The dispersion function for the movement of a cloud of tagged sediments with identical hydraulic characteristics under the action of a steady uniform flow may be expressed in terms of some pertinent correlation statistics that can be obtained from the results of single radio-active tracer experiments. These statistics were derived in terms of the auto and cross serial correlations of step lengths and rest durations of single particle motions. The long term dispersion function was examined taking into account the auto and cross serial correlations of the step lengths and rest durations. Comparisons were also made for the case with independent step lengths, independent rest durations and with the step lengths independent of the rest durations.