Cent. Eur. J. Math. • 12(11) • 2014 • 1674-1686 DOI: 10.2478/s11533-014-0436-8 Central European Journal of Mathematics Precise small deviations in L 2 of some Gaussian processes appearing in the regression context Research Article Alisa A. Kirichenko 1∗ , Ya. Yu. Nikitin 2,3† 1 Korteweg-de Vries Institute for Mathematics,University of Amsterdam, P. O. Box 19268 1000 GG Amsterdam, The Netherlands 2 Department of Mathematics and Mechanics, Saint-Petersburg State University, Universitetski pr., 28, St. Petersburg, 198504, Russia 3 National Research University - Higher School of Economics, Souza Pechatnikov, 16, St. Petersburg, 190008, Russia Received 19 September 2013; accepted 28 February 2014 Abstract: We find precise small deviation asymptotics with respect to the Hilbert norm for some special Gaussian processes connected to two regression schemes studied by MacNeill and his coauthors. In addition, we also obtain precise small deviation asymptotics for the detrended Brownian motion and detrended Slepian process. MSC: 60G15, 60J65, 62J05 Keywords: Gaussian process • Small deviations • Precise asymptotics © Versita Sp. z o.o. 1. Introduction Small deviation theory for Gaussian processes has been in intensive development in recent years (see the complete bibliography in [21]). Its progress is stimulated by numerous links with such important mathematical domains as the accuracy of discrete approximation for random processes, the calculation of the metric entropy for functional sets and the Chung law of the iterated logarithm, see the review paper [18]. The small deviation theory is closely related to various topics in Statistics, e. g., functional data analysis [8], nearest-neighbour density estimation [4], and Bayesian nonparametrics [30]. ∗ E-mail: alice.kirichenko@gmail.com † E-mail: yanikit47@gmail.com 1674 Unauthenticated Download Date | 12/30/16 9:28 AM