A Random Walk Approach to Galton-Watson Trees

Journal of Theoretical Probability

Volumn 13, Issue 3, 2000, Pages 777-803

  Bennies, Jürgen ^a   Kersting, Götz ^a  

^a GOETHE UNIVERSITY   (Germany)

Author keywords

Branching processes; Functional limit theorem; Galton Watson trees; Random walk bridges; Random walk excursions

Indexed keywords

EID: 0034560041     PISSN: 08949840     EISSN: None     Source Type: Journal    
DOI: 10.1023/A:1007862612753     Document Type: Article

References (38)

1. Aldous, D. J. (1990). The random walk construction of uniform spanning trees and uniform labelled trees. SIAM J. Disc. Math. 3, 450-465.
2. Aldous, D. J. (1991). Asymptotic fringe distributions for general families of random trees. Ann. Appl. Prob. 1, 228-266.
3. Aldous, D. J. (1991). The continuum random tree II: An overview. In Barlow, M., and Bingham, N. (eds.), Stochastic Analysis, Cambridge University Press, pp. 23-70.
4. Aldous, D. J. (1993). The continuum random tree III. Ann. Prob. 21, 248-289.
5. Bennies, J. (1995). Zur Konstruktion von zufälligen Bäumen durch Irrfahrten, Diplom Thesis, Frankfurt/Main.
6. Berg, S., and Mutafchiev, L. R. (1990). Random mappings with an attracting center: Lagrangian distributions and a regression function. J. Appl. Prob. 27, 622-636.
7. Borovkov, K. A., and Vatutin, V. A. (1996). On distribution tails and expectations of maxima in critical branching processes. J. Appl. Prob. 33, 614-622.
8. Durrett, R. T., Iglehart, D. L., and Miller, D. R. (1977). Weak convergence to Brownian meander and Brownian excursion. Ann. Prob. 5, 117-129.
9. Dwass, M. (1969). The total progeny in a branching process and a related random walk. J. Appl. Prob. 6, 682-686.
10. Feller, W. (1966). An Introduction to Probability Theory and Its Applications, Vol. 2, Wiley, Pinceton.
11. Geiger, J. (1997). Growing conditioned Galton-Watson trees from the top, Preprint.
12. Geiger, J., and Kersting, G. (1995). Depth-first search of random trees, and Poisson point processes. In Athreya and Jagers (eds.), Classical and Modern Branching Process, pp. 111-127.
13. Grimmett, G. R. (1980). Random labelled trees and their branching networks. J. Austral. Math. Soc. Ser. A 30, 229-237.
14. Harris, T. E. (1952). First passage and recurrence distributions. Trans. Amer. Math. Soc. 73, 471-486.
15. Harris, T. E. (1963). The Theory of Branching Processes, Springer, Heidelberg.
16. Joffe, A., and Waugh, W. A. O'N. (1982). Exact distribution of kin numbers in a Galton-Watson process. J. Appl. Prob. 19, 767-776.
17. Kaigh, W. D. (1976). An invariance principle for random walk conditioned by a late return to zero. Ann. Prob. 4, 115-121.
18. Kennedy, D. P. (1975). The Galton-Watson process conditioned on the total progeny. J. Appl. Prob. 12, 800-806.
19. Kersting, G. (1998). On the height profile of a conditioned Galton-Watson tree, preprint.
20. Kolchin, V. F. (1986). Random Mappings, Optimization Software, New York.
21. Kolmogorov, A. (1938). Zur Lösung einer biologischen Aufgabe, Izv. NII. mathem. i mek. Tomskogo un. 2, 1-12.
22. Le Gall, J. F. (1989). Marches aléatoires, mouvement brownien et processus de branchement, Seminaire de Probabilites XXIII, Lecture Notes in Math., Vol. 1372, Springer, pp. 258-274.
23. Le Gall, J. F., and Le Jan, Y. (1998). Branching processes in Lévy-processes: The exploration process. Ann. Prob. 26, 213-252.
24. Lindvall, T. (1976). On the maximum of a branching process. Scand. J. Statist. Theory Appl. 3, 209-214.
25. Lyons, R., Pemantle, R., and Peres, Y. (1995). Conceptual proofs of L log L criteria for mean behaviour of branching processes. Ann. Prob. 23, 1125-1138.
26. Neveu, J. (1986). Arbres et processus de Galton-Watson. Ann. Inst. Henri Poincaré 22, 199-207.
27. Pitman, J. W. (1975). One-dimensional Brownian motion and the three-dimensional Bessel process. Adv. Appl. Prob. 7, 511-526.
28. Pitman, J. W. (1997). Enumerations of Trees and Forests related to Branching Processes and Random Walks, Technical Report No. 503, Berkely.
29. Roger, C., and Williams, D. (1987). Markoff Processes, and Martingales, Vol. 2, Wiley, Chichester.
30. Spencer, J. (1993). Nine lectures on random graphs. In Hennequin, P. L. (ed.), Ecole d'été de Probabilités de Saint-Flour XXXI, 1991, Lecture Notes in Mathematics, Vol. 1541, Springer.
31. Spitzer, F. (1964). Principles of Random Walk, Wiley, Princeton.
32. Takács, L. (1977). Combinatorial Methods in the Theory of Stochastic Processes, Krieger Publ., New York.
33. Takács, L. (1990). Counting forests. Discrete Mathematics 84, 323-326.
34. Takács, L. (1993). Limit distributions of queues and random rooted trees. J. Appl. Math. and Stoch. Anal. 6, 189-216.
35. Tierney, L. (1994). Markov chains for exploring Posterior Distributions. Ann. Stat. 4, 1701-1762.
36. Vervaat, W. (1979). A relation between Brownian bridges and Brownian excursions. Ann. Prob. 7, 143-149.
37. Viskov, O. V. (1970). Some comments on branching processes. Math. Zametki 8(4), 701-705.
38. Waymire, E. C. (1991). On network structure function computations. In Brillinger, D., Rosenblatt, M., and Taqqu, M. (eds.), Time Series, Springer, Heidelberg.