A note on central limit theorem for Toeplitz type quadratic forms in stationary Gaussian variables
Define \(L_n = \sum^n_{k,j = 1} a_{k - j} X(k) X(j)\) with \(a_k = \int^\pi_{-\pi} e^{i \lambda k} g(\lambda) d \lambda\), \(k = 0\), \(\pm 1, \dots\), where \(g:\) \([- \pi, \pi] \to \mathbb{R}\) is even and integrable and \(X(t)\), \(t = 0\), \(\pm 1, \dots\), is a mean zero stationary Gaussian sequence possessing a spectral density \(f (\lambda)\), \(\lambda \in [- \pi, \pi]\), \(f(- \lambda) = f (\lambda)\). Under either of the following conditions: \[ f(x) \in L_{p_1},\;g(x) \in L_{p_2} \quad \text{where} \quad {1 \over p_1} + {1 \over p_2} \leq {1 \over 2}; \tag{1} \] \[ f(x) < c |x |^{-\alpha},\;\bigl |g(x) \bigr |< c |x |^{- \beta} \quad \text{where} \quad \alpha, \beta < 1,\;\alpha + \beta < {1 \over 2}, \tag{2} \] \(n^{- 1/2} (L_n - {\mathcal E} L_n)\) converges in distribution to the \(N(0, \sigma^2)\) distribution, where \[ \sigma^2 = 16 \pi^3 \int^\pi_{-\pi} f^2 (\lambda) g^2 (\lambda) d \lambda. \]
scientific article; zbMATH DE number 795440