Probability Theory and Related Fields

, Volume 129, Issue 4, pp 469–494

Exact L2-small ball behavior of integrated Gaussian processes and spectral asymptotics of boundary value problems

Article

Abstract.

We find the exact small deviation asymptotics for the L2-norm of various m-times integrated Gaussian processes closely connected with the Wiener process and the Ornstein – Uhlenbeck process. Using a general approach from the spectral theory of linear differential operators we obtain the two-term spectral asymptotics of eigenvalues in corresponding boundary value problems. This enables us to improve the recent results from [15] on the small ball asymptotics for a class of m-times integrated Wiener processes. Moreover, the exact small ball asymptotics for the m-times integrated Brownian bridge, the m-times integrated Ornstein – Uhlenbeck process and similar processes appear as relatively simple examples illustrating the developed general theory.

Key words or phrases: Integrated Wiener process Integrated Ornstein Uhlenbeck process Small deviations Boundary value problem Green function; Spectral asymptotics 

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References

  1. 1.
    Lifshits, M.A: Asymptotic behavior of small ball probabilities. Prob. Theory and Math. Stat. B. Grigelionis, et al., (eds.), VSP/TEV, 1999, pp. 453–468Google Scholar
  2. 2.
    Li, W.V., Shao, Q.M.: Gaussian processes: Inequalities, Small Ball Probabilities and Applications. Stochastic Processes: Theory and Methods, Handbook of Statistics, 19, C.R. Rao, D.Shanbhag, (eds.), 2001, pp. 533–597Google Scholar
  3. 3.
    Sytaya, G.N.: On some asymptotic representations of the Gaussian measure in a Hilbert space. Theory of Stochastic Processes 2, 93–104 (1974) Kiev (in Russian)Google Scholar
  4. 4.
    Zolotarev, V.M: Gaussian measure asymptotic in l2 on a set of centered spheres with radii tending to zero. 12th Europ.Meeting of Statisticians, Varna, 1979, p. 254Google Scholar
  5. 5.
    Zolotarev, V.M.: Asymptotic behavior of Gaussian measure in l2. J. Sov. Math. 24, 2330–2334 (1986)Google Scholar
  6. 6.
    Dudley, R.M., Hoffmann-Jørgensen, J., Shepp, L.A.: On the lower tail of Gaussian seminorms. Ann. Prob. 7, 319–342 (1979)MathSciNetMATHGoogle Scholar
  7. 7.
    Ibragimov, I.A.: The probability of a Gaussian vector with values in a Hilbert space hitting a ball of small radius. J. Sov. Math. 20, 2164–2174 (1982)MATHGoogle Scholar
  8. 8.
    Csáki, E.: On small values of the square integral of a multiparameter Wiener process. Statistics and Probability, Proc. of the 3rd Pannonian Symp. on Math. Stat. D. Reidel, Boston, 1982, pp. 19–26Google Scholar
  9. 9.
    Dunker, T., Lifshits, M.A., Linde, W.: Small deviations of sums of independent variables. Proc. Conf. High Dimensional Probability. Ser. Progress in Probability, Birkhäuser, 43, 1998, pp. 59–74Google Scholar
  10. 10.
    Ibragimov, I.A.: On estimation of the spectral function of a stationary Gaussian process. Theor. Probab. Appl. 8, 366–401 (1963)MATHGoogle Scholar
  11. 11.
    Li, W.V.: Comparison results for the lower tail of Gaussian seminorms. J. Theor. Prob. 5, 1–31 (1992)MathSciNetGoogle Scholar
  12. 12.
    Nazarov, A.I: On the sharp constant in the small ball asymptotics of some Gaussian processes under L2-norm. Nonlinear equations and mathematical analysis (Problems of Math. Anal., 26 (2003)), T. Rozhkovskaya, Novosibirsk, pp. 179–214Google Scholar
  13. 13.
    Khoshnevisan, D., Shi, Z.: Chung’s law for integrated Brownian motion, Trans. Am. Math. Soc. 350, 4253–4264 (1998)CrossRefMathSciNetMATHGoogle Scholar
  14. 14.
    Chen, X., Li, W.V.: Quadratic functionals and small ball probabilities for the m-fold integrated Brownian motion. Ann. Prob. 28, 1052–1077 (2000)Google Scholar
  15. 15.
    Gao, F., Hannig, J., Torcaso, F.: Integrated Brownian motions and Exact L2-small balls. Ann. Prob. 31, 1320–1337 (2003)CrossRefMATHGoogle Scholar
  16. 16.
    Dunford, N., Schwartz, J.T: Linear operators. Part III: Spectral operators. With the assistance of William G. Bade and Robert G. Bartle. Wiley-Interscience, N.Y., 1971Google Scholar
  17. 17.
    Beghin, L., Nikitin, Ya.Yu., Orsingher, E.: Exact small ball constants for some Gaussian processes under the L2-norm. Zapiski Seminarov POMI 298, 5–21 (2003)Google Scholar
  18. 18.
