Long-range Dependency - Short-range Dependence Vs. Long-range Dependence

Short-range Dependence Vs. Long-range Dependence

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.

A formal statement of difference between SRD and LRD is given as follows: "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 to 1. A value of 0.5 indicates the absence of long-range dependence. The closer H is to 1, the greater the degree of persistence or long-range dependence.