Criticality for branching processes in random environment

Annals of Probability

Volumn 33, Issue 2, 2005, Pages 645-673

  Afanasyev, V I ^a   Geiger, J ^b   Kersting, G ^c   Vatutin, V A ^a  

^a STEKLOV MATHEMATICAL INSTITUTE   (Russian Federation)

^b University of Koblenz Landau   (Germany)

^c GOETHE UNIVERSITY   (Germany)

Author keywords

Branching process; Conditioned random walk; Functional limit theorem; Random environment; Random walk; Spitzer's condition; Tanaka decomposition

Indexed keywords

EID: 17044388623     PISSN: 00911798     EISSN: None     Source Type: Journal    
DOI: 10.1214/009117904000000928     Document Type: Article

References (23)

1. AFANASYEV, V. I. (1993). A limit theorem for a critical branching process in random environment. Discrete Math. Appl. 5 45-58. (In Russian.)
2. AFANASYEV, V. I. (1997). A new theorem for a critical branching process in random environment. Discrete Math. Appl. 7 497-513.
3. AFANASYEV, V. I. (2001). A functional limit theorem for a critical branching process in a random environment. Discrete Math. Appl. 11587-606.
4. AGRESTI, A. (1975). On the extinction times of varying and random environment branching processes. J. Appl. Probab. 12 39-46.
5. ATHREYA, K. B. and KARLIN, S. (1971). On branching processes with random environments: I, II. Ann. Math. Statist. 42 1499-1520, 1843-1858.
6. ATHREYA, K. B. and NEY, P. (1972). Branching Processes. Springer, Berlin.
7. BERTOIN, J. and DONEY, R. A. (1994). On conditioning a random walk to stay nonnegative. Ann. Probab. 22 2152-2167.
8. BINGHAM, N., GOLDIE, C. and TEUGELS, J. (1987). Regular Variation. Cambridge Univ. Press.
9. DONEY, R. A. (1985). Conditional limit theorems for asymptotically stable random walks. Z. Wahrsch. Verw. Gebiete 70 351-360.
10. DONEY, R. A. (1995). Spitzer's condition and ladder variables in random walks. Probab. Theory Related Fields 101 577-580.
11. DURRETT, R. (1978). Conditioned limit theorems for some null recurrent Markov processes. Ann. Probab. 6 798-828.
12. DYAKONOVA, E. E., GEIGER, J. and VATUTIN, V. A. (2004). On the survival probability and a functional limit theorem for branching processes in random environment. Markov Process. Related Fields 10 289-306.
13. FELLER, W. (1971). An Introduction to Probability Theory and Its Applications II. Wiley, New York.
14. GEIGER, J. and KERSTING, G. (2000). The survival probability of a critical branching process in a random environment. Theory Probab. Appl. 45 517-525.
15. JAGERS, P. (1974). Galton-Watson processes in varying environments. J. Appl. Probab. 11 174-178.
16. JIRINA, M. (1976). Extinction of non-homogeneous Galton-Watson processes. J. Appl. Probab. 13 132-137.
17. KOZLOV, M. V. (1976). On the asymptotic behavior of the probability of non-extinction for critical branching processes in a random environment. Theory Probab. Appl. 21 791-804.
18. KOZLOV, M. V. (1995). A conditional function limit theorem for a critical branching process in a random medium. Dokl. Akad. Nauk 344 12-15. (In Russian.)
19. PETROV, V. V. (1975). Sums of Independent Random Variables. Springer, Berlin.
20. SMITH, W. L. and WILKINSON, W. E. (1969). On branching processes in random environments. Ann. Math. Statist. 40 814-827.
21. TANAKA, H. (1989). Time reversal of random walks in one dimension. Tokyo J. Math. 12 159-174.
22. VATUTIN, V. A. (2002). Reduced branching processes in random environment: The critical case. Theory Probab. Appl. 47 99-113.
23. VATUTIN, V. A. and DYAKONOVA, E. E. (2003). Galton-Watson branching processes in random environment, I: Limit theorems. Tear. Veroyatnost. i Primenen. 48 274-300. (In Russian.)