A Brief History of Long Memory: Hurst, Mandelbrot and the Road to ARFIMA, 1951–1980

Abstract

Long memory plays an important role in many fields by determining the behaviour and predictability of systems; for instance, climate, hydrology, finance, networks and DNA sequencing. In particular, it is important to test if a process is exhibiting long memory since that impacts the accuracy and confidence with which one may predict future events on the basis of a small amount of historical data. A major force in the development and study of long memory was the late Benoit B. Mandelbrot. Here, we discuss the original motivation of the development of long memory and Mandelbrot's influence on this fascinating field. We will also elucidate the sometimes contrasting approaches to long memory in different scientific communities.Keywords: long-range dependence; Hurst effect; fractionally differenced models; Mandelbrot IntroductionIn many fields, there is strong evidence that a phenomenon called "long memory" plays a significant role, with implications for forecast skill, low frequency variations, and trends. In a stationary time series, the term "long memory"-sometimes "long range dependence" (LRD) or "long term persistence"-implies that there is non-negligible dependence between the present and all points in the past. To dispense quickly with some technicalities, we clarify here that our presentation follows the usual convention in statistics [1,2] and define a stationary finite variance process to have long memory when its two-sided autocorrelation function (ACF) diverges:lim N→∞ ∑ N k=−N ρ(k) → ∞. This is equivalent to its power spectrum having a pole at zero frequency [1,2]. In practice, this means the ACF and the power spectrum both follow a power-law, because the underlying process does not have any characteristic decay timescale. This is in striking contrast to many standard (stationary) stochastic processes where the effect of each data point decays so fast that it rapidly becomes indistinguishable from noise. The study of long memory processes is important because they exhibit nonintuitive properties where many familiar mathematical results fail to hold, and because of the numerous datasets [1,2] where evidence for long memory has been found. In this paper, we will give a historical account of three key aspects of long memory: (1) The environmetric observations in the 1950s which first

On the continuing relevance of Mandelbrot’s non-ergodic fractional renewal models of 1963 to 1967

“…[12,18]); -1/f noise: the observation of singular low frequency behaviour in the empirical periodogram of a time series; -long range dependence (LRD): a property of a stationary model by construction. This can only be inferred to be a property of an empirical time series if certain additional conditions are known to be met, including the important one of stationarity.…”

“…In the 1950s an analogous time domain effect (the Hurst phenomenon) was seen in the statistical growth with the observation time scale τ of Hurst's rescaled range R/S(τ ) calculated on the minima of levels of the Nile river [12]. Rather than the expected dependence ∼τ must asymptotically have J = 1/2.…”

“…Skepticism about fGn-and another related LRD process, Granger and Hosking's autoregressive fractionally integrated moving average (ARFIMA, [12]) as a universal explanation for observed Hurst effects remained (and remains) considerable, though, because of the highly nonMarkovian properties of these processes. Many authors, particularly in statistics and econometrics, have found models based on change points to be better motivated, not least because many datasets are unambiguously known to have change points that need to be handled.…”

“…The physicist's "problem of 1/f noise" has been with us since the pioneering work of Schottky and Johnson in the early 20th century on fluctuating currents in vacuum tubes [11][12][13]. The paradox is usually framed in power spectral terms, i.e.…”

“…The LRD fGn process is the formal differential of H-self similar fractional Brownian motion (fBm), and was subsequently developed by him with Van Ness and Wallis, particularly in a hydrological context [12,14]. fGn is a stationary ergodic process, for which a power spectral density is a natural, well-defined concept, the paradox here residing in the singular behaviour of S (f ) at zero frequency (in the H > 1/2 case).…”