Autocorrelation function simple random walks

In the science of complexity the system response X(t) is expected to depart from the totally random condition of the simple random walk model, since such fluctuations are expected to have memory and correlation. In the physics literature, anomalous diffusion has been associated with phenomena with longtime memory such that the autocorrelation function is... 

A Levy walk, on the other hand, takes into account the fact that longer steps take longer times to complete than do shorter steps. The recognition of this simple fact ties the distribution of step sizes to the distribution of time intervals, which in the case of turbulence was determined by the fluctuations in the fluid velocity [62]. In the present example the continuum form of the Levy walk process is described by Eq. (42), with the autocorrelation function for the random driver being given by the inverse power law Eq. (66) and W is the constant speed of the walker. The asymptotic form of the second moment for this process is... 

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