We previously demonstrated that there is an abrupt (rather than smooth) transition between reactive and predictive modes of eye-movement tracking of target lights (a phase transition). We also found evidence that the sequence of eye movements in the reactive mode was independent, while those in the predictive mode were correlated and possibly formed a random fractal sequence. Here we confirm the finding of fractal structure by quantifying the rate of decay of nonlinear forecasting when applied to these data. We also estimate the window over which consecutive trials are correlated, and show that the duration of this window is fixed in time rather than number of trials. These results have implications for the neural mechanisms that drive predictive movements.