Precise small deviations in L 2 of some Gaussian processes appearing in the regression context
Open Mathematics (2014)
- Volume: 12, Issue: 11, page 1674-1686
- ISSN: 2391-5455
Access Full Article
topAbstract
topHow to cite
topReferences
top- [1] Adler, R. J. An introduction to continuity, extrema, and related topics for general Gaussian processes. IMS Lect. Notes, 12 (1990), Hayword. Zbl0747.60039
- [2] Ai, X., Li, W. V., Liu, G. Karhunen-Loève expansions for detrended Brownian motion. Statist. & Probab. Lett., 2012, 82(7), 1235–1241. http://dx.doi.org/10.1016/j.spl.2012.03.007 Zbl1259.60093
- [3] Beghin, L., Nikitin, Ya. Yu., Orsingher, E. Exact small ball constants for some Gaussian processes under the L 2-norm. Journ. of Math. Sci., 2005, 128(1), 2493–2502. http://dx.doi.org/10.1007/s10958-005-0197-9 Zbl1078.60028
- [4] Berlinet, A. F., Servien, R. Necessary and sufficient condition for the existence of a limit distribution of the nearestneighbour density estimator. Journ. of Nonparam. Statist., 2011, 23(3), 633–643. http://dx.doi.org/10.1080/10485252.2011.567334 Zbl1284.62248
- [5] Dunker T., Lifshits M. A., Linde W. Small deviations of sums of independent variables. In: High Dimensional Probability. Progress in Probability, 1998, 43, Birkhaåuser, Basel, 59–74. http://dx.doi.org/10.1007/978-3-0348-8829-5_4 Zbl0902.60039
- [6] Fatalov, V. R. Ergodic means for large values of T and exact asymptotics of small deviations for a multi-dimensional Wiener process. Izvestiya: Mathematics, 2013, 77(6), 1224–1259. http://dx.doi.org/10.1070/IM2013v077n06ABEH002675 Zbl1288.60030
- [7] Fatalov, V. R. Small deviations for two classes of Gaussian stationary processes and L p-functionals, 0 < p ≤ ∞. Problems Inform. Transmission, 2010, 46(1), 62–85. http://dx.doi.org/10.1134/S0032946010010060 Zbl1205.93140
- [8] Ferraty F., Vieu Ph. Nonparametric functional data analysis. Berlin: Springer, 2006. Zbl1119.62046
- [9] Fill, J. A., Torcaso, F. Asymptotic analysis via Mellin transforms for small deviations in L 2-norm of integrated Brownian sheets. Probab. Theory Relat. Fields, 2003, 130(2), 259–288. Zbl1052.60027
- [10] Gao, F., Hannig, J., Lee, T. -Y., Torcaso, F. Laplace transforms via Hadamard factorization with applications to small ball probabilities. Electr. Journ. of Probab., 2003, 8(13), 1–20. Zbl1064.60061
- [11] Gao, F., Hannig, J., Torcaso, F., Comparison Theorems for Small Deviations of random series. Electr. Journ. of Probab. 2003, 8(1), 1–17. Zbl1065.60040
- [12] Gao, F., Hannig, J., Torcaso, F. Integrated Brownian motions and Exact L 2-small balls. Ann. of Probab. 2003, 31(3), 1320–1337. http://dx.doi.org/10.1214/aop/1055425782 Zbl1047.60030
- [13] Gao, F., Hannig, J., Lee, T. -Y., Torcaso, F. Exact L 2-small balls of Gaussian processes. Journ. of Theoret. Probab., 2004, 17(2), 503–520. http://dx.doi.org/10.1023/B:JOTP.0000020705.28185.4c Zbl1049.60028
- [14] Jandhyala, V. K., Jiang, P. L. Eigenvalues of a Fredholm integral operator and applications to problems of statistical inference. Journ. Integr. Eq. Appl., 1996, 8(4), 413–427. http://dx.doi.org/10.1216/jiea/1181075971 Zbl0885.45001
