Advertisement

Lithuanian Mathematical Journal

, Volume 58, Issue 4, pp 341–359 | Cite as

Karhunen–Loève expansion for a generalization of Wiener bridge

  • Mátyás BarczyEmail author
  • Rezső L. Lovas
Article
  • 28 Downloads

Abstract

We derive a Karhunen–Loève expansion of the Gauss process \( {B}_t-g(t){\int}_0^1{g}^{\hbox{'}}(u)\mathrm{d}{B}_u,t\in \left[0,1\right] \), where (Bt)t ∈ [0, 1] is a standardWiener process, and g : [0, 1] →  is a twice continuously differentiable function with g(0) = 0 and \( {\int}_0^1{\left(g\hbox{'}(u)\right)}^2\mathrm{d}u=1 \). This process is an important limit process in the theory of goodness-of-fit tests. We formulate two particular cases with the functions \( g(t)=\left(\sqrt{2}/\pi \right)\sin \left(\pi t\right),t\in \left[0,1\right] \), and g(t) = t, t ∈ [0, 1]. The latter corresponds to the Wiener bridge over [0, 1] from 0 to 0.

Keywords

Gauss process Karhunen–Loève expansion integral operator Wiener bridge 

MSC

60G15 60G12 34B60 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    M.B. Abdeddaiem, On goodness-of-fit tests for parametric hypotheses in perturbed dynamical systems using a minimum distance estimator, Stat. Inference Stoch. Process., 19(3):259–287, 2016.MathSciNetCrossRefGoogle Scholar
  2. 2.
    R.J. Adler, An Introduction to Continuity, Extrema, and Related Topics for General Gaussian Processes, IMS Lect. Notes, Monogr. Ser., Vol. 12, IMS, Hayward, CA, 1990.Google Scholar
  3. 3.
    X. Ai, A note on Karhunen–Loève expansions for the demeaned stationary Ornstein–Uhlenbeck process, Stat. Probab. Lett., 117:113–117, 2016.MathSciNetCrossRefGoogle Scholar
  4. 4.
    X. Ai, W.V. Li, and G. Liu, Karhunen–Loève expansions for the detrended Brownian motion, Stat. Probab. Lett., 82(7):1235–1241, 2012.CrossRefGoogle Scholar
  5. 5.
    X. Ai and Y. Sun, Karhunen–Loève expansion for the additive two-sided Brownian motion, Commun. Stat., Theory Methods, 47(13):3085–3091, 2018.CrossRefGoogle Scholar
  6. 6.
    R.B. Ash and M.F. Gardner, Topics in Stochastic Processes, Probability and Mathematical Statistics, Vol. 27, Academic Press, New York, San Francisco, London, 1975.zbMATHGoogle Scholar
  7. 7.
    M. Barczy and E. Iglói, Karhunen–Loève expansions of alpha-Wiener bridges, Cent. Eur. J. Math., 9(1):65–84, 2011.MathSciNetCrossRefGoogle Scholar
  8. 8.
    M. Barczy and P. Kern, Representations of multidimensional linear process bridges, Random Oper. Stoch. Equ., 21(2):159–189, 2013.MathSciNetCrossRefGoogle Scholar
  9. 9.
    M. Barczy and L.R. Lovas, Karhunen–Loève expansion for a generalization of Wiener bridge, preprint, 2016, arXiv:1602.05084.Google Scholar
  10. 10.
    S. Corlay and G. Pagès, Functional quantization-based stratified sampling methods, Monte Carlo Methods Appl., 21(1):1–32, 2015.MathSciNetCrossRefGoogle Scholar
  11. 11.
    D.A. Darling, The Kolmogorov–Smirnov, Cramér–von Mises Tests, Ann. Math. Stat., 28:823–838, 1957.MathSciNetCrossRefGoogle Scholar
  12. 12.
    P. Deheuvels, Karhunen–Loève expansions of mean-centered Wiener processes, in E. Giné, V. Koltchinskii, W. Li, and J. Zinn (Eds.), High Dimensional Probability. Proceedings of the Fourth International Conference, IMS Lecture Notes Monogr. Ser., Vol. 51, IMS, Beachwood, OH, 2006, pp. 62–76.CrossRefGoogle Scholar
  13. 13.
