The Structure of Self-similar Stable Processes with Stationary Increments

具有平稳增量的自相似稳定过程的结构

• 批准号: 0102410
• 负责人: Murad Taqqu
• 金额: --
• 依托单位: Trustees of Boston University
• 依托单位国家: 美国
• 项目类别: Continuing grant
• 财政年份: 2001
• 资助国家: 美国
• 起止时间: 2001-08-01 至 2005-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0102410&HistoricalAwards=false
• 关键词: Structure; Self; similar; Stable; Processes

0102410Taqqu The focus of this research is on a special class of stochastic processes with the following three properties: stationary increments, self-similar and stable non-Gaussian probability laws (stable sssi processes, in short). Unlike the Gaussian case, there are infinitely many different stable sssi processes. This overwhelming variety may be regarded as a fundamental problem. One now has to understand how these processes are different or what it is that they have in common. However, non-Gaussianity also brings to the picture new tools that were unavailable in the Gaussian case. It has been known for quite some time now that non-Gaussian stable processes having some invariance property, like self-similarity or stationarity of the increments, can be associated with nonsingular flows. It is then based on some properties of these flows that one can describe the structure of the corresponding stable processes. The focus will be at first on an important subclass of stable sssi processes called self-similar mixed moving averages. The connection of self-similar mixed moving averages to nonsingular flows allows one to decompose them into separate, independent processes and then explore each part in the decomposition separately. The purpose of this research is to better understand a class of random processes that have characteristics that one encounters in many areas of applications. Examples of such processes include the limit of the so-called renewal reward processes applied in telecommunications and "random wavelet expansions" introduced in probabilistic modeling of images. These processes are fractal-like. They display scale invariance and tend to take often extreme values that deviate greatly from the mean. Their mathematical structure is complex. The goal of this research is to develop tools that can be used to analyze that structure

0102410 Taqqu本研究的重点是一类特殊的随机过程，具有以下三个性质：平稳增量，自相似和稳定的非高斯概率律（简称稳定sssi过程）。与高斯情形不同，有无穷多个不同的稳定sssi过程。这种压倒性的多样性可以被视为一个根本问题。我们现在必须了解这些过程是如何不同的，或者它们有什么共同之处。然而，非高斯性也为图像带来了高斯情况下不可用的新工具。它已经知道了相当长的一段时间，现在，非高斯稳定的过程具有一定的不变性，如自相似性或平稳性的增量，可以与非奇异流。然后，基于这些流的一些性质，人们可以描述相应的稳定过程的结构。重点将首先放在稳定sssi过程的一个重要子类上，称为自相似混合移动平均线。自相似混合移动平均线与非奇异流的连接允许将它们分解为单独的、独立的过程，然后分别探索分解中的每个部分。本研究的目的是更好地理解一类随机过程，这些过程具有在许多应用领域中遇到的特征。这样的过程的例子包括限制所谓的更新奖励过程中应用的电信和“随机小波扩展”中引入的图像的概率建模。这些过程是分形的。它们显示出尺度不变性，并且往往采用与平均值相差很大的极值。它们的数学结构很复杂。这项研究的目标是开发可以用来分析这种结构的工具