Abstract We study the (approximate) self-affine regimes of random processes with finite domain power-law power spectra for arbitrary values of the spectral exponent α. In particular, we focus on regimes that are imprinted in the structure function of the process. For this function we obtain exact series and asymptotic expansions. We also provide expressions for the usual statistical parameters: variance, correlation time, and introduce a specific parameter, called chronothesy which measures the intensity of the self-affine fluctuations. There are essentially three qualitatively different self-affine regimes, corresponding to values of α in the following intervals: 1 < α < 3, α ≥ 3, and 0 < α <- 1. When in the first regime, the process exhibits a statistical, however, only approximate self-affinity. The self-affinity is embodied in the leading asymptotic term which represents the familiar ideal fractal behavior. If α > 3, the process shows again approximate self-affinity, which with increasing α saturates and leads to a self-affinity exponent of ≈ 2. For 0 < α < 1, the approximate self-affine behavior, which in this case is characterized by a negative exponent, is blurred by intense oscillations to the extent that, in practice, it cannot be observed. As an application, it is shown that the time series generated by sampling X and Y coordinates of the Lorenz system are characterized by blurred self-affinity.