Solutions to complex smoothing equations

• 成果类型: Article
• 署名作者: Meiners, Matthias; Mentemeier, Sebastian
• 署名单位: Technical University of Darmstadt; Dortmund University of Technology
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-016-0709-1
• 发表日期: 2017
• 页码: 199-268
• 关键词: multitype branching-processes; fixed-points; martingale convergence; LIMIT-THEOREMS; kinetic-models; asymptotics; distributions; Transforms; EXISTENCE

摘要

We consider smoothing equations of the form X law = Sigma(j >= 1) TjXj + C where is a given sequence of random variables and are independent copies of X and independent of the sequence . The focus is on complex smoothing equations, i.e., the case where the random variables are complex-valued, but also more general multivariate smoothing equations are considered, in which the are similarity matrices. Under mild assumptions on , we describe the laws of all random variables X solving the above smoothing equation. These are the distributions of randomly shifted and stopped L,vy processes satisfying a certain invariance property called -stability, which is related to operator (semi)stability. The results are applied to various examples from applied probability and statistical physics.