Infinitely Divisible Processes and Related Topics

9704744 Rosinski教授Jan Rosinski打算在关于变换组的平稳性的一般框架下研究非高斯无限可分过程的表示。为了实现这一目标，他建议在随机过程理论和遍历理论、群表示，特别是在Mackey意义下诱导的群表示和泛函分析之间建立新的联系。这些一般结果可应用于平稳、自相似和各向同性无限可分随机场的结构分析。他还打算继续研究随机过程的渐近独立性、样本路径连续性和多重随机积分。无限可分随机过程自然而然地出现在理论和应用概率、通信、网络、数学金融和统计学的许多领域。粗略地说，如果一个过程的行为是由大量相互独立的随机因素决定的，那么它是无限可分的。平稳性是一个描述随机过程统计对称性的概念。平稳的无穷可分过程可能表现出很高的变异性，目前还不能很好地理解它，因为传统的高斯过程方法不适用于这里。该项目旨在开发分析无限可分过程的工具和方法，这些工具和方法可用于高度可变现象的建模。

9704744 Rosinski Professor Jan Rosinski intends to investigate representations of non-Gaussian infinitely divisible processes under a general framework of stationarity with respect to groups of transformations. To accomplish this goal, he proposes to develop new connections between the theory of stochastic processes and ergodic theory, group representations, particularly those induced in the sense of Mackey, and functional analysis. These general results can be applied in the structural analysis of stationary, self-similar, and isotropic infinitely divisible random fields. He also intends to continue research on asymptotic independence of stochastic processes, sample path continuity, and multiple stochastic integrals. Infinitely divisible random processes appear naturally in many areas of theoretical and applied probability, communications, networking, mathematical finance, and statistics. Roughly speaking, a process is infinitely divisible if its behavior is determined by a large number of mutually independent random factors. Stationarity is a notion describing statistical symmetries of a random process. Stationary infinitely divisible processes, which may exhibit high variability, are not well understood at present because the traditional methodology of Gaussian processes is not applicable here. This project is intended to develop tools and methods for analysis of infinitely divisible processes which can be used in modeling of highly variable phenomena.