Asymptotic Theory for Long-Memory Processes

Abstract

In Section 2.2 we already mentioned stationary processes whose autocovariance functions converge to zero with power law decay. Because this rate of convergence is slower than that of the usual AR, MA, and ARMA processes, we call them long-memory processes (or processes with long-range dependence). The phenomenon of long-range dependence was known long before suitable statistical models were introduced. Hurst (1951) studied the records of water flows through the Nile and through other rivers, the price of wheat, and meteorological series such as rainfall, temperature, and so on. His empirical conclusion was that the range (to be defined in Section 5.1) of the records shows long-range dependence. Motivated by Hurst’s results, Mandelbrot and Van Ness (1968) introduced fractional Brownian motions and fractional noises (to be defined in Section 5.1), and related works (Mandelbrot and Wallis (1968, 1969a, b)) claimed that Hurst’s findings could be modeled by them. Since then, a lot of probabilistic and statistical methods have been brought in long-memory processes (see Beran (1994a) and Robinson (1994a)). Interestingly, the illuminated results are often different from those for ordinary short-memory processes. Also the applications have been extended from hydrology to a variety of fields such as economics, engineering, enviromental sciences, and physics (see Beran (1994a)). Thus long-memory has become a central component of time series analysis. This chapter is devoted to presenting a concise and modern review of statistical analysis for long-memory processes.