RNA free energy landscapes are analyzed by means of ``time-series'' that are obtained from random walks restricted to excursion sets. The power spectra, the scaling of the jump size distribution, and the scaling of the curve length measured with different yard stick lengths are used to describe the structure of these ``time-series''. Although they are stationary by construction, we find that their local behavior is consistent with both AR(1) and self-affine processes. Random walks confined to excursion sets (i.e., with the restriction that the fitness value exceeds a certain threshold at each step) exhibit essentially the same statistics as free random walks. We find that an AR(1) time series is in general approximately self-affine on time scales up to approximately the correlation length. We present an empirical relation between the correlation parameter $\rho$ of the AR(1) model and the exponents characterizing self-affinity.