Implication of a power-law power-spectrum for self-affinity

We examine numerically the self-affine scaling of time series with an imposed power-law power spectrum P(ω)=C${\mathrm{\omega }}^{\mathrm{-}\mathrm{\alpha }}$, for different exponents 1≤α≤3, and for different sequences of phases. We use two different criteria for testing self-affinity, a fractal dimension of the graph of the time series, and a more sensitive test based on the scaling of moments of probability distributions. For α≠2, our results suggest that time series with a power-law spectrum are only approximately self-affine, even in the best case of long-time series with high-dimensional, δ-function-correlated, uniformly distributed phases. Scaling curves are most sensitive to phases with long correlation times, are weakly dependent on the shape of the phase probability distribution, and are independent of the fractal dimension of the phases.