On large deviations formulas for quadratic functions of Gaussian random variables. (English) Zbl 0972.60010

Ibramhalilov, I.; Skorokhod, Anatoly V.

Random Oper. Stoch. Equ. 9, No. 1, 23-28 (2001).

The method of large deviations proposed by H. Cramer [Actuar. Sci. Indust. 736, 5-23 (1938)] for the central limit theorem, was essentially generalized by 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.

Reviewer: A.V.Swishchuk (Kyïv)

MSC:

60F10	Large deviations
60G15	Gaussian processes
62H15	Hypothesis testing in multivariate analysis

Keywords:

large deviations; Gaussian random variables; testing hypothesis; estimates for probabilities

Citations:

Zbl 0566.60097