Criticality for branching processes in random environment

• 成果类型: Article
• 署名作者: Afanasyev, VI; Geiger, J; Kersting, G; Vatutin, VA
• 署名单位: Russian Academy of Sciences; Steklov Mathematical Institute of the Russian Academy of Sciences; Goethe University Frankfurt; University of Kaiserslautern
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/009117904000000928
• 发表日期: 2005
• 页码: 645-673
• 关键词: limit-theorems; extinction

摘要

We study branching processes in an i.i.d. random environment, where the associated random walk is of the oscillating type. This class of processes generalizes the classical notion of criticality. The main properties of such branching processes are developed under a general assumption, known as Spitzer's condition in fluctuation theory of random walks, and some additional moment condition. We determine the exact asymptotic behavior of the survival probability and prove conditional functional limit theorems for the generation size process and the associated random walk. The results rely on a stimulating interplay between branching process theory and fluctuation theory of random walks.