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Small values of Gaussian processes and functional laws of the iterated logarithm
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  • Published: June 1995

Small values of Gaussian processes and functional laws of the iterated logarithm

  • Ditlev Monrad1,3 &
  • Holger Rootzén2,3 

Probability Theory and Related Fields volume 101, pages 173–192 (1995)Cite this article

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Summary

We estimate small ball probabilities for locally nondeterministic Gaussian processes with stationary increments, a class of processes that includes the fractional Brownian motions. These estimates are used to prove Chung type laws of the iterated logarithm.

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References

  1. deAcosta, A.: Small deviations in the functional central limit theorem with applications to functional laws of the iterated logarithm. Ann. Probab.11, 78–101 (1983)

    Google Scholar 

  2. Anderson, T.W.: The integral of a symmetric unimodal function over a symmetric convex set and some probability inequalities. Proc. Am. Math. Soc.6, 170–176 (1955)

    Google Scholar 

  3. Berman S.M.: Gaussian sample functions: uniform dimension and Hölder conditions nowhere. Nagoya Math. J.46, 63–86 (1972)

    Google Scholar 

  4. Berman S.M.: Local nondeterminism and local times for Gaussian processes. Indiana Univ. J. Math.23, 69–94 (1973)

    Google Scholar 

  5. Berman S.M.: Gaussian processes with biconvex covariances. J. Multivar. Anal.8, 30–44 (1978)

    Google Scholar 

  6. Borell, C.: A note on Gauss measures which agree on small balls. Ann. Inst. Henri Poincaré, Sect. B, XIII-3, 231–238 (1977)

    Google Scholar 

  7. Chung, K.L.: On the maximum partial sums of sequences of independent random variables. Trans. Am. Math. Soc.64, 205–233 (1948)

    Google Scholar 

  8. Csáki, E.: A relation between Chung's and Strassen's laws of the iterated logarithm. Z. Wahrscheinlichkeitstheor. Verw. Geb.54, 287–301 (1980)

    Google Scholar 

  9. Cuzick, J., Du Preez, J.: Joint continuity of Gaussian local times. Ann. Probab.10, 810–817 (1982)

    Google Scholar 

  10. Goodman, V., Kuelbs, J.: Rates of clustering for some Gaussian self-similar processes. Probab. Theory Relat. Fields88, 47–75 (1991)

    Google Scholar 

  11. Grill, K.: A lim inf result in Strassen's law of the iterated logarithm. Probab. Theory Relat. Fields89, 149–157 (1991)

    Google Scholar 

  12. Jain, N.C., Marcus, M.B.: Continuity of subgaussian processes. Adv. Probab.4, 81–196 (1978)

    Google Scholar 

  13. Jain, N.C., Pruitt, W.E.: The other law of the iterated logarithm. Ann. Probab.3, 1046–1049 (1975)

    Google Scholar 

  14. Kuelbs, J.: The law of the iterated logarithm and related strong convergence theorems for Banach space valued random variables. (Lect. Notes Math., 539, pp. 224–314) Berlin Heidelberg New York: Springer (1976)

    Google Scholar 

  15. Kuelbs, J., Li, W.V., Talagrand, M.: Lim inf results for Gaussian samples and Chung's functional LIL. (to appear 1992)

  16. Loeve, M.: Probability Theory I. Berlin Heidelberg New York: Springer 1977

    Google Scholar 

  17. Mandelbrot, B.B., van Ness, J.W.: Fractional Brownian motions, fractional noises and applications. SIAM Review10, 422–437 (1968)

    Google Scholar 

  18. Marcus, M.B.: Gaussian processes with stationary increments possessing discontinuous sample paths. Pac. J. Math.26, 149–157 (1968)

    Google Scholar 

  19. Oodaira, H.: On Strassen's version of the law of the iterated logarithm for Gaussian processes. Z. Wahrscheinlichkeitstheor. Verw. Geb.21, 289–299 (1972)

    Google Scholar 

  20. Pitt, L.D.: Local times for Gaussian vector fields. Indiana Univ. J. Math.27, 309–330 (1978)

    Google Scholar 

  21. Pitt, L.D., Tran, L.T.: Local sample path properties of Gaussian fields. Ann. Probab.7, 477–493 (1979)

    Google Scholar 

  22. Šidák, Z.: On multivariate normal probabilities of rectangles: their dependence on correlations. Ann. Math. Stat.39, 1425–1434 (1968)

    Google Scholar 

  23. Strassen, V.: An invariance principle for the law of the iterated logarithm. Z. Wahrscheinlichkeitstheor. Verw. Geb.3, 211–226 (1964)

    Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Statistics, University of Illinois, 725 South Wright Street, 61820, Champaign, IL, USA

    Ditlev Monrad

  2. Department of Mathematics, Chalmers University, S-41296, Goteborg, Sweden

    Holger Rootzén

  3. Center for Stochastic Processes, University of North Carolina at Chapel Hill, 27599, Chapel Hill, NC, USA

    Ditlev Monrad & Holger Rootzén

Authors
  1. Ditlev Monrad
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  2. Holger Rootzén
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Additional information

Research supported by the United States Air Force office of Scientific Research, Contract No. 91-0030

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Monrad, D., Rootzén, H. Small values of Gaussian processes and functional laws of the iterated logarithm. Probab. Th. Rel. Fields 101, 173–192 (1995). https://doi.org/10.1007/BF01375823

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  • Received: 16 November 1992

  • Revised: 19 July 1994

  • Issue Date: June 1995

  • DOI: https://doi.org/10.1007/BF01375823

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Mathematics Subject Classification (1991)

  • 60F15
  • 60G15
  • 60G17
  • 60G18
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