= Power-law velocity-correlation dispersion
{title2=$\langle\Delta X^2\rangle=2\sigma_v^2\int_0^t(t-s)(1+s)^{-\alpha}\,ds$}
For a zero-mean stationary <velocity> process of variance $\sigma_v^2$ and normalized <autocorrelation> $(1+|s|)^{-\alpha}$, integrating velocity twice gives the displayed expression, exactly rather than just asymptotically. For $\alpha>1$ the long-time <diffusion coefficient> is $\sigma_v^2/(\alpha-1)$. At $\alpha=1$, the displacement variance is asymptotic to $2\sigma_v^2t\log t$. For $0<\alpha<1$, it is asymptotic to $2\sigma_v^2t^{2-\alpha}/[(1-\alpha)(2-\alpha)]$. Linear variance growth does not by itself imply a <central limit theorem> or a <normal distribution> for displacement.
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