Abstract
The Large Deviation Principle (LDP) is derived for several quadratic additive functionals of centered stationary Gaussian processes. For example, the rate function corresponding to \(1/T\int {_0^T } X_t^2 dt\) is the Fenchel-Legendre transform of \(L(y) = - (1/4\pi )\int {_{ - \infty }^\infty } \log (1 - 4\pi yf(s))ds\) where X t is a continuous time process with the bounded spectral density f(s). This spectral density condition is strictly weaker than the one necessary for the LDP to hold for all bounded continuous functionals. Similar results are obtained for the energy of multivariate discrete-time Gaussian processes and in the regime of moderate deviations, the latter yielding the corresponding Central Limit Theorems.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
REFERENCES
Avram, F. (1988). On bilinear forms in Gaussian random variables and Toeplitz matrices, Prob. Th. Rel. Fields 79, 37–45.
Benitz, G. R., and Bucklew, J. A. (1990). Large deviation rate calculations for nonlinear detectors in Gaussian noise, IEEE Trans. Inform. Th. 36, 358–371.
Bradley, R. C. (1992). On the spectral density and asymptotic normality of weakly dependent random fields, J. Th. Prob. 5, 355–373.
Bryc, W. (1993). A remark on the connection between the large deviation principle and the central limit theorem, Stat. Prob. Lett. 18, 253–256.
Bryc, W., and Dembo, A. (1995). On large deviations of empirical measures for stationary Gaussian processes, Stoch. Proc. Appl. 58, 23–34.
Bryc, W., and Dembo, A. (1996). Large deviations and strong mixing, Ann. Inst. Henri Poincaré 32, 549–569.
Bryc, W., and Dembo, A. (1993). Large Deviations for quadratic functionals of Gaussian processes (expanded version) IMA Preprint Series #1179.
Bryc, W., and Smolenski, W. (1993). On the large deviation principle for a quadratic functional of the autoregressive process, Stat. Prob. Lett. 17, 281–285.
Bucklew, J. A. (1990). Large Deviations Techniques in Decision, Simulation, and Estimation, John Wiley.
Bucklew, J. A., and Sadowsky, J. S. (1993). A contribution to the theory of Chernoff's bounds, IEEE Trans. Inform. Th. 39, 249–254.
Coursol, J., and Dacunha-Castelle, D. (1979). Sur la formule de Chernoff pour deux processus gaussiens stationnaires, C. R. Acad. Sci. Paris, Ser. A–B 288, A769–A770.
Dembo, A., and Zeitouni, O. (1993). Large Deviations Technique and Applications, Jones and Barlett, Boston, Massachusetts.
Dembo, A., and Zeitouni, O. (1995). Large deviations via parameter dependent change of measure and an application to the lower tail of Gaussian processes. In Bolthausen, E., Dozzi, M., and Russo, F. (eds.), Progress in Probability, Vol. 36 Birkhauser, pp. 111–121.
Deuschel, J. (1989). Invariance principle and empirical mean large deviations of the critical Ornstein-Uhlenbeck process, Ann. Prob. 17, 74–90.
Donsker, M. D., and Varadhan, S. R. S. (1985). Large deviations for stationary Gaussian processes, Commun. Math. Phys. 97, 187–210.
Dunford, N., and Schwartz, J. T. (1958–1971). Linear Operators, Interscience Publishers, New York.
Fox, R., and Taqqu, M. S. (1987). Central Limit Theorems for quadratic forms in random variables having long-range dependence, Prob. Th. Rel. Fields 74, 213–240.
Grenander, U., and Szegö, G. (1958). Toeplitz Forms and Their Applications, University of California Press.
Koopmans, T. (1942). Serial correlation and quadratic forms in normal variables, Ann. Math. Stat. 13, 14–33.
Mazo, J. E., and Salz, J. (1965). Probability of error for quadratic detectors, Bell Syst. Tech. J. XLIV, 2165–2186.
Roenblatt, M. (1985b). Stationary Sequences and Random Fields, Birkhäuser, Boston.
Steinberg, Y., and Zeitouni, O. (1992). On tests for normality, IEEE Trans. Infor. Th. 38, 1779–1787.
Terrin, N., and Taqqu, M. S. (1990). A noncentral limit theorem for quadratic forms of Gaussian stationary sequences, J. Th. Prob. 3, 449–475.
Rights and permissions
About this article
Cite this article
Bryc, W., Dembo, A. Large Deviations for Quadratic Functionals of Gaussian Processes. Journal of Theoretical Probability 10, 307–332 (1997). https://doi.org/10.1023/A:1022656331883
Issue Date:
DOI: https://doi.org/10.1023/A:1022656331883