time intervals $(t, t+\Delta t)$.

(iii)

:   Velocity correlations are evanescent. For time steps appreciably longer than $m/f$, which for Brownian particles is actually a quite short time, particle displacements in a series of time steps are very nearly independent from each other.

Conclusion (ii) corresponds to the statement that $x(t)$ is the sum of a series of identically-distributed random variables. Conclusion (iii) corresponds to the independent statement that the time evolution of $x(t)$ is described by a Markoff process. In this very special case, the distribution of particle displacements is described by Doob’s Theorem[@doob1942a]. Doob’s theorem is closely related to the central limit theorem. Doob’s theorem treats random processes such as $\Delta x(t)$, while the central limit theorem treats random variables. For the Langevin model, Doob’s Theorem shows that the distribution of particle displacements is a Gaussian $$P(\Delta x) =   \left(2\pi  \langle \Delta x^{2} \rangle \right)^{-1/2} \exp(- (\Delta x(t))^{2}/ 2 \langle \Delta x^{2} \rangle ).
    \label{eq:gaussianform}$$ For this special case, the incoherent scattering function reduces to $$g^{(1s)}(q,t) = \exp(- q^{2} \langle (\Delta x(t))^{2} \rangle/2).
 \label{eq:g1swrong}$$ Equation \[eq:g1swrong\] is quite accurate for the systems considered in Ref. [@berne1976a], namely highly dilute solutions of monodisperse objects in simple Newtonian solvents.

However, 