On large deviations formulas for quadratic functions of Gaussian random variables (Q2722273)

The method of large deviations proposed by \textit{H. Cramer} [Actuar. Sci. Indust. 736, 5-23 (1938)] for the central limit theorem, was essentially generalized by \textit{R. S. Ellis} [``Entropy, large deviations, and statistical mechanics'' (1985; Zbl 0566.60097)]. The Ellis approach is used to investigate the large deviations for a sequence of the random variables of the form \(Z_{n}=\sum_{i,k=-n}^{n}(a^{n}_{i,k}-\delta_{i-k})X_{i}X_{k}\), where \((X_{k}, k=0,\pm 1,\pm 2,...)\) is a sequence of independent Gaussian random variables with \(EX_{k}=0, EX_{k}^{2}=1\), and \(a^{n}_{i,k}\) are elements of the matrix \(B^{-1}_{n},\) where \(B_{n}\) is a \((2n+1)\times (2n+1)\) matrix with the elements \(b_{i,k}=b_{i-k}\) and \(\{b_{k}, k=0,\pm 1,\dots\}\) is the correlation function of a stationary stochastic process with discrete time, \(\delta_{0}=1, \delta_{k}=0\) for \(k\not=0.\) The result is applied for testing a hypothesis on the correlation function of a Gaussian stationary stochastic process with discrete time.