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

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.

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References

  1. [1]

    Adler, R. J. An introduction to continuity, extrema, and related topics for general Gaussian processes. IMS Lect. Notes, 12 (1990), Hayword.

  2. [2]

    Ai, X., Li, W. V., Liu, G. Karhunen-Loève expansions for detrended Brownian motion. Statist. & Probab. Lett., 2012, 82(7), 1235–1241.

  3. [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.

  4. [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.

  5. [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.

  6. [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.

  7. [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.

  8. [8]

    Ferraty F., Vieu Ph. Nonparametric functional data analysis. Berlin: Springer, 2006.

  9. [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.

  10. [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.

  11. [11]

    Gao, F., Hannig, J., Torcaso, F., Comparison Theorems for Small Deviations of random series. Electr. Journ. of Probab. 2003, 8(1), 1–17.

  12. [12]

    Gao, F., Hannig, J., Torcaso, F. Integrated Brownian motions and Exact L 2-small balls. Ann. of Probab. 2003, 31(3), 1320–1337.

  13. [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.

  14. [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.

  15. [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.

  16. [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.

  17. [17]

    Li, W. V. Comparison results for the lower tail of Gaussian seminorms. Journ. Theor. Prob., 1992, 5(1), 1–31.

  18. [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.)

  19. [19]

    Lifshits, M. A. Gaussian Random Functions. Dordrecht: Kluwer, 1995.

  20. [20]

    Lifshits, M. Lectures on Gaussian processes. SpringerBriefs in Mathematics, Springer, 2012.

  21. [21]

    Lifshits, M. A. Bibliography on small deviation probabilities, 2014. Available at http://www.proba.jussieu.fr/pageperso/smalldev/biblio.pdf.

  22. [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.

  23. [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.

  24. [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.

  25. [25]

    Nazarov, A. I. On a set of transformations of Gaussian random functions. Theor. Probab. Appl., 2009, 54(2), 203–216.

  26. [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.

  27. [27]

    Nikitin, Ya. Yu., Pusev, R. S. Exact Small Deviation Asymptotics for Some Brownian Functionals. Theor. Probab. Appl., 2013, 57(1), 60–81.

  28. [28]

    Slepian, D. First passage time for a particular Gaussian process. Ann. Math. Stat., 1961, 32(2), 610–612.

  29. [29]

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

  30. [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.

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Author information

Correspondence to Alisa A. Kirichenko.

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Kirichenko, A.A., Nikitin, Y.Y. Precise small deviations in L 2 of some Gaussian processes appearing in the regression context. centr.eur.j.math. 12, 1674–1686 (2014). https://doi.org/10.2478/s11533-014-0436-8

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MSC

  • 60G15
  • 60J65
  • 62J05

Keywords

  • Gaussian process
  • Small deviations
  • Precise asymptotics