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2014 | 12 | 11 | 1674-1686

Tytuł artykułu

Precise small deviations in L 2 of some Gaussian processes appearing in the regression context

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Języki publikacji

EN

Abstrakty

EN
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.

Wydawca

Czasopismo

Rocznik

Tom

12

Numer

11

Strony

1674-1686

Opis fizyczny

Daty

wydano
2014-11-01
online
2014-06-29

Bibliografia

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  • [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
  • [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
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  • [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
  • [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
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  • [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
  • [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
  • [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
  • [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
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  • [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
  • [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
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  • [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
  • [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
  • [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
  • [29] Titchmarsh, E. C. The theory of functions. 2nd ed. London: Oxford University Press, 1939.
  • [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

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Bibliografia

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