Definition of Long Memory

A long time before suitable stochastic processes were available, deviations from independence that were noticeable far beyond the usual time horizon were observed, often even in situations where independence would have seemed a natural assumption. For instance, the Canadian–American astronomer and mathematician Simon Newcomb (Astronomical constants (the elements of the four inner planets and the fundamental constants of astronomy), 1895) noticed that in astronomy errors typically affect whole groups of consecutive observations and therefore drastically increase the “probable error” of estimated astronomical constants so that the usual $\sigma/\sqrt{n}$ -rule no longer applies. Although there may be a number of possible causes for Newcomb’s qualitative finding, stationary long-memory processes provide a plausible “explanation”. Similar conclusions were drawn before by Peirce (Theory of errors of observations, pp. 200–204, 1873) (see also the discussion of Peirce’s data by Wilson and Hilferty (Proc. Natl. Acad. Sci. USA 15(2):120–125, 1929) and later in the book by Mosteller and Tukey (Data analysis and regression: a second course in statistics, 1977) in a section entitled “How $\sigma/\sqrt{n}$ can mislead”). Newcomb’s comments were confirmed a few years later by Pearson (Philosophical transactions of the royal society of London, pp. 235–299, 1902), who carried out experiments simulating astronomical observations. Using an elaborate experimental setup, he demonstrated not only that observers had their own personal bias, but also each individual measurement series showed persisting serial correlations.