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# What is long-range dependency?

Module by: Brandon Hodgson. E-mail the author

## What is long-range dependence?

Self-similartraffic exhibits dependencies over a long range of time scales (Wikipedia 2006). Thus, self-similar processes are said to exhibit long-range dependency (Wikipedia 2006).

Long-range dependent processes are characterised by their autocovariancefunctions (Wikipedia 2006). In long-range processes there is a strong coupling between values at different times (Wikipedia 2006). This means that the decay of the autocovariance function is hyperbolic and decays slower than exponentially, and that the area under the curve is infinite (Wikipedia 2006).

Also, in long-range processes, the correlations at longer time scales are more significant.

• The area under the autocorrelationfunction summed over all lags is infinite (Wikipedia 2006).
• The decay of the autocorrelationfunction is often assumed to have the specific functional form,

(1)(Wikipedia 200).

where ρ(k) is the autocorrelation function at a lag k, α is a parameter in the interval (0,1) and the ~ means asymptotically proportional to as k approaches infinity (Wikipedia 2006).

Exercise: Does a Mandelbrotset exhibit lond-range dependency? Answer

References:

Wikipedia. "Long-range dependency", Wikimedia Foundation Inc,  http://en.wikipedia.org/wiki/Long-range_dependency, Last accessed 11 February 2006.

Brandon Hodgson

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