In recent years it has become clear that a nonlinear formulation of test-particle theories has to be preferred over the traditional quasilinear approach. Nonlinear transport theories such as the nonlinear guiding-centre theory or the weakly nonlinear theory are based on certain assumptions which cannot be derived systematically. One of the key inputs into these theories is the velocity correlation function. In the current paper the Taylor–Green–Kubo formulation is used to deduce a general relation between the mean square displacement of the particle, the running diffusion coefficient and the velocity correlation function. This relation can be used to extract velocity correlation function from test-particle simulations and from transport theories. The latter possibility is the subject of the current paper. These results, which are essential for the improvement of nonlinear transport theories, are compared with standard models applied previously. An additional result of this paper is that for realistic wave spectra, perpendicular diffusion is recovered even within the magnetostatic slab model.