Abstract
For T > 0, we prove theorems concerning sharp asymptotics of the probabilities
as u → ∞, where wj(t), j = 1, . . . , n, are independent Wiener processes and wj0,T (t), j = 1, . . . , n, are independent Brownian bridges on the segment [0, T]. Our research method is the double sum method for the Gaussian processes and fields. We also give an application of the obtained results to the statistical tests for the homogeneity hypothesis of k one-dimensional samples.
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References
I. G. Abrahamson, “Exact Bahadur efficiencies for the Kolmogorov–Smirnov and Kuiper one- and two-sample statistics,” Ann. Math. Statist., 38, No. 5, 1475–1490 (1967).
Yu. K. Belyaev and V. I. Piterbarg, “The asymptotic behavior of the average number of the A-points of upcrossings of a Gaussian field beyond a high level,” in: Belyaev Yu. K., ed., Upcrossings of Random Fields [in Russian], Izd. Mosk. Univ., Moscow (1972), pp. 62–89.
A. N. Borodin and P. Salminen, Handbook of Brownian Motion [in Russian], Lan’, St.-Petersburg (2000).
V. R. Fatalov, “Asymptotics of large deviations of Gaussian fields,” Izv. AN Armenii. Matem., 27, No. 6, 59–81. (1992).
V. R. Fatalov, “Asymptotics of large deviations of Gaussian fields. Applications,” Izv. AN Armenii. Matem., 28, No. 5, 32–55. (1993).
V. R. Fatalov, “Double sum method for the Gaussian fields with parameter set from lp space,” Fundam. Prikl. Matem., 2, No. 4, 1117–1141. (1996).
V. R. Fatalov, “Large deviations for Gaussian processes in Hölder norm,” Izv. Math., 67, No. 5, 1061–1079 (2003).
V. R. Fatalov, “Occupation times and exact asymptotics of small deviations of Bessel processes for Lp-norms with p > 0,” Izv. Math., 71, No. 4, 721–752 (2007).
V. R. Fatalov, “Ergodic means for large values of T and exact asymptotics of small deviations for a multi-dimensional Wiener process,” Izv. Math., 77, No. 6, 1224–1259 (2013).
N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, North Holland, Amsterdam (1981).
J. L. Kelly, General Topology, Springer, New York (1975).
J. Kiefer, “K-sample analogues of the Kolmogorov–Smirnov and Cramér–v. Mises tests,” Ann. Math. Statist., 30, No. 2, 420–447 (1959).
H. J. Landau and L. A. Shepp, “On the supremum of a Gaussian process,” Sankhyà, A32, No. 4, 369–378 (1971).
M. Leadbetter, G. Lindgren, and H. Rootzen, Extremes and Related Properties of Random Sequences and Processes, Springer, New York (1983).
M. A. Lifshits, Gaussian Random Functions, Kluwer, Dordrecht (1995).
M. B. Marcus and L. A. Shepp, “Continuity of Gaussian processes,” Trans. Am. Math. Soc., 151, 377–392 (1970).
A. S. Mishchenko and A. T. Fomenko, A Course in Differential Geometry and Topology [in Russian], Izd. Mosk. Univ., Moscow (1980).
Y. Nikitin, Asymptotic Efficiency of Nonparametric Tests, Cambridge Univ. Press, Cambridge (1995).
A. A. Novikov, “Small deviations of Gaussian process,” Mat. Zametki, 29, 150–155 (1981).
V. Piterbarg, “High deviations for multidimensional stationary Gaussian process with independent coordinates,” in: Univ. of Lund and Lund Inst. of Technology, Dept. of Math. Statistics. Reports, Vol. 6, Lund (1991), pp. 1–34.
V. I. Piterbarg, “High excursions for nonstationary generalized chi-square processes,” Stoch. Proc. Appl., 53, No. 2, 307–337 (1994).
V. I. Piterbarg, Asymptotic Methods in the Theory of Gaussian Processes and Fields, Amer. Math. Soc., Providence (2012).
V. I. Piterbarg and V. R. Fatalov, “The Laplace method for probability measures in Banach spaces,” Russ. Math. Surv., 50, No. 6, 1151–1239 (1995).
D. Revuz and M. Yor, Continuous Martingales and Brownian Motion, Springer, Berlin (1999).
W. Rudin, Principles of Mathematical Analysis, McGraw-Hill, New York (1976).
S. M. Rytov, Introduction to Statistical Radiophysics [in Russian], Nauka, Moscow, (1966).
B. Simon, Functional Integration and Quantum Physics, Academic Press, New York (1979).
A. D. Ventzel and M. I. Freidlin, Fluctuations in Dynamic Systems under the Action of Small Random Perturbations [in Russian], Nauka, Moscow (1979).
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V. R. Fatalov is deceased.
Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 23, No. 1, pp. 219–257, 2020.
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Fatalov, V.R. Supremum of the Euclidean Norms of the Multidimensional Wiener Process and Brownian Bridge: Sharp Asymptotics of Probabilities of Large Deviations. J Math Sci 262, 546–573 (2022). https://doi.org/10.1007/s10958-022-05836-6
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DOI: https://doi.org/10.1007/s10958-022-05836-6