Stoch. Proc. Appl., 2012, Volume 122, Issue 7, Pages 2594–2609
Subcritical branching processes in a random environment without the Cramer condition
V. Vatutina, X. Zhengb
a Department of Discrete Mathematics, Steklov Mathematical Institute, 8 Gubkin Street, 119 991 Moscow, Russia
b Department of ISOM, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon,
Hong Kong Special Administrative Region
A subcritical branching process in random environment (BPRE) is considered whose associated random walk does not satisfy the Cramer condition. The asymptotics for the survival probability of the process is investigated, and a Yaglom type conditional limit theorem is proved for the number of particles up to moment n given survival to this moment. Contrary to other types of subcritical BPRE, the limiting distribution is not discrete. We also show that the process survives for a long time owing to a single big jump of the associate random walk accompanied by a population explosion at the beginning of the process.
|Russian Foundation for Basic Research
|First authorís research partially supported by the Russian Foundation for Basic Research, grant 11-01-00139.
Second authorís research partially supported by GRF 606010 of the HKSAR.
MSC: 60J80, 60K37, 60G50, 60F17
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