On the lengths of excursions of some Markov processes

Pitman, James W.; Yor, Marc

Séminaire de probabilités de Strasbourg, Tome 31 (1997), pp. 272-286.

[1] J. Azéma. Sur les fermés aléatoires. In Séminaire de Probabilités XIX, pages 397-495. Springer, 1985. Lecture Notes in Math. 1123. | Numdam | MR | Zbl

[2] H. Dym and H.P. Mckean. Gaussian Processes, Function Theory and the Inverse Spectral Problem. Academic Press, 1976. | MR | Zbl

[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 | Zbl

[4] R.K. Getoor. Excursions of a Markov process. Annals of Probability, 7 :244 - 266, 1979. | MR | Zbl

[5] J. Hawkes and A. Truman. Statistics of local time and excursions for the Ornstein-Uhlenbeck process. In M. Barlow and N. Bingham, editors, Stochastic Analysis, volume 167 of Lect. Note Series, pages 91-101. London Math. Soc., 1991. | MR | Zbl

[6] Y. Hu, Z. Shi, and M. Yor. Some applications of Lévy's area formula to pseudo-Brownian and pseudo-Bessel bridges. In Exponential functionals and principal values of Brownian motion. Biblioteca de la Revista Matematica Ibero-Americana, Madrid, 1996/1997. | MR | Zbl

[7] K. Itô. Poisson point processes attached to Markov processes. In Proc. 6th Berk. Symp. Math. Stat. Prob., volume 3, pages 225-240, 1971. | MR | Zbl

[8] K. Itô and H.P. Mckean. Diffusion Processes and their Sample Paths. Springer, 1965. | Zbl

[9] I.S. Kac and M.G. Krein. On the spectral function of the string. Amer. Math. Society Translations, 103:19-102, 1974. | Zbl

[10] O. Kallenberg. Canonical representations and convergence criteria for processes with interchangeable increments. Z. Wahrsch. Verw. Gebiete, 27:23-36, 1973. | MR | Zbl

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

[12] J.F.C. Kingman. Poisson Processes. Clarendon Press, Oxford, 1993. | MR | Zbl

[13] G.N. Kinkladze. A note on the structure of processes the measure of which is absolutely continuous with respect to the Wiener process modulus measure. Stochastics, 8:39 - 44, 1982. | MR | Zbl

[14] F.B. Knight. Characterization of the Lévy measure of inverse local times of gap diffusions. In Seminar on Stochastic Processes, 1981, pages 53-78. Birkhäuser, Boston, 1981. | MR | Zbl

[15] 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 | Zbl

[16] S. Kotani and S. Watanabe. Krein's spectral theory of strings and generalized diffusion processes. In M. Fukushima, editor, Functional Analysis in Markov Processes, pages 235-249. Springer, 1982. Lecture Notes in Math. 923. | MR | Zbl

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

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

[19] 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 | Zbl

[20] J. Pitman. Stationary excursions. In Séminaire de Probabilités XXI, pages 289-302. Springer, 1986. Lecture Notes in Math. 1247. | Numdam | MR | Zbl

[21] J. Pitman. Cyclically stationary Brownian local time processes. To appear in Probability Theory and Related Fields, 1996. | MR | Zbl

[22] J. Pitman and M. Yor. A decomposition of Bessel bridges. Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete, 59:425-457, 1982. | MR | Zbl

[23] J. Pitman and M. Yor. Sur une décomposition des ponts de Bessel. In M. Fukushima, editor, Functional Analysis in Markov Processes, pages 276-285. Springer, 1982. Lecture Notes in Math. 923. | MR | Zbl

[24] 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 | Zbl

[25] 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 | Zbl

[26] J. Pitman and M. Yor. On the relative lengths of excursions derived from a stable subordinator. Technical Report 469, Dept. Statistics, U.C. Berkeley, 1996. To appear in Séminaire de Probabilités XXXI. | Numdam | MR | Zbl

[27] C. Rainer. Projection d'une diffusion sur sa filtration lente. In J. Azéma, M. Emery, and M. Yor, editors, Séminaire de Probabilités XXX, pages 228-242. Springer, 1996. Lecture Notes in Math. 1626. | Numdam | MR | Zbl

[28] D. Revuz and M. Yor. Continuous martingales and Brownian motion. Springer, Berlin-Heidelberg, 1994. 2nd edition. | MR | Zbl

[29] T. Shiga and S. Watanabe. Bessel diffusions as a one-parameter family of diffusion processes. Z. Wahrsch. Verw. Gebiete, 27:37-46, 1973. | MR | Zbl

[30] A. Truman and D. Williams. A generalised arc sine law and Nelson's stochastic mechanics of one-dimensional time homogeneous diffusions. In M. A. Pinsky, editor, Diffusion Processes and Related Problems in Analysis, Vol.1, volume 22 of Progress in Probability, pages 117-135. Birkhäuser, 1990. | MR | Zbl

[31] A. Truman and D. Williams. Excursions and Itô calculus in Nelson's stochastic mechanics. In A. Boutet de Monvel et. al. editor, Recent Developments in Quantum Mechanics : Proceedings of the Brasov Conference, Poiana Brasov 1989, volume 12 of Mathematical Physics Studies. Kluwer Academic, 1991. | MR | Zbl

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

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