On the relative lengths of excursions derived from a stable subordinator

Jim Pitman; Marc Yor

Séminaire de probabilités de Strasbourg (1997)

  • Volume: 31, page 287-305

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Pitman, Jim, and Yor, Marc. "On the relative lengths of excursions derived from a stable subordinator." Séminaire de probabilités de Strasbourg 31 (1997): 287-305. <http://eudml.org/doc/113965>.

@article{Pitman1997,
author = {Pitman, Jim, Yor, Marc},
journal = {Séminaire de probabilités de Strasbourg},
keywords = {excursions of a recurrent Markov process; inverse local time process; stable subordinator; generalized arc-sine laws},
language = {eng},
pages = {287-305},
publisher = {Springer - Lecture Notes in Mathematics},
title = {On the relative lengths of excursions derived from a stable subordinator},
url = {http://eudml.org/doc/113965},
volume = {31},
year = {1997},
}

TY - JOUR
AU - Pitman, Jim
AU - Yor, Marc
TI - On the relative lengths of excursions derived from a stable subordinator
JO - Séminaire de probabilités de Strasbourg
PY - 1997
PB - Springer - Lecture Notes in Mathematics
VL - 31
SP - 287
EP - 305
LA - eng
KW - excursions of a recurrent Markov process; inverse local time process; stable subordinator; generalized arc-sine laws
UR - http://eudml.org/doc/113965
ER -

References

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  1. [1] M. Barlow. Skew brownian motion and a one-dimensional differential equation. Stochastics, 25:1-2, 1988. Zbl0657.60075MR1008231
  2. [2] M. Barlow, J. Pitman, and M. Yor. Une extension multidimensionnelle de la loi de l'arc sinus. In Séminaire de Probabilités XXIII, pages 294-314. Springer, 1989. Lecture Notes in Math.1372. Zbl0738.60072MR1022918
  3. [3] E.B. Dynkin. Some limit theorems for sums of independent random variables with infinite mathematical expectations. IMS-AMS Selected Translations in Math. Stat. and Prob., 1:171-189, 1961. Zbl0112.10105MR116376
  4. [4] B. Fristedt and S.J. Taylor. Constructions of local time for a Markov process. Z. Wahrsch. Verw. Gebiete, 62:73 - 112, 1983. Zbl0519.60078MR684210
  5. [5] P. Greenwood and J. Pitman. Construction of local time and Poisson point processes from nested arrays. Journal of the London Mathematical Society, 22:182-192, 1980. Zbl0427.60048MR579823
  6. [6] J.M. Harrison and L.A. Shepp. On skew Brownian motion. The Annals of Probability, 9:309 - 313, 1981. Zbl0462.60076MR606993
  7. [7] J.F.C. Kingman. Random discrete distributions. J. Roy. Statist. Soc. B, 37:1-22, 1975. Zbl0331.62019MR368264
  8. [8] F.B. Knight. On the duration of the longest excursion. In E. Cinlar, K.L. Chung, and R.K. Getoor, editors, Seminar on Stochastic Processes, pages 117-148. Birkhäuser, 1985. Zbl0622.60083MR896740
  9. [9] J. Lamperti. An occupation time theorem for a class of stochastic processes. Trans. Amer. Math. Soc., 88:380 - 387, 1958. Zbl0228.60046MR94863
  10. [10] J. Lamperti. An invariance principle in renewal theory. Ann. Math. Stat., 33:685 - 696, 1962. Zbl0106.33902MR137176
  11. [11] P. Lévy. Sur certains processus stochastiques homogènes. Compositio Math., 7:283-339, 1939. Zbl0022.05903MR919JFM65.1346.02
  12. [12] M. Perman. Order statistics for jumps of normalized subordinators. Stoch. Proc. Appl., 46:267-281, 1993. Zbl0777.60070MR1226412
  13. [13] M. Perman, J. Pitman, and M. Yor. Size-biased sampling of Poisson point processes and excursions. Probability Theory and Related Fields, 92:21-39, 1992. Zbl0741.60037MR1156448
  14. [14] J. Pitman. Partition structures derived from Brownian motion and stable subordinators. Technical Report 346, Dept. Statistics, U.C. Berkeley, 1992. To appear in Bernoulli. Zbl0882.60081MR1466546
  15. [15] J. Pitman and M. Yor. Arcsine laws and interval partitions derived from a stable subordinator. Proc. London Math. Soc. (3), 65:326-356, 1992. Zbl0769.60014MR1168191
  16. [16] J. Pitman and M. Yor. The two-parameter Poisson-Dirichlet distribution derived from a stable subordinator. Technical Report 433, Dept. Statistics, U.C. Berkeley, 1995. To appear in The Annals of Probability. Zbl0880.60076MR1434129
  17. [17] J. Pitman and M. Yor. Random discrete distributions derived from self-similar random sets. Electronic J. Probability, 1:Paper 4, 1-28, 1996. Zbl0891.60042MR1386296
  18. [18] J. Pitman and M. Yor. Some conditional expectations given an average of a stationary or self-similar random process. Technical Report 438, Dept. Statistics, U.C. Berkeley, 1996. In preparation. 
  19. [19] C.L. Scheffer. The rank of the present excursion. Stoch. Proc. Appl., 55:101-118, 1995. Zbl0819.60069MR1312151
  20. [20] M.S. Taqqu. A bibliographical guide to self-similar processes and long-range dependence. In Dependence in Probab. and Stat.: A Survey of Recent Results; Ernst Eberlein, Murad S. Taqqu (Ed.), pages 137-162. Birkhäuser (Basel, Boston), 1986. Zbl0596.60054MR899989
  21. [21] J. Walsh. A diffusion with a discontinuous local time. In Temps Locaux, volume 52-53 of Astérisque, pages 37-45. Soc. Math. de France, 1978. 
  22. [22] S. Watanabe. On time inversion of one-dimensional diffusion processes. Z. Wahrsch. Verw. Gebiete, 31:115-124, 1975. Zbl0286.60035MR365731
  23. [23] S. Watanabe. Generalized arc-sine laws for one-dimensional diffusion processes and random walks. In Proceedings of Symposia in Pure Mathematics, volume 57, pages 157-172. A.M.S., 1995. Zbl0824.60080MR1335470
  24. [24] J.G. Wendel. Zero-free intervals of semi-stable Markov processes. Math. Scand., 14:21 - 34, 1964. Zbl0132.12802MR171319
  25. [25] M. Yor. Some Aspects of Brownian Motion. Lectures in Math., ETH Zürich. Birkhaüser, 1992. Part I: Some Special Functionals. Zbl0779.60070MR1193919
  26. [26] M. Yor. Random Brownian scaling and some absolute continuity relationships. In E. Bolthausen, M. Dozzi, and F. Russo, editors, Seminar on Stochastic Analysis, Random Fields and Applications. Centro Stefano Franscini, Ascona, 1993, pages 243-252. Birkhäuser, 1995. Zbl0827.60010MR1360280

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