I wonder if there's a timescale-dependence of the appearance of randomness here. Subject the data to a series of Gaussian high bandpass filters in Fourier space and repeat the analysis. Does the randomness get higher the higher the frequency?
I'm not exactly sure what your asking, but basically randomness in returns decreases as you increase the sampling rate (i.e. annual returns are more normal than say minutely returns). This is due basically to the fact that the more activity happens between measurements. (I could be misunderstanding your question.) High frequency measurements of prices often exhibit regularities that result from the trading mechanism e.g. bid-ask bounce.