Evolution of branching processes in a random environment
This review paper presents the known results on the asymptotics of the survival probability and limit theorems conditioned on survival of critical and subcritical branching processes in independent and identically distributed random environments. This is a natural generalization of the time-inhomogeneous branching processes. The key assumptions of the family of population models in question are nonoverlapping generations and discrete time. The reader should be aware of the fact that there are many very interesting papers covering other issues in the theory of branching processes in random environments which are not mentioned here.
KeywordsLimit Theorem STEKLOV Institute Random Environment Annealed Approach Subcritical Case
Unable to display preview. Download preview PDF.
- 1.V. I. Afanasyev, “Limit theorems for a conditional random walk and some applications,” Cand. Sci. (Phys.-Math.) Dissertation (Moscow State Univ., Moscow, 1980).Google Scholar
- 9.V. I. Afanasyev, C. Böinghoff, G. Kersting, and V. A. Vatutin, “Conditional limit theorems for intermediately subcritical branching processes in random environment,” Ann. Inst. Henri Poincaré, Probab. Stat. (in press); arXiv: 1108.2127 [math.PR].Google Scholar
- 28.E. Dyakonova, “On subcritical multi-type branching process in random environment,” in Algorithms, Trees, Combinatorics and Probabilities: Proc. Fifth Colloquium on Mathematics and Computer Science (DMTCS, Nancy, 2008), pp. 401–408.Google Scholar
- 38.M. V. Kozlov, “On the asymptotic behavior of the probability of non-extinction for critical branching processes in a random environment,” Teor. Veroyatn. Primen. 21(4), 813–825 (1976) [Theory Probab. Appl. 21, 791–804 (1977)].Google Scholar
- 40.Q. Liu, “On the survival probability of a branching process in a random environment,” Ann. Inst. Henri Poincaré, Probab. Stat. 32, 1–10 (1996).Google Scholar
- 41.Sevastyanov B.A., Branching Processes (Nauka, Moscow, 1971) [in Russian].Google Scholar
- 49.V. A. Vatutin and E. E. Dyakonova, “Reduced branching processes in random environment,” in Mathematics and Computer Science. II: Algorithms, Trees, Combinatorics and Probabilities, Ed. by B. Chauvin, P. Flajolet, D. Gardy, and A. Mokkadem (Birkhäuser, Basel, 2002), pp. 455–467.Google Scholar
- 52.V. Vatutin and E. Dyakonova, “Yaglom type limit theorem for branching processes in random environment,” in Mathematics and Computer Science. III: Algorithms, Trees, Combinatorics and Probabilities, Ed. by M. Drmota, P. Flajolet, D. Gardy, and B. Gittenberger (Birkhäuser, Basel, 2004), pp. 375–385.CrossRefGoogle Scholar
- 57.V. A. Vatutin and A. E. Kyprianou, “Branching processes in random environment die slowly,” in Algorithms, Trees, Combinatorics and Probabilities: Proc. Fifth Colloquium on Mathematics and Computer Science (DMTCS, Nancy, 2008), pp. 379–400.Google Scholar