On the relative lengths of excursions derived from a stable subordinator
Séminaire de probabilités de Strasbourg, Volume 31  (1997), p. 287-305
@article{SPS_1997__31__287_0,
     author = {Pitman, Jim and Yor, Marc},
     title = {On the relative lengths of excursions derived from a stable subordinator},
     journal = {S\'eminaire de probabilit\'es de Strasbourg},
     publisher = {Springer - Lecture Notes in Mathematics},
     volume = {31},
     year = {1997},
     pages = {287-305},
     zbl = {0884.60072},
     mrnumber = {1478738},
     language = {en},
     url = {http://www.numdam.org/item/SPS_1997__31__287_0}
}
Pitman, Jim; Yor, Marc. On the relative lengths of excursions derived from a stable subordinator. Séminaire de probabilités de Strasbourg, Volume 31 (1997) , pp. 287-305. http://www.numdam.org/item/SPS_1997__31__287_0/

[1] M. Barlow. Skew brownian motion and a one-dimensional differential equation. Stochastics, 25:1-2, 1988. | MR 1008231 | Zbl 0657.60075

[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. | Numdam | MR 1022918 | Zbl 0738.60072

[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. | MR 116376 | Zbl 0112.10105

[4] B. Fristedt and S.J. Taylor. Constructions of local time for a Markov process. Z. Wahrsch. Verw. Gebiete, 62:73 - 112, 1983. | MR 684210 | Zbl 0519.60078

[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. | MR 579823 | Zbl 0427.60048

[6] J.M. Harrison and L.A. Shepp. On skew Brownian motion. The Annals of Probability, 9:309 - 313, 1981. | MR 606993 | Zbl 0462.60076

[7] J.F.C. Kingman. Random discrete distributions. J. Roy. Statist. Soc. B, 37:1-22, 1975. | MR 368264 | Zbl 0331.62019

[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. | MR 896740 | Zbl 0622.60083

[9] J. Lamperti. An occupation time theorem for a class of stochastic processes. Trans. Amer. Math. Soc., 88:380 - 387, 1958. | MR 94863 | Zbl 0228.60046

[10] J. Lamperti. An invariance principle in renewal theory. Ann. Math. Stat., 33:685 - 696, 1962. | MR 137176 | Zbl 0106.33902

[11] P. Lévy. Sur certains processus stochastiques homogènes. Compositio Math., 7:283-339, 1939. | JFM 65.1346.02 | Numdam | MR 919 | Zbl 0022.05903

[12] M. Perman. Order statistics for jumps of normalized subordinators. Stoch. Proc. Appl., 46:267-281, 1993. | MR 1226412 | Zbl 0777.60070

[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. | MR 1156448 | Zbl 0741.60037

[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. | MR 1466546 | Zbl 0882.60081

[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. | MR 1168191 | Zbl 0769.60014

[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. | MR 1434129 | Zbl 0880.60076

[17] J. Pitman and M. Yor. Random discrete distributions derived from self-similar random sets. Electronic J. Probability, 1:Paper 4, 1-28, 1996. | MR 1386296 | Zbl 0891.60042

[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] C.L. Scheffer. The rank of the present excursion. Stoch. Proc. Appl., 55:101-118, 1995. | MR 1312151 | Zbl 0819.60069

[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. | MR 899989 | Zbl 0596.60054

[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] S. Watanabe. On time inversion of one-dimensional diffusion processes. Z. Wahrsch. Verw. Gebiete, 31:115-124, 1975. | MR 365731 | Zbl 0286.60035

[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. | MR 1335470 | Zbl 0824.60080

[24] J.G. Wendel. Zero-free intervals of semi-stable Markov processes. Math. Scand., 14:21 - 34, 1964. | MR 171319 | Zbl 0132.12802

[25] M. Yor. Some Aspects of Brownian Motion. Lectures in Math., ETH Zürich. Birkhaüser, 1992. Part I: Some Special Functionals. | MR 1193919 | Zbl 0779.60070

[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. | MR 1360280 | Zbl 0827.60010