Asymptotic properties of supercritical age-dependent branching processes and homogeneous branching random walks

Stochastic Processes and their Applications

Volumn 82, Issue 1, 1999, Pages 61-87

  Liu, Quansheng ^a  

^a UNIVERSITÉ DE RENNES 1   (France)

Author keywords

60J80; Absolute continuity; Age dependent branching processes; Branching random walks; Decay rate and analiticity of characteristic function; Exponential moments; Functional equation; Left tails; Martingales; Moments; Moments of negative orders; Right tail

Indexed keywords

EID: 0002649784     PISSN: 03044149     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0304-4149(99)00008-3     Document Type: Article

References (26)

1. Athreya K.B. On the supercritical one dimensional age dependent branching processes. Ann. Math. Statist. 40:1969;743-763.
2. Athreya K.B., Kaplan N. Convergence of the age distribution in the one-dimensional supercritical age-dependent branching process. Ann. Probab. 4(1):1976;38-50.
3. Athreya, K.B., Ney, P.E., 1972. Branching Processes. Springer, Berlin.
4. Barral, J., 1997. Continuté, Moments d'ordres négatifs et Analyse Multifractale des cascades multiplicatives de Mandelbrot. Thèse, Univ. Paris-Sud, Orsay.
5. Bellman R., Harris T. On age-dependent binary branching processes. Ann. Math. 55:1952;280-295.
6. Biggins J.D., Grey D.R. Continuity of limit random variables in the branching random walk. J. Appl. Probab. 16:1979;740-749.
7. Biggins J.D., Kyprianou A.E. Seneta-Heyde norming in the branching random walk. Ann. Probab. 25:1997;337-360.
8. Bingham N.H. On the limit of a supercritical branching process. J. Appl. Probab. 25A:1988;215-228.
9. Bingham N.H., Doney R.A. Asymptotic properties of supercritical branching processes II: Crump-Mode and Jirina processes. Adv. Appl. Probab. 7:1975;66-82.
10. Cohn H. Norming constants for the finite mean supercritical Bellman-Harris process. Z. Wahrsch. 61:1982;189-205.
11. Dubuc S. La densité de la loi-limite d'un processus en cascade expansif. Z. Wahrsch. Verw. Gebeite. 19:1971;281-290.
12. Feller, W., 1970. An Introduction to Probability Theory and its Applications, second ed., vol. II. Wiley, New York.
13. Grintsevichyus One limit distribution for a random walk on the line. Lithunian Math. Trans. 15:1975;580-589.
14. Harris T.E. Branching processes. Ann. Math. Statist. 19:1948;474-494.
15. Harris, T.E., 1963. The Theory of Branching Processes. Springer, Berlin.
16. Kasahara Y. Tauberian theorems of exponential type. J. Math. Kyoto Univ. 18:1978;209-219.
17. Kesten H. Random difference equations and renewal theory for products of random matrices. Acta Math. 131:1973;207-248.
18. Kyprianou, A., 1999. Slow variation and uniqueness of solutions to the functional equation in the branching random walk. J. Appl. Probab., in press.
19. Liu, Q.S., 1996. The growth of an entire characteristic function and the tail probabilities of the limit of a tree martingale. In: B. Chauvin, S. Cohen, A. Rouault (Eds.), Trees. Progress in Probability, vol. 40. Birkhäuser, Basel, pp. 51-80.
20. Liu, Q.S., 1997a. Sur une équation fonctionnelle et ses applications: une extension du théorème de Kesten-Stigum concernant des processus de branchement. Adv. Appl. Probab. 29, 353-373.
21. Liu, Q.S., 1997b. Self-similar cascades and the branching random walk. Prépublication 97-03, Univ. Rennes 1.
22. Liu Q.S. Fixed points of a generalized smoothing transformation and applications to the branching random walk. Adv. Appl. Probab. 30:1998;85-112.
23. Liu, Q.S., 1999a. An extension of a functional equation of Mandelbrot and Poincaré. Prépublication, Univ. Rennes 1.
24. Liu, Q.S., 1999b. Extended Multiplicative cascades and the general branching random walk: absolute continuity and asymptotic properties. Prépublications, IRMAR, Univ. Rennes 1.
25. Rösler U. A fixed point theorem for distributions. Stochastic Process. Appl. 42:1992;195-214.
26. Schuh H.J. Seneta constants for the supercritical Bellman-Harris process. Adv. Appl. Probab. 14:1982;732-751.