Abstract
In this paper we calculate the exact asymptotics of the probability P{‖w(t)+uct‖ p >u},u→∞, wherew(t) is the standard Wiener process and ‖x‖ p is the ordinary norm in the spaceL p[0,1],p≥2. The result is obtained on the basis of a general theorem due to the author on the asymptotics of the Gaussian measureP(uD),u→∞, for a Borel setD belonging to a separable Banach space.
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References
V. Dobric, M. B. Marcus, and M. Weber, “The distribution of large values of the supremum of Gaussian processes,”Astérisque,157–158, 95–127 (1988).
W. Linde, “Gaussian measures of large balls in\(\ell ^p \),”Ann. Probab.,19, No. 3, 1264–1279 (1991).
M. A. Lifshits, “Gaussian large deviations of a smooth seminorm,”Zap. Nauchn. Sem. LOMI,194, 106–113 (1992).
V. R. Fatalov, “Large deviations of Gaussian measures on the spaces\(\ell ^p \) andL p, p≥2”,Teor. Veroyatnost. i Primenen. [Theory Probab. Appl.] 41, No. 3, 682–689 (1996).
A. A. Borovkov and A. A. Mogul’skii, “On the probabilities of small deviations for random processes,”Trudy Instit. Math. SO AN SSSR,13, 147–168 (1989).
V. I. Piterbarg and V. R. Fatalov, “The Laplace method for probability measures on Banach spaces,”Uspekhi Mat. Nauk [Russian Math. Surveys],50, No. 6, 57–150 (1995).
V. R. Fatalov, “Exact asymptotics of large deviations for Gaussian measures on a Hilbert space,”Izv. Akad. Nauk Armen. Mat.,27, No. 5, 43–57 (1992).
Handbook of Mathematical Functions With Formulas, Graphs, and Mathematical Tables, (M. Abramowitz and I. Stegun, editors) National Bureau of Standards, Washington, D.C. (1964).
H. Bateman and A. Erdélyi,Higher Transcendental Functions, Vol. 1, McGraw-Hill, New York-Toronto-London (1953).
E. Hobson,The Theory of Spherical and Ellipsoidal Harmonics, Cambridge (1931).
L. Robin,Fonctions sphériques de Legendre et fonctions sphéroidales, Vol. 3, Gauthier-Villars, Paris (1959).
N. N. Vakhaniya, V. I. Tarieladze, and S. A. Chobanyan,Probability Distributions in Banach spaces [in Russian], Nauka, Moscow (1985).
H.-H. Kuo,Gaussian Measures in Banach spaces Springer-Verlag, Heidelberg (1976).
B. S. Rajput, “Gaussian measures onL p spaces, 1≤p<∞”,J. Multivariate Anal.,2, No. 4, 382–403 (1972).
R. Bonic and J. Frampton, “Smooth functions on Banach manifolds,”J. Math. Mech.,15, No. 5, 877–898 (1966).
V. M. Alekseev, V. M. Tikhomirov, and S. V. Fomin,Optimal Control [in Russian], Nauka, Moscow (1979).
A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev,Integrals and Series. Elementary Functions [in Russian], Nauka, Moscow (1981).
E. Kamke,Differentialgleichungen. Lösungsmethoden und Lösungen I. Gewöhnliche Differentialgleichungen, 6. Verbesserte Auflage, Leipzig (1959).
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Translated fromMatematicheskie Zametki, Vol. 65, No. 3, pp. 429–436, March, 1999.
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Fatalov, V.R. Large deviations of theL p-norm of a wiener process with drift. Math Notes 65, 358–364 (1999). https://doi.org/10.1007/BF02675079
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DOI: https://doi.org/10.1007/BF02675079