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Sharp large deviations for Gaussian quadratic forms with applications. (English) Zbl 0939.60013
Under regularity assumptions, we establish a sharp large deviation principle for Hermitian quadratic forms of stationary Gaussian processes. Our result is similar to the well-known Bahadur-Rao theorem [R. R. Bahadur and R. R. Rao, Ann. Math. Stat. 31, 1015-1027 (1960; Zbl 0101.12603)] on the sample mean. We also provide several examples of application such as the sharp large deviation properties of the Neyman-Pearson likelihood ratio test, of the sum of squares, of the Yule-Walker estimator of the parameter of a stable autoregressive Gaussian process, and finally of the empirical spectral repartition function.

MSC:
60F10 Large deviations
47B35 Toeplitz operators, Hankel operators, Wiener-Hopf operators
11E25 Sums of squares and representations by other particular quadratic forms
60G15 Gaussian processes
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References:
[1] Azencott R. and Dacunha-Castelle D., Séries d’observations irrégulières. Masson ( 1984). Zbl0546.62060 MR746133 · Zbl 0546.62060
[2] Bahadur R. and Ranga Rao R., On deviations of the sample mean. Ann. Math. Statist. 31 ( 1960) 1015-1027. Zbl0101.12603 MR117775 · Zbl 0101.12603
[3] Barndoff-Nielsen O.E. and Cox D.R., Asymptotic techniques for uses in statistics. Chapman and Hall, Londres ( 1989). Zbl0672.62024 MR1010226 · Zbl 0672.62024
[4] Barone P., Gigli A. and Piccioni M., Optimal importance sampling for some quadratic forms of A.R.M.A. processes. IEEE Trans. Inform. Theory 41 ( 1995) 1834-1844. Zbl0845.62064 MR1385583 · Zbl 0845.62064
[5] Basor E., A localization theorem for Toeplitz determinants. Indiana Univ. Math. J. 28 ( 1979) 975-983. Zbl0396.47018 MR551161 · Zbl 0396.47018
[6] Basor E., Asymptotic formulas for Toeplitz and Wiener-Hopf operators. Integral Equations Operator Theory 5 ( 1982) 659-665. Zbl0519.47014 MR697009 · Zbl 0519.47014
[7] Bercu B., Gamboa F. and Rouault A., Large deviations for quadratic forms of stationary Gaussian processes. Stochastic Process. Appl. 71 ( 1997) 75-90. Zbl0941.60050 MR1480640 · Zbl 0941.60050
[8] Book S.A., Large deviation probabilities for weighted sums. Ann. Math. Statist. 43 ( 1972) 1221-1234. Zbl0243.60020 MR331486 · Zbl 0243.60020
[9] Bottcher A. and Silbermann. Analysis of Toeplitz operators. Springer, Berlin ( 1990). Zbl0732.47029 MR1071374 · Zbl 0732.47029
[10] Bouaziz M., Testing Gaussian sequences and asymptotic inversion of Toeplitz operators. Probab. Math. Statist. 14 ( 1993) 207-222. Zbl0819.62038 MR1321761 · Zbl 0819.62038
[11] Bryc W. and Dembo A., Large deviations for quadratic functionals of Gaussian processes. J. Theoret. Probab. 10 ( 1997) 307-332. Zbl0894.60026 MR1455147 · Zbl 0894.60026
[12] Bryc W. and Smolenski W., On large deviation principle for a quadratic functional of the autoregressive process. Statist. Probab. Lett. 17 ( 1993) 281-285. Zbl0786.60025 MR1237769 · Zbl 0786.60025
[13] Bucklew J.A., Large deviations techniques in decision, simulation, and estimation. Wiley ( 1990). MR1067716
