The exploration process of inhomogeneous continuum random trees, and an extension of Jeulin's local time identity

Probability Theory and Related Fields

Volumn 129, Issue 2, 2004, Pages 182-218

  Aldous, David ^a   Miermont, Grégory ^b   Pitman, Jim ^a  

^a UNIVERSITY OF CALIFORNIA   (United States)

^b SORBONNE UNIVERSITÉ   (France)

Author keywords

Continuum random tree; Exchangeable increments; Exploration process; L vy process; Weak convergence

Indexed keywords

EID: 2942558363     PISSN: 01788051     EISSN: None     Source Type: Journal    
DOI: 10.1007/s00440-003-0334-7     Document Type: Article

References (34)

1. Aldous, D.J.: The continuum random tree I. Ann. Probab. 19, 1-28 (1991)
2. Aldous, D.J.: The continuum random tree II: an overview. In: M.T. Barlow and N.H. Bingham, (eds), Stochastic Analysis. Cambridge University Press, pp. 23-70, 1991
3. Aldous, D.J.: The continuum random tree III. Ann. Probab. 21, 248-289 (1993)
4. Bridge Asymptotics for Random p-Mappings. Electr. J. Probab. 9, 37-56 (2004)
5. Aldous, D.J., Miermont, G., Pitman, J.: Weak Convergence of Random p-Mappings and the Exploration Process of the Inhomogeneous Continuum Random Tree. arXiv:math.pr/0401115
6. Aldous, D.J., Pitman, J.: Brownian bridge asymptotics for random mappings. Random Struct. Algorithms 5, 487-512 (1994)
7. Aldous, D.J., Pitman, J.: The standard additive coalescent. Ann. Probab 26, 1703-1726 (1998)
8. Aldous, D.J., Pitman, J.: A family of random trees with random edge-lengths. Random Struct. Algorithms 15, 176-195 (1999)
9. Aldous, D.J., Pitman, J.: Inhomogeneous continuum random trees and the entrance boundary of the additive coalescent. Probab. Th. Rel. Fields 118, 455-482 (2000)
10. Aldous, D.J., Pitman, J.: Invariance principles for non-uniform random mappings and trees. In: V. Malyshev and A.M. Vershik, (eds.), Asymptotic Combinatorics with Applications in Mathematical Physics. Kluwer Academic Publishers, pp. 113-147, 2002
11. Bertoin, J.: On the local rate of growth of Lévy processes with no positive jumps. Stoch. Proc. App. 55, 91-100 (1995)
12. Bertoin, J.: A fragmentation process connected to Brownian motion. Probab. Theory Relat. Fields 117, 289-301 (2000)
13. Bertoin, J.: Eternal additive coalescent and certain bridges with exchangeable increments. Ann. Probab. 29, 344-360 (2001)
14. Biane, P., Yor, M.: Sur la loi des temps locaux browniens pris en un temps exponentiel. In: Séminaire de Probabilités XXII. Lecture Notes in Math. 1321, Springer, pp. 454-466, 1988
15. Camarri, M., Pitman, J.: Limit distributions and random trees derived from the birthday problem with unequal probabilities. Electron. J. Probab. 5(2), 1-18 (2000)
16. Chassaing, P., Louchard, G.: Phase transition for parking blocks. Brownian excursion and coalescence. Random Struct. Algorithms 21, 76-119 (2002)
17. Duquesne, T., Le Gall, J.-F.: Random trees. Lévy processes and spatial branching processes. Astérisque 281, (2002)
18. Durrett, R.: Probability: theory and examples. Duxbury Press, Belmont, CA, second edition, 1996
19. Jeulin, T.: Application de la théorie du grossissement à l'étude des temps locaux browniens. In: T. Jeulin and M. Yor (eds.), Grossissements de filtrations: exemples et applications. Lecture notes in Maths 1118, Springer, Berlin, 1985
20. Kallenberg, O.: Canonical representations and convergence criteria for processes with interchangeable increments. Z. Wahrsch. Verw. Gebiete 27, 23-36 (1973)
21. Kallenberg, O.: Path properties for processes with independent and interchangeable increments. Z. Wahrsch. Verw. Gebiete 28, 257-271 (1974)
22. Kersting, G.: On the height profile of a conditioned Galton-Watson tree. Preprint, 1998
23. Knight, F.B.: The uniform law for exchangeable and Lévy process bridges. Hommage à P. A. Meyer et J. Neveu. Astérisque 236, 171-188 (1996)
24. Le Gall, J.-F.: The uniform tree in a Brownian excursion. Probab. Th. Rel. Fields 96, 369-383 (1993)
25. Le Gall, J.-F.: Brownian excursions, trees and measure-valued branching processes. Ann. Probab. 19, 1299-1439 (1991)
26. Le Gall, J.-F., Le Jan, Y.: Branching processes in Lévy processes: The exploration process. Ann. Probab. 26, 213-252 (1998)
27. Marckert, J.-F., Mokkadem, A.: The depth-first processes of Galton-Watson trees converge to the same Brownian excursion. Ann. Probab. 31, 1655-1678 (2003)
28. Miermont, G.: Ordered additive coalescent and fragmentations associated to Lévy processes with no positive jumps. Electr. J. Probab. 6, paper 14, 33 pp. (2001)
29. Millar, P.W.: Comparison theorems for sample function growth. Ann. Probab. 9, 330-334 (1981)
30. O'Cinneide, C.A., Pokrovskii, A.V.: Nonuniform random transformations. Ann. Appl. Probab. 10(4), 1151-1181 (2000)
31. Pitman, J.: Random mappings, forests and subsets associated with Abel-Cayley-Hurwitz multinomial expansions. Séminaire Lotharingien de Combinatoire, Issue 46, 45 pp., (2001) www.mat.univie.ac.at/~slc/
32. Pitman, J., Yor, M.: Arcsine laws and interval partitions derived from a stable subordinator. Proc. London Math. Soc. 3 (65), 326-356 (1992)
33. Revuz, D., Yor, M.: Continuous martingales and Brownian motion. Springer, Berlin-Heidelberg, third edition, 1999
34. Vervaat, W.: A relation between Brownian bridge and Brownian excursion. Ann. Probab. 7, 143-149 (1979)