The assumption that the earth's reflectivity is random (i.e. has a white spectrum) is only necessary for the statistical extraction of a wavelet from the data itself via the amplitude spectrum of the autocorrelation function. The convolutional model still works regardless of the reflectivity pattern, although getting a wavelet may be more difficult if the "white reflectivity" assumption does not hold.
On the other hand, in most sedimentary basins, earth's reflectivity pattern tends to not be random. Sedimentary deposition happens in nested cycles forming smaller cyclic layers within large cycle layers. This fractal nature is observed in the amplitude spectrum of real (i.e. measured) reflectivity series. Instead of a having a flat (i.e. white) amplitude spectrum, real reflectivity series tend to have low amplitudes in the lower frequencies and high amplitudes in the high frequencies (i.e. blue) while generally conforming to a consistent positive slope line when ploted in octave vs dB. The slope of this line is a characteristic of the local geology and contains information about the fractal nature of the layering.
This concept is used in a technique called spectral blueing (and also similarily in coloured inversion), where seismic data is forced to match the characteristic reflectivity amplitude spectrum line causing an increase in apparent bandwidth.