The depth first processes of Galton-Watson trees converge to the same Brownian excursion

Annals of Probability

Volumn 31, Issue 3, 2003, Pages 1655-1678

  Marckert, Jean François ^a   Mokkadem, Abdelkader ^a  

^a LABORATOIRE DES SCIENCES DU CLIMAT ET DE L ENVIRONNEMENT   (France)

Author keywords

Brownian excursion; Depth; Galton Watson trees; Ladder variable; Moderate deviations; Simple tree; Subtree

Indexed keywords

EID: 0038373017     PISSN: 00911798     EISSN: None     Source Type: Journal    
DOI: 10.1214/aop/1055425793     Document Type: Article

References (28)

1. ALDOUS, D. (1991). The continuum random tree. II. An overview. In Stochastic Analysis (M. T. Barlow and N. H. Bingham, eds.) 23-70. Cambridge Univ. Press.
2. ALDOUS, D. (1993). The continuum random tree. III. Ann. Probab. 21 248-289.
3. ALDOUS, D. (1998). Brownian excursion conditioned on its local time. Elect. Comm. Probab. 3 79-90.
4. CHASSAING, P., MARCKERT, J. F. and YOR, M. (2000). The height and width of simple trees. In Mathematics and Computer Science: Algorithms, Trees, Combinatorics and Probabilities (D. Gardy and A. Mokkadem, eds.) 17-30. Birkhäuser, Boston.
5. CHASSAING, P. and MARCKERT, J. F. (2001). Parking functions, empirical processes, and the width of rooted labeled trees. Elec. J. Combin. 8 R14.
6. CORMEN, T. H., LEISERSON, C. E. and RIVEST, R. L. (1990). Introduction to Algorithms. MIT Press.
7. CSAKI, E. and MOHANTY, S. G. (1981). Excursion and meander in random walk. Canad. J. Statist. 9 57-70.
8. DRMOTA, M. (1994). The height distribution of leaves in rooted trees. Discrete Math. Appl. 4 45-58.
9. DRMOTA, M. (1996). On nodes of given degree in random trees. In Probabilistic Methods in Discrete Mathematics (V. F. Kolchin, V. Ya. Kozlov, V. V. Mazalov, Yu. L. Pavlov and Yu. V. Prokhorov, eds.). VSP, Zeist, The Netherlands.
10. DRMOTA, M. and GITTENBERGER, B. (1997). On the profile of random trees. Random Structures Algorithms 10 421-451.
11. DUQUESNE, T. and LE GALL, J. F. (2002). Random Trees, Lévy Processes and Spatial Branching Processes. Astérisque 281.
12. DURRETT, R. T., IGLEHART, R. D. L. and MILLER, D. R. (1977). Weak convergence to Brownian meander and Brownian excursion. Ann. Probab. 5 117-129.
13. FELLER, W. (1971). An Introduction to Probability Theory and Its Applications 2, 2nd ed. Wiley, New York.
14. FLAJOLET, P., GOURDON, X. and MARTINEZ, C. (1997). Patterns in random binary search trees. Random Structures Algorithms 11 223-244.
15. FUK, D. H. and NAGAEV, S. V. (1971). Probability inequalities for sums of independent random variables. Theory Probab. Appl. 16 643-660.
16. GUTJAHR, W. and PFLUG, G. C. (1992). The asymptotic contour process of a binary tree is a Brownian excursion. Stochastic Process. Appl. 41 69-89.
17. KENNEDY, D. P. (1976). The distribution of the maximum Brownian excursion. J. Appl. Probab. 13 371-376.
18. LE GALL, J. F. (2000). Lévy processes and branching processes. Courses Notes.
19. LE GALL, J. F. and LE JAN, Y. (1998). Branching processes in Lévy processes: The exploration process. Ann. Probab. 26 213-252.
20. LIMIC, V. (2000). On the behavior of LIFO preemptive resume queues in heavy traffic. Electron. Comm. Probab. 5 13-27.
21. LIMIC, V. (2001). A LIFO queue in heavy traffic. Ann. Appl. Probab. 11 301-331.
22. MARCKERT, J. F. (2000). The contour of size n general planar trees. Preprint 53-00, Univ. Versailles.
23. MEIR, A. and MOON, J. W. (1978). On the altitude of nodes in random trees. Canad. J. Math. 30 997-1015.
24. OTTER, R. (1949). The multiplicative process. Ann. Math. Statist. 20 206-224.
25. PETROV, V. V. (1975). Sums of Independant Random Variables. Springer, Berlin.
26. PORT, S. C. (1994). Theoretical Probability for Applications. Wiley, New York.
27. STEYAERT, J. M. and FLAJOLET, P. (1983). Patterns and pattern-matching in trees: An analysis. Inform. Control 58 19-58.
28. TAKÁCS, L. (1993). Limit distributions for queues and random rooted trees. J. Appl. Math. Stochastic Anal. 6 189-216.