Advertisement

Hirsch’s integral test for the iterated Brownian motion

  • Jean Bertoin
  • Zhan Shi
Chapter
Part of the Lecture Notes in Mathematics book series (LNM, volume 1626)

Abstract

We present an analogue of Hirsch’s integral test to decide whether a function belongs to the lower class of the supremum process of an iterated Brownian motion.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Bibliography

  1. Bertoin, J. (1996), Iterated Brownian motion and stable(1/4) subordinator, Statist. Probab. Letters. (to appear).Google Scholar
  2. Burdzy, K. (1993), Some path properties of iterated Brownian motion, in: E. Çinlar, K.L. Chung and M. Sharpe, eds, Seminar on stochastic processes 1992 (Birkhäuser) pp. 67–87.Google Scholar
  3. Burdzy, K. and Khoshnevisan, D. (1995), The level sets of iterated Brownian motion, Séminaire de Probabilités XXIX pp. 231–236, Lecture Notes in Math. 1613, Springer.MathSciNetzbMATHGoogle Scholar
  4. Csáki, E. (1978), On the lower limits of maxima and minima of Wiener process and partial sums, Z. Wahrscheinlichkeitstheorie verw. Gebiete 43, 205–221.MathSciNetCrossRefzbMATHGoogle Scholar
  5. Csáki, E., Csörgő, M., Földes, A. and Révész, P. (1989), Brownian local time approximated by a Wiener sheet, Ann. Probab. 17, 516–537.MathSciNetCrossRefzbMATHGoogle Scholar
  6. Csáki, E., Csörgő, M., Földes, A. and Révész, P. (1995), Global Strassen-type theorems for iterated Brownian motions, Stochastic Process. Appl. 59, 321–341.MathSciNetCrossRefzbMATHGoogle Scholar
  7. Csáki, E., Földes, A. and Révész P. (1995), Strassen theorems for a class of iterated processes, preprint.Google Scholar
  8. Deheuvels, P. and Mason, D.M. (1992), A functional LIL approach to pointwise Bahadur-Kiefer theorems, in: R.M. Dudley, M.G. Hahn and J. Kuelbs, eds, Probability in Banach spaces 8 (Birkhäuser) pp. 255–266.Google Scholar
  9. Feller, W. E. (1971), An introduction to probability theory and its applications, 2nd edn, vol. 2. Wiley, New York.zbMATHGoogle Scholar
  10. Funaki, T. (1979), A probabilistic construction of the solution of some higher order parabolic differential equations, Proc. Japan Acad. 55, 176–179.MathSciNetCrossRefzbMATHGoogle Scholar
  11. Hu, Y., Pierre Loti Viaud, D. and Shi, Z. (1995), Laws of the iterated logarithm for iterated Wiener processes, J. Theoretic. Prob. 8, 303–319.MathSciNetCrossRefzbMATHGoogle Scholar
  12. Hu, Y. and Shi, Z. (1995), The Csörgő-Révész modulus of non-differentiability of iterated Brownian motion, Stochastic Process. Appl. 58, 267–279.MathSciNetCrossRefzbMATHGoogle Scholar
  13. Khoshnevisan, D. and Lewis, T.M. (1996), The uniform modulus of iterated Brownian motion, J. Theoretic. Prob. (to appear).Google Scholar
  14. Khoshnevisan, D. and Lewis, T.M. (1996), Chung’s law of the iterated logarithm for iterated Brownian motion, Ann. Inst. Henri Poincaré (to appear)Google Scholar
  15. Pitman, J.W. and Yor, M. (1993). Homogeneous functionals of Brownian motion (unpublished manuscript).Google Scholar
  16. Shi, Z. (1995), Lower limits of iterated Wiener processes, Statist. Probab. Letters. 23, 259–270.MathSciNetCrossRefzbMATHGoogle Scholar
  17. Spitzer, F. (1964). Principles of random walks. Van Nostrand, Princeton.CrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag 1996

Authors and Affiliations

  • Jean Bertoin
    • 1
  • Zhan Shi
    • 2
  1. 1.Laboratoire de ProbabilitésUniversité Pierre et Marie CurieParis Cedex 05France
  2. 2.LSTAUniversité Pierre et Marie CurieParis Cedex 05France

Personalised recommendations