Reduced subcritical Galton-Watson processes in a random environment

Advances in Applied Probability

Volumn 31, Issue 1, 1999, Pages 88-111

  Fleischmann, Klaus ^a   Vatutin, Vladimir A ^b  

^a WEIERSTRASS INSTITUTE FOR APPLIED ANALYSIS AND STOCHASTICS   (Germany)

^b STEKLOV MATHEMATICAL INSTITUTE   (Russian Federation)

Author keywords

[No Author keywords available]

Indexed keywords

FUNCTIONS; THEOREM PROVING; TREES (MATHEMATICS);

CONDITIONAL LIMIT THEOREM; GALTON-WATSON PROCESSES; GENEALOGICAL TREES;

RANDOM PROCESSES;

EID: 85038063528     PISSN: 00018678     EISSN: None     Source Type: Journal    
DOI: 10.1017/S0001867800008971     Document Type: Article

References (27)

1. AFANASYEV, V. I. (1980). Limit theorems for a conditional random walk and some applications. Diss. Cand. Sci., Moscow, MSU.
2. AFANASEV, V. I. (1990). On a maximum of a transient random walk in random environment. Theory Prob. Appl. 35, 205-215.
3. AFANASYEV, V. I. (1998). Limit theorems for a moderately subcritical branching process in a random environment. Discr. Math. Appl. 8, 35-52.
4. AGRESTI, A. (1975). On the extinction time of varying and random environment branching processes. J. Appl. Prob. 12, 39-46.
5. ATHREYA, K. B. AND KARLIN, S. (1972). On branching processes in random environment: II. Limit theorems. Ann. Math. Statist. 42, 1843-1858.
6. ATHREYA, K. B. AND NEY, P. E. (1912). Branching Processes. Springer, Berlin.
7. BOROVKOV, K. A. AND VATUTIN, V. A. (1997). Reduced critical branching processes in random environment. Stoch. Proc. Appl. 71, 225-240.
8. DURRETT, R. (1978). The genealogy of critical branching processes. Stoch. Proc. Appl. 8, 101-116.
9. DWASS, M. (1969). The total progeny in a branching process and related random walk. J. Appl. Prob. 6, 682-686.
10. FELLER, W. (1971). An Introduction to Probability Theory and its Applications. Vol. II, 2nd edn. Wiley, New York.
11. FLEISCHMANN, K. AND PREHN, U. (1974). Ein Grenzwertsatz für subkritische Verzweigungsprozesse mit endlich vielen Typen von Teilchen. Math. Nachr. 64, 357-362.
12. FLEISCHMANN, K. AND SIEGMUND-SCHULTZE, R. (1977). The structure of reduced critical Galton-Watson processes. Math. Nachr. 79, 233-241.
13. JAGERS, P., NERMAN, O. AND TAIB, Z. (1991). When did Joe's great ...grandfather live? Or on the timescale of evolution. In Selected Proceedings of the Sheffield Symposium on Applied Probability, eds. I. V. Basawa and R. L. Taylor (IMS Lecture Notes. Monogr. Ser. 18). Inst. Math. Statist., Hayward, CA.
14. KAO, P. (1978). Limiting diffusion for random walks with drift conditioned to stay positive. J. Appl. Prob. 15, 280-291.
15. KESTEN, H., KOZLOV, M. V. AND SPITZER, F. (1975). A limit theorem for random walk in a random environment. Composito Mathematica 30, 145-168.
16. KOZLOV, M. V. (1973). Random walk in a one-dimensional random environment. Theory Prob. Appl. 18, 387-389.
17. KOZLOV, M. V. (1976). On the asymptotic probability of non-extinction for a critical branching process in random environment. Theory Prob. Appl. 21, 791-804.
18. O'CONNELL, N. (1995). The genealogy of branching processes and the age of our most recent common ancestor. Adv. Appl. Prob. 27, 418-42.
19. PREHN, U. (1979). Die mittlere Quellzeit subkritischer Verzweigungsprozesse. Math Nachr. 88, 409-410.
20. SAGITOV, S. (1995). Three limit theorems for reduced critical branching processes. Russian Math. Surv. 50, 1025-1043.
21. SINAI, J. G. (1982). Limit behavior of one-dimensional random walks in a random medium. Theory Prob. Appl. 27, 256-268.
22. SMITH, W. L. AND WILKINSON, W. E. (1969). On branching processes in random environments. Ann. Math. Statist. 40, 814-827.
23. VATUTIN, V. A. (1979). Distance to the nearest common ancestor in Bellman-Harris branching processes. Math. Notes 25, 378-387.
24. VATUTIN, V. A. AND D'YAKONOVA, E. E. (1998). On the extinction probability at a given moment for a class of branching processes in a random environment (in Russian). Diskretnaya Mathematika 9, 100-126.
25. VERAVERBEKE, N. AND TEUGELS, J. L. (1976). The exponential rate of convergence of the distribution of the maximum of a random walk. J. Appl. Prob. 13, 733-740.
26. VIGILANT, L., PENNINGTON, R., HARPENDING, H., KOCHER, T. D. AND WILSON, A. (1989).Mitochondrial DNA sequences in single hairs from a southern African population. Proc. Nat. Acad. Sci. USA 86, 9350-9354.
27. ZUBKOV, A. M. (1975). Limiting distributions for the distance to the closest mutual ancestor. Theory Prob. Appl. 20, 602-612.