- [15] Jandhyala, V. K., MacNeill, I. B. Residual partial sum limit process for regression models with applications to detecting parameter changes at unknown times. Stoch. Proc. Appl., 1989, 33(2), 309–323. http://dx.doi.org/10.1016/0304-4149(89)90045-8 Zbl0679.62056
- [16] Kharinski, P. A., Nikitin, Ya. Yu. Sharp small deviation asymptotics in L 2-norm for a class of Gaussian processes. Journ. Math. Sci., 2006, 133(3), 1328–1332. http://dx.doi.org/10.1007/s10958-006-0042-9
- [17] Li, W. V. Comparison results for the lower tail of Gaussian seminorms. Journ. Theor. Prob., 1992, 5(1), 1–31. http://dx.doi.org/10.1007/BF01046776 Zbl0743.60009
- [18] Li W. V., Shao Q. M. Gaussian processes: inequalities, small ball probabilities and applications. Stochastic Processes: Theory and Methods. Amsterdam: North-Holland, 2001, 533–597. (Handbook Statist., v. 19.) Zbl0987.60053
- [19] Lifshits, M. A. Gaussian Random Functions. Dordrecht: Kluwer, 1995. http://dx.doi.org/10.1007/978-94-015-8474-6
- [20] Lifshits, M. Lectures on Gaussian processes. SpringerBriefs in Mathematics, Springer, 2012. http://dx.doi.org/10.1007/978-3-642-24939-6
- [21] Lifshits, M. A. Bibliography on small deviation probabilities, 2014. Available at http://www.proba.jussieu.fr/pageperso/smalldev/biblio.pdf.
- [22] MacNeill, I. B. Properties of sequences of partial sums of polynomial regression residuals with applications to tests for change of regression at unknown times. Ann. Stat., 1978, 6(2), 422–433. http://dx.doi.org/10.1214/aos/1176344133 Zbl0375.62064
- [23] Nazarov, A. I. On the sharp constant in the small ball asymptotics of some Gaussian processes under L 2-norm. Journ. of Math. Sci., 2003, 117(3), 4185–4210. http://dx.doi.org/10.1023/A:1024868604219
- [24] Nazarov, A. I. Exact L 2-small ball asymptotics of Gaussian processes and the spectrum of boundary-value problems. Journ. of Theor. Prob., 2009, 22(3), 640–665. http://dx.doi.org/10.1007/s10959-008-0173-7 Zbl1187.60025
- [25] Nazarov, A. I. On a set of transformations of Gaussian random functions. Theor. Probab. Appl., 2009, 54(2), 203–216. http://dx.doi.org/10.1137/S0040585X97984103 Zbl1214.60011
- [26] Nazarov, A. I., Nikitin, Ya. Yu. Exact L 2-small ball behavior of integrated Gaussian processes and spectral asymptotics of boundary value problems. Probab. Theor. Relat. Fields, 2004, 129(4), 469–494. http://dx.doi.org/10.1007/s00440-004-0337-z Zbl1051.60041
- [27] Nikitin, Ya. Yu., Pusev, R. S. Exact Small Deviation Asymptotics for Some Brownian Functionals. Theor. Probab. Appl., 2013, 57(1), 60–81. http://dx.doi.org/10.1137/S0040585X97985790 Zbl1278.60072
- [28] Slepian, D. First passage time for a particular Gaussian process. Ann. Math. Stat., 1961, 32(2), 610–612. http://dx.doi.org/10.1214/aoms/1177705068 Zbl0113.12403
- [29] Titchmarsh, E. C. The theory of functions. 2nd ed. London: Oxford University Press, 1939. Zbl0022.14602
- [30] van der Vaart A. W., van Zanten H. Rates of contraction of posterior distributions based on Gaussian process priors. Ann. Statist., 2008, 36(3), 1435–1463. http://dx.doi.org/10.1214/009053607000000613 Zbl1141.60018