Cycles in finite samples and cumulative processes of higher orders

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Abstract
The process formed by a sequence of cumulative departures from the mean or from some other constant (residual mass curve, cusum chart) is a popular tool for the representation and analysis of time series in many sciences, for example, in hydrology, climatology, economics, game theory. In these and other natural and social sciences, similar cumulative processes also often arise naturally; examples include fluctuations of storage in a dam with a constant release rate, lake levels, volume of glaciers, biomass, inventories, and bank accounts. Moreover, many natural economic and other phenomena may represent, or contain, components of cumulative processes of higher orders, i.e., cumulative processes of cumulative processes. In this paper we show that for a sample {yt(0)}≡{xt} of any finite size N, the pure cumulative process of nth order, yt(n)≡∑i=1t(yi(n−1) − μ(n−1)), where μ(n−1) is the sample mean of {yt(n−1)} and t=1, 2, …, N, converges for n→∞ to a sine wave with a period equal to an integral fraction of the sample size N. This happens for any initial sample {yt(0)} and the convergence is of an exponential order. For samples from most stochastic as well as deterministic processes, the period of the limiting sine wave is equal to the sample size N. This behavior is demonstrated by examples involving samples from various processes ranging from pure random series to various deterministic series and including time series of some natural processes such as streamflow, lake levels, and glacier volumes. The paper includes a demonstration of effects of noise superimposed on, and of error in the value of, sample mean on the rate of convergence, and a discussion of some practical implications of the phenomenon described; it brings together some aspects of the work of Slutzky (1937), Hurst (1951), and Yule (1926).

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