[14] Bucklew J. and Sadowsky J., A contribution to the theory of Chernoff bounds. IEEE Trans. Inform. Theory 39 ( 1993) 249-254. Zbl0765.60017 MR1211504 · Zbl 0765.60017
[15] Coursol J. and Dacunha-Castelle D., Sur la formule de Chernoff pour deux processus gaussiens stationnaires. C. R. Acad. Sci. Sér. I Math. 288 ( 1979) 769-770. Zbl0397.62056 MR535808 · Zbl 0397.62056
[16] Cramér H., Random variables and probability distributions. Cambridge University Press ( 1970). Zbl0184.40101 MR254895 JFM63.1080.01 · Zbl 0184.40101
[17] Dacunha-Castelle D., Remarque sur l’étude asymptotique du rapport de vraisemblance de deux processus gaussiens. C. R. Acad. Sci. Sér. I Math. 288 ( 1979) 225-228. Zbl0397.62055 MR525929 · Zbl 0397.62055
[18] Dembo A. and Zeitouni O., Large deviations techniques and applications. Jones and Barblett Pub. Boston ( 1993). Zbl0793.60030 MR1202429 · Zbl 0793.60030
[19] Esseen C., Fourier analysis of distribution functions. Acta Math. 77 ( 1945) 1-25. Zbl0060.28705 MR14626 · Zbl 0060.28705
[20] Gamboa F. and Gassiat E., Sets of superresolution and the maximum entropy method on the mean. SIAM J. Math. Anal. 27 ( 1996) 1129-1152. Zbl0936.62005 MR1393430 · Zbl 0936.62005
[21] Gamboa F. and Gassiat E., Bayesian methods for ill posed problems. Ann. Statist. 25 ( 1997) 328-350. Zbl0871.62010 MR1429928 · Zbl 0871.62010
[22] Golinskii B. and Ibragimov I., On Szegös limit theorem. Math. USSR- Izv. 5 ( 1971) 421-444. Zbl0249.42012 · Zbl 0249.42012
[23] Grenander V. and Szegö G., Toeplitz forms and their applications. University of California Press ( 1958). Zbl0080.09501 MR94840 · Zbl 0080.09501
[24] Guyon X., Random fields on a network/ modeling, statistics and applications. Springer ( 1995). Zbl0839.60003 MR1344683 · Zbl 0839.60003
[25] Hartwig R.E. and Fisher M.E., Asymptotic behavior of Toeplitz matrices and determinants. Arch. Rational Mech. Anal. 32 ( 1969) 190-225. Zbl0169.04403 MR236593 · Zbl 0169.04403
[26] Howland J., Trace class Hankel operators. Quart. J. Math. Oxford Ser. (2) 22 ( 1971) 147-159. Zbl0216.41903 MR288630 · Zbl 0216.41903
[27] Jensen J.L., Saddlepoint Approximations. Oxford Statist. Sci. Ser. 16 ( 1995).
[28] Johansson K., On Szegös asymptotic formula for Toeplitz determinants and generalizations. Bull. Sci. Math. 112 ( 1988) 257-304. Zbl0661.30001 MR975365 · Zbl 0661.30001
[29] Lavielle M., Detection of changes in the spectrum of a multidimensional process. IEEE Trans. Signal Process. 42 ( 1993) 742-749. Zbl0825.93760 · Zbl 0825.93760
[30] Lehmann E.L., Testing statistical hypotheses. John Wiley and Sons, New-York ( 1959). Zbl0089.14102 MR107933 · Zbl 0089.14102
[31] Rudin W., Real and complex analysis. McGraw Hill International Editions ( 1987). Zbl0925.00005 MR924157 · Zbl 0925.00005
[32] Taniguchi M., Higher order asymptotic theory for time series analysis. Springer, Berlin ( 1991). Zbl0729.62086 MR1177599 · Zbl 0729.62086
[33] Widom H., On the limit block Toeplitz determinants. Proc. Amer. Math. Soc. 50 ( 1975) 167-173. Zbl0312.47027 MR370254 · Zbl 0312.47027
[34] Widom H., Asymptotic behavior of block Toeplitz matrices and determinants IIAdv. Math. 21 ( 1976). Zbl0344.47016 MR409512 · Zbl 0344.47016
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