Abstract:
We find some logarithmic and exact small deviation asymptotics for the L2-norm of certain Gaussian processes closely connected with the Wiener process. In particular the processes obtained by centering and integrating Brownian motion and Brownian bridge are examined.
Citation:
L. Beghin, Ya. Yu. Nikitin, E. Orsingher, “Exact small ball constants for some Gaussian processes under L2-norm”, Probability and statistics. Part 6, Zap. Nauchn. Sem. POMI, 298, POMI, St. Petersburg, 2003, 5–21; J. Math. Sci. (N. Y.), 128:1 (2005), 2493–2502
\Bibitem{BegNikOrs03}
\by L.~Beghin, Ya.~Yu.~Nikitin, E.~Orsingher
\paper Exact small ball constants for some Gaussian processes under $L^2$-norm
\inbook Probability and statistics. Part~6
\serial Zap. Nauchn. Sem. POMI
\yr 2003
\vol 298
\pages 5--21
\publ POMI
\publaddr St.~Petersburg
\mathnet{http://mi.mathnet.ru/znsl1148}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=2038861}
\zmath{https://zbmath.org/?q=an:1078.60028}
\transl
\jour J. Math. Sci. (N. Y.)
\yr 2005
\vol 128
\issue 1
\pages 2493--2502
\crossref{https://doi.org/10.1007/s10958-005-0197-9}
Linking options:
https://www.mathnet.ru/eng/znsl1148
https://www.mathnet.ru/eng/znsl/v298/p5
This publication is cited in the following 25 articles:
L. V. Rozovskii, “Small deviation probabilities for sums of independent positive random variables”, Vestn. St. Petersbg. Univ., Math., 7:3 (2020), 295–307
A.A. Khartov, M. Zani, “Approximation complexity of sums of random processes”, Journal of Complexity, 54 (2019), 101399
V. R. Fatalov, “Integrals of Bessel processes and multi-dimensional Ornstein–Uhlenbeck processes:
exact asymptotics for Lp-functionals”, Izv. Math., 82:2 (2018), 377–406
Nazarov A.I., Nikitin Ya.Yu., “On Small Deviation Asymptotics in l-2 of Some Mixed Gaussian Processes”, 6, no. 4, 2018, 55
Ibragimov I.A., Lifshits M.A., Nazarov A.I., Zaporozhets D.N., “On the History of St. Petersburg School of Probability and Mathematical Statistics: II. Random Processes and Dependent Variables”, Vestn. St Petersb. Univ.-Math., 51:3 (2018), 213–236
Xiaohui Ai, Yang Sun, “Karhunen–Loeve expansion for the additive two-sided Brownian motion”, Communications in Statistics - Theory and Methods, 47:13 (2018), 3085
Yu. P. Petrova, “Spectral Asymptotics for Problems with Integral Constraints”, Math. Notes, 102:3 (2017), 369–377
Xiaohui Ai, “Karhunen–Loeve expansion for the additive detrended Brownian motion”, Communications in Statistics - Theory and Methods, 46:16 (2017), 8210
Xiaohui Ai, “A note on Karhunen–Loève expansions for the demeaned stationary Ornstein–Uhlenbeck process”, Statistics & Probability Letters, 117 (2016), 113
Kirichenko A.A., Nikitin Ya.Yu., “Precise Small Deviations in l-2 of Some Gaussian Processes Appearing in the Regression Context”, Cent. Eur. J. Math., 12:11 (2014), 1674–1686
Ai XiaoHui, Li V W., “Karhunen-Loeve Expansions for the M-Th Order Detrended Brownian Motion”, Sci. China-Math., 57:10 (2014), 2043–2052
Liu J.V., Huang Z., Mao H., “Karhunen-Loeve Expansion for Additive Slepian Processes”, Stat. Probab. Lett., 90 (2014), 93–99
V. R. Fatalov, “Ergodic means for large values of T and exact asymptotics of small deviations for a multi-dimensional Wiener process”, Izv. Math., 77:6 (2013), 1224–1259
Liu J.V., “Karhunen-Loeve Expansion for Additive Brownian Motions”, Stoch. Process. Their Appl., 123:11 (2013), 4090–4110
Ya. Yu. Nikitin, R. S. Pusev, “The exact asymptotic of small deviations for a series of Brownian functionals”, Theory Probab. Appl., 57:1 (2013), 60–81
Ai X., Li W.V., Liu G., “Karhunen-Loeve Expansions for the Detrended Brownian Motion”, Stat. Probab. Lett., 82:7 (2012), 1235–1241
Mikhail Lifshits, SpringerBriefs in Mathematics, Lectures on Gaussian Processes, 2012, 1
R. S. Pusev, “Asymptotics of small deviations of the Bogoliubov processes with respect to a quadratic norm”, Theoret. and Math. Phys., 165:1 (2010), 1348–1357
A. I. Nazarov, “On one transformations family of Gaussian random functions”, Theory Probab. Appl., 54:2 (2010), 203–216
N. A. Serdyukova, “Dependence of the approximation complexity of random fields on dimension”, Theory Probab. Appl., 54:2 (2010), 272–284