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

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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.