Mathematical Sciences: Long Memory and Infinite Variance

9404093 Taqqu本研究关注具有长记忆和/或无穷方差的时间序列的研究。在有限方差的情况下，当协方差像幂一样趋于零并且谱密度在原点发散时，会发生长记忆。这种行为在许多现实生活中都可以观察到。有时长记忆与无穷大的方差一起出现，例如当底层过程是稳定的但不是高斯的时候。在这种情况下，分布的尾部像幂函数一样衰减。研究将集中在具有长记忆和/或无穷方差的时间序列的概率和统计特性。大多数现有的统计技术都假设要分析的数据不是很依赖。然而，在一些领域，如经济学和地球物理学，数据可以显示长期的依赖性，有时也可以从他们的平均值大幅波动。这项研究的目标是开发新的技术来建模和分析这种类型的数据。

9404093 Taqqu This research concerns the investigation of the time series with long memory and/or infinite variance. Long memory, in the finite variance case, occurs when the covariances tend to zero like a power and the spectral density diverges at the origin. This behavior has been observed in many real-life phenomena. Sometimes long memory occurs jointly with infinite variance, for example when the underlying process is stable but not Gaussian. In this case it is the tail of the distribution that decays like a power function. The research will focus on the probabilistic and statistical properties of time series with long memory and/or infinite variance. Most of the available statistical techniques assume that the data to be analyzed are not very dependent. However, in a number of areas such as economics and geophysics, the data can display long range dependence and can also sometimes fluctuate significantly from their mean value. The goal of this research is to develop new techniques for modeling and analyzing this type of data.