Exact small ball asymptotics in weighted \(L_2\)-norm for some Gaussian processes. (English. Russian original) Zbl 1288.60045
J. Math. Sci., New York 163, No. 4, 409-429 (2009); translation from Zap. Nauchn. Semin. POMI 364, 166-199 (2009).
Summary: We find the exact small ball asymptotics under weighted \(L_2\)-norm for a wide class of Gaussian processes which generate boundary-value problems for ordinary differential equations. Sharp constants in the asymptotics are derived for a number of processes connected with special functions.
MSC:
| 60G15 | Gaussian processes |
| 34L20 | Asymptotic distribution of eigenvalues, asymptotic theory of eigenfunctions for ordinary differential operators |
| 60F10 | Large deviations |
| 60J65 | Brownian motion |
References:
| [1] | W. V. Li and Q. M. Shao, ”Gaussian processes: inequalities, small ball probabilities, and applications,” Handbook Statist., 19, 533–597 (2001). · Zbl 0987.60053 · doi:10.1016/S0169-7161(01)19019-X |
| [2] | M. A. Lifshits, ”Asymptotic behavior of small ball probabilities,” in: Prob. Theor. Math. Stat. (1999), Proc.VII Int. Vilnius Conference, pp. 453–468. · Zbl 0994.60017 |
| [3] | G. N. Sytaya, ”On some asymptotic representations of the Gaussian measure in a Hilbert space,” Theory of Stochastic Processes, 2, 93–104 (1974). · Zbl 0282.60022 |
| [4] | V. M. Zolotarev, ”Gaussian measure asymptotics in l 2 on a set of centered spheres with radii tending to zero,” in: 12th Europ. Meeting of Statisticians (1979), p. 254. |
| [5] | R. M. Dudley, J. Hoffmann-Jørgensen, and L. A. Shepp, ”On the lower tail of Gaussian seminorms,” Ann. Probab., 7, 319–342 (1979). · Zbl 0424.60041 · doi:10.1214/aop/1176994953 |
| [6] | I. A. Ibragimov, ”The probability of a Gaussian vector with values in a Hilbert space hitting a ball of small radius,” Zap. Nauchn. Semin. LOMI, 85, 75–93 (1979). · Zbl 0417.60050 |
| [7] | W. V. Li, ”Comparison results for the lower tail of Gaussian seminorms,” J. Theor. Probab., 5(1), 1–31 (1992). · Zbl 0743.60009 · doi:10.1007/BF01046776 |
| [8] | T. Dunker, M. A. Lifshits, and W. Linde, ”Small deviations of sums of independent variables,” Progr. Probab., 43, 59–74 (1998). · Zbl 0902.60039 |
| [9] | J.-R. Pycke, ”Un lien entre le développement de Karhunen-Loève de certains processus gaussiens et le laplacien dans des espaces de Riemann,” Thèse de doctorat de l’Université Paris 6 (2003). |
| [10] | A. I. Nazarov and Ya. Yu. Nikitin, ”Exact L 2-small ball behavior of integrated Gaussian processes and spectral asymptotics of boundary-value problems,” Prob. Theor. Rel. Fields, 129(4), 469–494 (2004). · Zbl 1051.60041 |
| [11] | A. I. Nazarov, ”On the sharp constant in the small ball asymptotics of some Gaussian processes under L 2-norm,” Probl. Mat. Anal., 26, 179–214 (2003). · Zbl 1046.60035 |
| [12] | A. I. Nazarov, ”Exact L 2-small ball asymptotics of Gaussian processes and the spectrum of boundary-value problems. J. Theor. Prob., DOI 10.1007/S10959-008-0173-7 . · Zbl 1187.60025 |
| [13] | F. Gao, J. Hannig, T.-Y. Lee, and F. Torcaso, ”Laplace transforms via Hadamard factorization with applications to small ball probabilities,” Electron. J. Probab., 8(13), 1–20 (2003). · Zbl 1064.60061 |
| [14] | A. N. Borodin and P. Salminen, Handbook of Brownian Motion: Facts and Formulae, Birkhauser, Basel (1996). · Zbl 0859.60001 |
| [15] | E. Kamke, Differentialgleichungen. Lösungsmethoden und Lösungen. I, 6th ed., Leipzig (1959). |
| [16] | Ya. Yu. Nikitin and P. A. Kharinski, ”Sharp small deviation asymptotics in the L 2-norm for a class of Gaussian processes,” Zap. Nauchn. Semin. POMI, 311, 214–221 (2004). |
| [17] | E. C. Titchmarsh, The Theory of Functions, 2nd ed., Oxford Univ. Press (1975). · Zbl 0336.30001 |
| [18] | I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Sums, Series, and Products [in Russian], 5th ed. Moscow (1971). · Zbl 0918.65002 |
| [19] | M. L. Kleptsyna and A. Le Breton, ”A Cameron-Martin type formula for general Gaussian processes – a filtering approach,” Stochastics and Stochastics Rep., 72(3–4), 229–250 (2002). · Zbl 1002.60031 · doi:10.1080/10451120290019203 |
| [20] | A. I. Karol, A. I. Nazarov, and Ya. Yu. Nikitin, ”Small ball probabilities for Gaussian random fields and tensor products of compact operators,” Trans. Amer. Math. Soc., 360(3), 1443–1474 (2008). · Zbl 1136.60024 · doi:10.1090/S0002-9947-07-04233-X |
| [21] | P. Deheuvels and G. Martynov, ”Karhunen-Loève expansions for weighted Wiener processes and Brownian bridges via Bessel functions,” Progr. Probab., 55, 57–93 (2003). · Zbl 1048.60021 |
| [22] | M. A. Naimark, Linear Differential Operators [in Russian], 2nd. ed., Moscow (1969). · Zbl 0219.34001 |
| [23] | A. Lachal, ”Bridges of certain Wiener integrals. Prediction properties, relation with polynomial interpolation and differential equations. Application to goodness-of-fit testing,” in: Bolyai Math. Studies X, Limit Theorems, Balatonlelle (Hungary) (1999), Budapest (2002), pp. 1–51. |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
