Some New Statistical Approaches to the Analysis of Long Memory Processes

This thesis describes methods of analysis and synthesis of long memory processes. Long memory processes are those which exhibit correlations between events separated by a long period of time. This phenomenon is characterized in the frequency domain by a sharp peak in the spectral density function as the frequency approaches zero. This characteristic is observed in many physical time series, including those in the fields of geophysics, astronomy and finance. A class of models that captures such long memory behaviour are fractionally differenced processes, the simplest of these processes is obtained by differencing white noise a fractional number of times. We employ two methods of analyzing such processes: Multitaper spectral estimation and Wavelet analysis. Multitaper spectral analysis uses the average of several direct spectral estimators evaluated using orthogonal tapers. We look at two sets of tapers: the discrete prolate spheroidal sequences and sinusoidal tapers. This method of s..