Approximate, saturated and blurred self-affinity of random processes with finite domain power-law power spectrum

We study the (approximate) self-affine regimes of random processes with finite domain power-law power spectra for arbitrary values of the spectral exponent ct. 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 alpha in the following intervals: 1 < alpha < 3, alpha greater than or equal to 3, and 0 < alpha less than or equal to 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 alpha > 3, the process shows again approximate self-affinity, which with increasing ct saturates and leads to a self-affinity exponent of approximate to 2. For 0 < alpha less than or equal to 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.