Mathematical Sciences: Asymptotics for the Spectra of Long-Memory Processes

如果两个数据点之间的协方差只取决于两个观测点在时间上的距离，而不取决于时间本身，那么随着时间的推移所进行的一系列观测就是平稳的。当观测值之间的距离变大时，协方差衰减得很慢，以至于经典的平稳序列极限理论不适用时，平稳序列具有长记忆。从经济学到地球物理学，许多领域都观察到具有长记忆的数据。最近，研究全球变暖的气候学家利用长记忆建立了温度数据模型。在这个建议中概述的研究，旨在了解一些函数的渐近性质的长记忆过程，这是感兴趣的统计研究人员。研究中使用的主要工具之一是谱密度，它是协方差函数的傅里叶变换。它既抓住了长记忆的本质，又使许多难题在数学上易于处理。随着时间的推移，可以观察到许多类型的数据。例如每月出生率、每日股票价格和每小时温度读数。通常，数据序列（称为时间序列）包含相互关联的观测值。例如，股票明天的价值与其当前价值相关。分析时间序列的经典统计理论依赖于这样一个假设，即在时间上相距很远的观测结果并不密切相关。然而，在从经济学到地球物理学的许多领域，已经观察到不具有这种性质的数据。也就是说，在时间上相距甚远的观测结果可能密切相关。最近，气候学家在研究全球变暖时观察到了这种现象（称为“长记忆”）。因此，为了确定全球变暖是否确实发生了，需要研究长记忆序列的新方法。本提案中概述的研究旨在开发分析长记忆数据的技术。

A sequence of observations made over time is said to be stationary if the covariance between two data points depends only on how far apart in time the two observations were made, and not on the times themselves. A stationary sequence has long memory if, as the distance between observations gets larger, the covariances decay so slowly that the classical limit theory for stationary sequences is not applicable. Data with long memory have been observed in many fields, from economics to geophysics. Most recently, climatologists studying global warming have modeled temperature data using long memory. The research outlined in this proposal is aimed at an understanding of the asymptotic properties of some functions of long-memory processes which are of interest to statistical researchers. One of the primary tools to be employed in the proposed research is the spectral density, which is the Fourier transform of the covariance function. It both captures the essential nature of long memory, and makes many difficult problems mathematically tractable. Many types of data are observed over time. Examples are monthly birth rates, daily stock prices, and hourly temperature readings. Frequently sequences of data (referred to as time series) contain observations that are interrelated. For example, the value of a stock tomorrow is related to its current value. Classical statistical theory for analyzing time series relies on the assumption that observations taken far apart in time are not closely related. However, in many fields from economics to geophysics, data which does not have this property has been observed. That is, observations far apart in time may nevertheless be closely related. Most recently, climatologists have observed this phenomenon (called long memory') while studying global warming. Consequently, to determine whether global warming has indeed occurred, new methods for studying long-memory sequences are needed. The research outlined in this proposal is aimed at developing techniques for the analy sis of long-memory data.