The fixed points of the multivariate smoothing transform

• 成果类型: Article
• 署名作者: Mentemeier, Sebastian
• 署名单位: University of Wroclaw
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-015-0615-y
• 发表日期: 2016
• 页码: 401-458
• 关键词: general branching-processes; renewal theory; functional-equation; martingale convergence; random environment; limit-theorem; variance; trees

摘要

Let (T-1, T-2, . . .) be a sequence of random d x d matrices with nonnegative entries, and let Q be a random vector with nonnegative entries. Consider random vectors X with nonnegative entries, satisfying X =(L) Sigma(i >= 1) TiXi + Q, (*) where =(L) denotes equality of the corresponding laws, (X-i)(i >= 1) are i.i.d. copies of X and independent of (Q, T-1, T-2,...). For d = 1, this equation, known as fixed point equation of the smoothing transform, has been intensively studied. Under assumptions similar to the one-dimensional case, we obtain a complete characterization of all solutions X to (*) in the non-critical case, and existence results in the critical case.