Journal of Theoretical Probability

, Volume 21, Issue 1, pp 246–265 | Cite as

Time Change Approach to Generalized Excursion Measures, and Its Application to Limit Theorems

  • Patrick J. Fitzsimmons
  • Kouji Yano


It is proved that generalized excursion measures can be constructed via time change of Itô’s Brownian excursion measure. A tightness-like condition on strings is introduced to prove a convergence theorem of generalized excursion measures. The convergence theorem is applied to obtain a conditional limit theorem, a kind of invariance principle where the limit is the Bessel meander.


Itô’s excursion measure Random time change Conditional limit theorems Bessel meander 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Biane, P., Yor, M.: Valeurs principales associées aux temps locaux browniens et processus stables symétriques. C.R. Acad. Sci. Paris Sér. I Math. 300(20), 695–698 (1985) MATHMathSciNetGoogle Scholar
  2. 2.
    Biane, P., Yor, M.: Valeurs principales associées aux temps locaux browniens. Bull. Sci. Math. (2) 111(1), 23–101 (1987) MATHMathSciNetGoogle Scholar
  3. 3.
    Bingham, N.H., Goldie, C.M., Teugels, J.L.: Regular variation. Encyclopedia of Mathematics and its Applications, vol. 27. Cambridge University Press, Cambridge (1989) MATHGoogle Scholar
  4. 4.
    Blumenthal, R.M.: Excursions of Markov processes. Probability and Its Applications. Birkhäuser, Boston (1992) MATHGoogle Scholar
  5. 5.
    Bolthausen, E.: On a functional central limit theorem for random walks conditioned to stay positive. Ann. Probab. 4(3), 480–485 (1976) MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Elliott, J.: Eigenfunction expansions associated with singular differential operators. Trans. Am. Math. Soc. 78, 406–425 (1955) MATHCrossRefGoogle Scholar
  7. 7.
    Iglehart, D.L.: Functional central limit theorems for random walks conditioned to stay positive. Ann. Probab. 2, 608–619 (1974) MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Ikeda, N., Watanabe, S.: Stochastic differential equations and diffusion processes, 2nd edn. North-Holland Mathematical Library, vol. 24. North-Holland, Amsterdam (1989) MATHGoogle Scholar
  9. 9.
    Itô, K.: Poisson point processes attached to Markov processes. in: Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability, Univ. California, Berkeley, CA, 1970/1971, vol. III: Probability Theory, pp. 225–239 (1972) Google Scholar
  10. 10.
    Itô, K., McKean, H.P. Jr.: Diffusion processes and their sample paths. Die Grundlehren der mathematischen Wissenschaften, Band 125. Springer, Berlin (1974). Second printing, corrected MATHGoogle Scholar
  11. 11.
    Jeulin, T.: Semi-martingales et grossissement d’une filtration. Lecture Notes in Mathematics, vol. 833. Springer, Berlin (1980) MATHGoogle Scholar
  12. 12.
    Kasahara, Y.: Spectral theory of generalized second order differential operators and its applications to Markov processes. Jpn. J. Math. (N.S.) 1(1), 67–84 (1975/76) MathSciNetGoogle Scholar
  13. 13.
    Kasahara, Y., Watanabe, S.: Brownian representation of a class of Lévy processes and its application to occupation times of diffusion processes. Ill. J. Math. 50, 515–539 (2006) MATHMathSciNetGoogle Scholar
  14. 14.
    Kotani, S.: Krein’s strings with singular left boundary. Rep. Math. Phys. (2007, to appear) Google Scholar
  15. 15.
    Li, Z., Shiga, T., Tomisaki, M.: A conditional limit theorem for generalized diffusion processes. J. Math. Kyoto Univ. 43(3), 567–583 (2003) MathSciNetGoogle Scholar
  16. 16.
    Pitman, J., Yor, M.: A decomposition of Bessel bridges. Z. Wahrsch. Verw. Geb. 59(4), 425–457 (1982) MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Pitman, J., Yor, M.: Decomposition at the maximum for excursions and bridges of one-dimensional diffusions. In: Itô’s Stochastic Calculus and Probability Theory, pp. 293–310. Springer, Tokyo (1996) Google Scholar
  18. 18.
    Pitman, J.W., Yor, M.: Some divergent integrals of Brownian motion. Adv. Appl. Probab. (Suppl.), 109–116 (1986) Google Scholar
  19. 19.
    Revuz, D., Yor, M.: Continuous martingales and Brownian motion, 3rd edn. Grundlehren der Mathematischen Wissenschaften, vol. 293. Springer, Berlin (1999) MATHGoogle Scholar
  20. 20.
    Stone, C.: Limit theorems for random walks, birth and death processes, and diffusion processes. Ill. J. Math. 7, 638–660 (1963) MATHGoogle Scholar
  21. 21.
    Watanabe, S.: Generalized arc-sine laws for one-dimensional diffusion processes and random walks. In: Stochastic Analysis, Ithaca, NY, 1993. Proc. Sympos. Pure Math., vol. 57, pp. 157–172. Amer. Math. Soc., Providence (1995) Google Scholar
  22. 22.
    Williams, D.: Path decomposition and continuity of local time for one-dimensional diffusions, I. Proc. Lond. Math. Soc. (3) 28, 738–768 (1974) MATHCrossRefGoogle Scholar
  23. 23.
    Xue, X.X.: A zero-one law for integral functionals of the Bessel process. In: Séminaire de Probabilités, XXIV, 1988/89. Lecture Notes in Mathematics, vol. 1426, pp. 137–153. Springer, Berlin (1990) CrossRefGoogle Scholar
  24. 24.
    Yano, K.: Excursion measure away from an exit boundary of one-dimensional diffusion processes. Publ. RIMS 42(3), 837–878 (2006) MATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of CaliforniaSan DiegoUSA
  2. 2.Department of MathematicsOsaka UniversityOsakaJapan

Personalised recommendations