Journal of Theoretical Probability

, Volume 5, Issue 1, pp 1–31

Comparison results for the lower tail of Gaussian seminorms

  • Wenbo V. Li
Article

Abstract

Let ξ=(ξn) be i.i.d.N(0, 1) random variables andq(x), q′(x):R→[0, ∞) be seminorms. We investigate necessary and sufficient conditions that the ratio ofP(q(ξ)<ε) andP(q′(ξ)<ε) goes to a positive constant as ε→0+. We give satisfactory answers forl2-norms and also some results for sup-norms andlp-norms. Some applications are given to the rate of escape of infinite dimensional Brownian motion, and we give the lower tail of the Ornstein-Uhlenbeck process and a weighted Brownian bridge under theL2-norms.

Key Words

Lower tail Gaussian seminorms Gaussian processes infinite dimensional Brownian motion 

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References

  1. 1.
    Anderson, T. W., and Darling, D. A. (1952). Asymptotic theory of certain “goodness of fit” criteria based on stochastic processes.Ann. Math. Stat. 23, 193–212.Google Scholar
  2. 2.
    Ash, R. B., and Gardner, m. F. (1975).Topics in Stochastic Processes Academic Press, New York.Google Scholar
  3. 3.
    Bass, R. F. (1988). Probability estimates for multiparameter Brownian processes.Ann. Prob 16, 251–264.Google Scholar
  4. 4.
    Borell, C. (1975). The Brunn-Minkowski inequality in Gauss space.Invent. Math. 30, 207–216.Google Scholar
  5. 5.
    Cox, D. D. (1980). Normalized Brownian motion on Banach spaces. Ph.D. thesis, University of Washington, Seattle.Google Scholar
  6. 6.
    Cox, D. D. (1982). On the existence of natural rate of escape functions for infinite dimensional Brownian motions.Ann. Prob. 10, 623–638.Google Scholar
  7. 7.
    Csáki, E. (1982). 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, pp. 19–26.Google Scholar
  8. 8.
    Erickson, K. B. (1980). Rates of escape of infinite dimensional Brownian motion.Ann. Prob. 8, 325–338.Google Scholar
  9. 9.
    Fernique, X. (1970). Intégrabilité des vecteurs gaussiens.C. R. Acad. Sci. Paris 270, 1698–1699.Google Scholar
  10. 10.
    Hoffmann-Jørgensen, J., Shepp, L. A., and Dudley, R. M. (1979). On the lower tail of Gaussian seminorms.Ann. Prob. 7, 319–342.Google Scholar
  11. 11.
    Hwang, C. (1980). Gaussian measures of large balls in a Hilbert space.Proc. AMS 78, 107–110.Google Scholar
  12. 12.
    Ibragimov, I. A. (1982). On the probability that a Gaussian vector with values in a Hilbert space hits a sphere of small radius.J. Sov. Math. 20, 2164–2174.Google Scholar
  13. 13.
    Kac, M., and Siegert, A. J. F. (1947). An explicit representation of a stationary Gaussian process.Ann. Math. Stat. 18, 438–442.Google Scholar
  14. 14.
    Kamke, E. (1977).Differentialgleichungen. Teubner, Stuttgart.Google Scholar
  15. 15.
    Kuelbs, J. (1968). The invariance principle for a lattice of random variables.Ann. Math. Stat. 39, 382–389.Google Scholar
  16. 16.
    Kuelbs, J. (1978). Rates of growth for Banach space valued independent increments processes.Proc. 2nd Oberwolfach Conference on Probability on Banach Spaces. Lecture Notes in Mathematics, Vol. 709, pp. 151–169, Springer, Berlin.Google Scholar
  17. 17.
    Li, W. V. (1992). Lim inf results for the Wiener process and its increments under theL 2-norm.Probability Theory and Related Fields, to appear.Google Scholar
  18. 18.
    Marcus, M. B., and Shepp, L. A. (1972). Sample behavior of Gaussian processes.Proc. Sixth. Berkeley Symp. Math. Stat. Prob. 2, 423–441.Google Scholar
  19. 19.
    Sytaya, G. N. (1974). On some asymptotic representations of the Gaussian measure in a Hilbert space. InTheory of Stochastic Processes. Publication No. 2, pp. 93–104, Ukrainian Academy of Sciences, Republican Interdepartmental Collection (in Russian).Google Scholar
  20. 20.
    Watson, G. N. (1966).A Treatise on the Theory of Bessel Functions, Cambridge University Press, Cambridge, England.Google Scholar
  21. 21.
    Zolotarev, V. M. (1961). Concerning a certain probability problem.Theor. Probab. Appl. 6, 201–204.Google Scholar
  22. 22.
    Zolotarev, V. M. (1986). Asymptotic behavior of the Gaussian measure inl 2.J. Sov. Math. 24, 2330–2334.Google Scholar

Copyright information

© Plenum Publishing Corporation 1992

Authors and Affiliations

  • Wenbo V. Li
    • 1
  1. 1.Department of MathematicsUniversity of Wisconsin-MadisonMadison

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