    P. Deheuvels and G. Martynov, Karhunen–Loève expansions for weighted Wiener processes and Brownian bridges via Bessel functions, in J. Hoffmann-Jørgensen, Marcus M.B., and Wellner J.A. (Eds.), High Dimensional Probability III. Third International Conference on High Dimensional Probability, Sandjberg, Denmark, June 24–28, 2002. Selected papers, Prog. Probab., Vol. 55, Birkhäuser, Basel, 2003, pp. 57–93.Google Scholar
  14. 14.
    A. Gassem, Goodness-of-fit test for switching diffusion, Stat. Inference Stoch. Process., 13(2):97–123, 2010.MathSciNetCrossRefGoogle Scholar
  15. 15.
    V.K. Jandhyala and I.B. MacNeill, Residual partial sum limit process for regression models with applications to detecting parameter changes at unknown times, Stochastic Processes Appl., 33(2):309–323, 1989.MathSciNetCrossRefGoogle Scholar
  16. 16.
    M. Kac, J. Kiefer, and J. Wolfowitz, On tests of normality and other tests of goodness of fit based on distance methods, Ann. Math. Stat., 26:189–211, 1955.MathSciNetCrossRefGoogle Scholar
  17. 17.
    M.A. Lifshits, Bibliography of small deviation probabilities, 2016, available from: https://www.proba.jussieu.fr/pageperso/smalldev/biblio.pdf.
  18. 18.
    J.V. Liu, Karhunen–Loève expansion for additive Brownian motions, Stochastic Processes Appl., 123(11):4090–4110, 2013.MathSciNetCrossRefGoogle Scholar
  19. 19.
    J.V. Liu, Z. Huang, and H. Mao, Karhunen–Loève expansion for additive Slepian processes, Stat. Probab. Lett., 90:93–99, 2014.MathSciNetCrossRefGoogle Scholar
  20. 20.
    A.I. Nazarov, Exact L2-small ball asymptotics of Gaussian processes and the spectrum of boundary-value problems, J. Theor. Probab., 22(3):640–665, 2009.CrossRefGoogle Scholar
  21. 21.
    A.I. Nazarov, On a set of transformations of Gaussian random functions, Theory Probab. Appl., 54(2):203–216, 2010.MathSciNetCrossRefGoogle Scholar
  22. 22.
    A.I. Nazarov and Ya.Yu. Nikitin, Exact L2-small ball behavior of integrated Gaussian processes and spectral asymptotics of boundary value problems, Probab. Theory Relat. Fields, 129(4):469–494, 2004.Google Scholar
  23. 23.
    A.I. Nazarov and Y.P. Petrova, The small ball asymptotics in Hilbertian norm for the Kac–Kiefer–Wolfowitz processes (in Russian), Teor. Veroyatn. Primen., 60(3):482–505, 2015.CrossRefGoogle Scholar
  24. 24.
    A. Papoulis, Probability, Random Variables, and Stochastic Processes, McGraw-Hill, New York, Toronto, London, Sydney, 1965.Google Scholar
  25. 25.
    J.-R. Pycke, Une généralisation du développement de Karhunen–Loève du pont brownien, C. R. Acad. Sci., Paris, Sér. I, Math., 333(7):685–688, 2001.MathSciNetCrossRefGoogle Scholar
  26. 26.
    J.-R. Pycke, Multivariate extensions of the Anderson–Darling process, Stat. Probab. Lett., 63(4):387–399, 2003.MathSciNetCrossRefGoogle Scholar
  27. 27.
    J.-R. Pycke, Un lien entre le développement de Karhunen–Loève de certains processus gaussiens et le laplacien dans des espaces de Riemann, PhD dissertation, University of Paris 6, France, 2003.Google Scholar
  28. 28.
    I.C. Tsantili and D.T. Hristopulos, Karhunen–Loève expansion of Spartan spatial random fields, Probabilist. Eng. Mech., 43:132–147, 2016.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.MTA-SZTE Analysis and Stochastics Research Group, Bolyai InstituteUniversity of SzegedSzegedHungary
  2. 2.Institute of MathematicsUniversity of DebrecenDebrecenHungary

Personalised recommendations