Long-range dependency (abbreviated as LRD) is a phenomenon that may arise in the analysis of spatial or time series data. It relates to the rate of decay of statistical dependence, with the implication that this decays more slowly than an exponential decay, typically a power-like decay. Some self-similar processes may exhibit long-range dependence, but not all processes having long-range dependence are self-similar. LRD has been used in various field such as video traffic modelling, econometrics, hydrology and linguistics[1] Different definitions of LRD are used for different applications: a review is given by Samorodnitsky.[1]
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One way of characterising long-range and short-range dependent processes is in terms of their autocovariance functions. In short-range dependent processes, the coupling between values at different times decreases rapidly as the time difference increases. Either the autocovariance drops to zero after a certain time-lag, or it eventually has an exponential decay. In long-range processes there is much stronger coupling. The decay of the autocovariance function is power-like and so decays slower than exponentially.
A second way of characterising long- and short-range dependence is in terms of the properties of sums of consecutive values and, in particular, how the properties change as the number of terms in the summation increases. In long-range dependent processes the variance and range of the run-sums are larger and increase more rapidly, compared to properties of the marginal distribution, than for short-range dependence or independent processes. One way of examining this behaviour uses the rescaled range. This aspect of long-range dependence is important in the design of dams on rivers for water resources, where the summations correspond to the total inflow to the dam over an extended period.[2]
A formal statement of difference between SRD and LRD is given as follows:[3] "All short-range dependent processes are characterized by an autocorrelation function which decays exponentially fast; processes with long-range dependence exhibit a much slower decay of the correlations - their autocorrelation functions typically obey some power law."
The Hurst parameter H is a measure of the extent of long-range dependence in a time series. H takes on values from 0.5 to 1. A value of 0.5 indicates the absence of long-range dependence.[4] The closer H is to 1, the greater the degree of persistence or long-range dependence.
Stochastic process that exhibit long-range dependence are often represented by starting from models that have been devised to be either exactly self similar (a self-similar process), or approximately so. Additional components of dependence can then be added in the same way that short-term dependence models are derived from independence or white noise. Among stochastic models that can be used for long-range dependence behaviour are autoregressive fractionally integrated moving average models, which are defined for discrete-time processes, while continuous-time models might start from fractional Brownian motion.