    Gao, F., Hannig, J., Lee T.-Y., Torcaso, F.: Exact L2-small balls of Gaussian processes. J. Theor. Prob. To appearGoogle Scholar
  19. 19.
    Gao, F., Hannig, J., Lee, -Y., T., Torcaso, F.: Laplace Transforms via Hadamard Factorization with applications to Small Ball probabilities. Electronic J. Prob. 8(13), 1–20 (2003)Google Scholar
  20. 20.
    Linde, W.: Comparison results for the small ball behavior of Gaussian random variables. Probability in Banach spaces, 9, Hoffmann-Jørgensen, J., et al. (eds.), 35, Birkhäuser , Ser. Progress in Probability, Boston, MA, 1994, pp. 273–292Google Scholar
  21. 21.
    Safarov, Yu., Vassiliev, D.: The asymptotic distribution of eigenvalues of partial differential operators. Transl. Math. Monographs, 155. AMS: Providence, RI, 1996Google Scholar
  22. 22.
    Naimark, M.A.: Linear Differential Operators. Ed. 2. Moscow, Nauka, 1969 (in Russian). English transl. of the 1st ed.: Naimark M.A.: Linear Differential Operators. Part I: Elementary Theory of Linear Differential Operators. With add. material by the author. N.Y.: F. Ungar Publishing Company Co. XIII, 1967. Part II: Linear differential operators in Hilbert space. With add. material by the author. N.Y.: F. Ungar Publishing Company Co. XV, 1968Google Scholar
  23. 23.
    Reed, M., Simon, B.: Methods of modern mathematical physics. 1: Functional analysis. Academic Press, Inc. New York-London: XVII, 1972Google Scholar
  24. 24.
    Reed, M., Simon, B.: Methods of modern mathematical physics. II: Fourier analysis, self-adjointness. Academic Press, a subsidiary of Harcourt Brace Jovanovich Publishers, New York–San Francisco – London, XV, 1975Google Scholar
  25. 25.
    Birman, M.S., Solomyak, M.Z.: Spectral theory of self-adjoint operators in Hilbert space. Math. and Its Applic. Soviet Series, 5, Dordrecht, etc.: Kluwer Academic Publishers, 1987Google Scholar
  26. 26.
    Chang, C.-H., Ha, C.-W.: The Greens functions of some boundary value problems via the Bernoulli and Euler polynomials. Arch. Mat. 76, 360–365 (2001)MathSciNetMATHGoogle Scholar
  27. 27.
    Lachal, A.: Study of some new integrated statistics: computation of Bahadur efficiency, relation with non-standard boundary value problems. Math. Meth. Statist. 10, 73–104 (2001)MathSciNetMATHGoogle Scholar
  28. 28.
    Lachal, A: Bridges of certain Wiener integrals. Prediction properties, relation with polynomial interpolation and differential equations. Application to goodness-of-fit testing. Bolyai Math. Studies X, Limit Theorems, Balatonlelle (Hungary), 1999, Budapest, 2002, pp. 1–51Google Scholar
  29. 29.
    Henze, N., Nikitin, Ya.Yu: Watson-type goodness-of-fit tests based on the integrated empirical process. Math. Meth. Statist. 11, 183–202 (2002)MathSciNetMATHGoogle Scholar
  30. 30.
    Revuz, D., Yor, M.: Continuous Martingales and Brownian Motion. 3rd ed., Springer-Verlag, 1998Google Scholar
  31. 31.
    Smirnov, V.I: A course of higher mathematics. IV, GITTL: Moscow–Leningrad, 1951, Ed.2 (in Russian). English transl. ed. by I.N. Sneeddon: International Series of Monographs in Pure and Applied Mathematics, 61, Pergamon Press: Oxford–London–New York–Paris, 1964Google Scholar
  32. 32.
    Gradshteyn, I.S., Ryzhik, I.M.: Table of integrals, series, and products. Corr. and enl. ed. by Alan Jeffrey. Academic Press (Harcourt Brace Jovanovich, Publishers), New York–London–Toronto, 1980Google Scholar
  33. 33.
    Donati-Martin, C., Yor, M.: Fubini’s theorem for double Wiener integrals and the variance of the Brownian path. Ann. Inst. Henri Poincaré, Probab. Stat. 27, 181–200 (1991)Google Scholar
  34. 34.
    Li, W.V.: Small Ball Probabilities for Gaussian Markov Processes under the Lp-norm. Stoch. Proc. Appl. 92, 87–102 (2001)CrossRefMathSciNetGoogle Scholar
  35. 35.
    Birman, M.S., Solomyak, M.Z.: Spectral asymptotics of nonsmooth elliptic operators. I. Trans. Moscow Math. Soc. 27, 1–52 (1972)MATHGoogle Scholar
  36. 36.
    Weyl, H.: Das asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Differentialgleichungen. Math. Ann. 71, 441–479 (1912)MATHGoogle Scholar

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© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  1. 1.Department of Mathematics and MechanicsSt.Petersburg State UniversityRussia

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