Estimates for the distribution of the supremum of square-Gaussian stochastic processes defined on noncompact sets
Authors:
Yu. V. Kozachenko and T. V. Fedoryanich
Translated by:
Oleg Klesov
Journal:
Theor. Probability and Math. Statist. 73 (2006), 81-97
MSC (2000):
Primary 60G17; Secondary 60G07
DOI:
https://doi.org/10.1090/S0094-9000-07-00683-7
Published electronically:
January 17, 2007
MathSciNet review:
2213843
Full-text PDF Free Access
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Additional Information
Abstract: Estimates for the distribution of the supremum of square-Gaussian stochastic processes defined on $\mathbb {R}^+$ are found in the paper. Using these results, we find estimates for the deviation in the uniform metric between the correlogram and the correlation function of a real stationary Gaussian stochastic process. A criterion for testing a hypothesis concerning the correlation function is also constructed.
References
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- Yu. V. Kozachenko and A. I. Stadnik, Pre-Gaussian processes and convergence in $C(T)$ of estimates for covariance functions, Teor. Veroyatnost. i Mat. Statist. 45 (1991), 54–62 (Russian); English transl., Theory Probab. Math. Statist. 45 (1992), 51–57. MR 1168448
- O. O. Kurchenko, Some inequalities for the Orlicz norm of a linear form of independent pre-Gaussian random variables, Teor. Ĭmovīr. Mat. Stat. 50 (1994), 97–100 (Ukrainian, with Ukrainian summary); English transl., Theory Probab. Math. Statist. 50 (1995), 99–102. MR 1445522
- L. S. Ponomarenko, Estimation of distributions of normal quadratic forms of normally distributed random variables, Teor. Veroyatnost. i Primenen. 30 (1985), no. 3, 545–549 (Russian). MR 805308
- Yurij V. Kozachenko and Olexander V. Stus, Square-Gaussian random processes and estimators of covariance functions, Math. Commun. 3 (1998), no. 1, 83–94 (English, with English and Croatian summaries). MR 1648867
- Yuri Kozachenko and Oksana Moklyachuk, Large deviation probabilities for square-Gaussian stochastic processes, Extremes 2 (1999), no. 3, 269–293 (2000). MR 1781940, DOI https://doi.org/10.1023/A%3A1009907019950
- V. V. Buldygin and Yu. V. Kozachenko, Metric characterization of random variables and random processes, Translations of Mathematical Monographs, vol. 188, American Mathematical Society, Providence, RI, 2000. Translated from the 1998 Russian original by V. Zaiats. MR 1743716
- Yuri Kozachenko and Iryna Rozora, Simulation of Gaussian stochastic processes, Random Oper. Stochastic Equations 11 (2003), no. 3, 275–296. MR 2009187, DOI https://doi.org/10.1163/156939703771378626
- Yu. V. Kozachenko and T. V. Fedoryanich, A criterion for testing hypotheses on the covariance function of Gaussian stationary process, Teor. Ĭmovīr. Mat. Stat. 69 (2003), 79–88 (Ukrainian, with Ukrainian summary); English transl., Theory Probab. Math. Statist. 69 (2004), 85–94 (2005). MR 2110907, DOI https://doi.org/10.1090/S0094-9000-05-00616-2
References
- Yu. V. Kozachenko and T. A. Oleshko, On the distribution of the supremum of square-Gaussian random processes, Teor. Imovir. Mat. Statist. 47 (1992), 55–62; English transl. in Theory Probab. Math. Statist. 47 (1993), 57–64. MR 1272223 (95a:60070)
- Yu. V. Kozachenko and A. I. Stadnik, Pre-Gaussian processes and convergence in $C(T)$ of estimates for covariance functions, Teor. Imovir. Mat. Statist. 45 (1991), 54–62; English transl. in Theory Probab. Math. Statist. 45 (1992), 51–57. MR 1168448 (93d:60065)
- O. O. Kurchenko, Some inequalities for the Orlicz norm of a linear form of independent pre-Gaussian random variables, Teor. Imovir. Mat. Statist. 50 (1994), 97–100; English transl. in Theory Probab. Math. Statist. 50 (1995), 99–102. MR 1445522 (98a:60019)
- L. S. Ponomarenko, On estimating distributions of normalized quadratic forms of normally distributed random variables, Teor. Veroyatnost. i Primenen. 30 (1985), no. 3, 545–549; English transl. in Theory Probab. Appl. 30 (1986), no. 3, 580–584. MR 805308 (87c:60026)
- Yu. V. Kozachenko and O. V. Stus, Square-Gaussian random processes and estimators of covariance functions, Math. Commun. 3 (1998), 83–94. MR 1648867 (2000b:60099)
- Yu. V. Kozachenko and O. Moklyachuk, Large deviation probabilities for square-Gaussian stochastic processes, Statistical Theory and Applications in Science, Engineering and Economics 2 (1999), no. 3, 269–295. MR 1781940 (2001k:60037)
- V. V. Buldygin and Yu. V. Kozachenko, Metric Characterization of Random Variables and Random Processes, TViMS, Kyiv, 1998; English transl., American Mathematical Society, Providence, RI, 2000. MR 1743716 (2001g:60089)
- Yu. Kozachenko and I. Rozora, Simulation of stochastic Gaussian processes, Random Oper. Stochastic Equations 11 (2003), no. 3, 275–296. MR 2009187 (2004i:60050)
- Yu. V. Kozachenko and T. V. Fedoryanich, A criterion for testing hypotheses on the covariance function of Gaussian stationary process, Teor. Imovir. Mat. Stat. 69 (2003), 79–88; English transl. in Theory Probab. Math. Statist. 69 (2004), 85–94. MR 2110907 (2005i:60075)
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Additional Information
Yu. V. Kozachenko
Affiliation:
Department of Probability Theory and Mathematical Statistics, Faculty for Mechanics and Mathematics, National Taras Shevchenko University, Academician Glushkov Avenue 6, Kiev 03127, Ukraine
Email:
yvk@univ.kiev.ua
T. V. Fedoryanich
Affiliation:
Department of Probability Theory and Mathematical Statistics, Faculty for Mechanics and Mathematics, National Taras Shevchenko University, Academician Glushkov Avenue 6, Kiev 03127, Ukraine
Email:
fedoryanich@ukr.net
Keywords:
Square-Gaussian processes,
correlograms,
stationary processes,
a criterion for testing a hypothesis
Received by editor(s):
November 15, 2004
Published electronically:
January 17, 2007
Additional Notes:
The first author was supported in part by the NATO Grant PST.CLG. 980408
Article copyright:
© Copyright 2007
American Mathematical Society
