The spatial behavior of stochastic acceleration of particles in the expanding solar wind plasma is considered. For this, a standard model is used representing a radially evolving, driven solar wind turbulence. On the basis of a turbulence correlation timescale approach, an analytical model is developed describing the radial dependence of the correlation length of the turbulence driven by stream interactions and the production of pickup ions. This analytical model, which might also be beneficial for other interesting problems such as solar modulation of cosmic rays, is used to consider the radial variation of Fermi-II acceleration of pickup ions in a dissipative, fully driven solar wind turbulence on their way to the heliospheric termination shock. This might be of particular interest for the injection of pickup ions into the process of diffusive shock acceleration and their subsequent conversion to anomalous cosmic rays.