadratic forms of stationary sequences. Theory Probability and its Applications

Let X(t), t = 0,±1,..., be a real-valued stationary Gaussian sequence with spectral density function f(λ). The paper considers a question of applicability of central limit theorem (CLT) for 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 CLT for standard normalized quadratic form Qn in the critical case α+ β = 1/2. We also show that CLT is not valid under the single condition that the asymptotic variance of Qn is separated from zero and infinity. Key words and phrases. Stationary Gaussian sequence, spectral density, Toeplitz type quadratic forms, central limit theorem, asymptotic variance, regularly varying func-tions. AMS 2000 subject classifications. Primary 60G10, 60F05; Secondary 60G15