Wavelet transform

Traditional spectral decomposition techniques often fail when applied to a self similar signal. This is due to the fact that such a signal is typically the superposition of bursts occurring at many different timescales. Although rare, long bursts having a very large amplitude provide much of the energy content of the signal, while short bursts with small amplitude, although very frequent, give small contribution to the energy of the signal. The energy is therefore fairly well localized in time and the signal is non-stationary. This localization prevents Fourier Analisys to work as an effective tool for this kind of signal.
However, a scale invariant signal can be efficiently analyzed by using the renormalization group approach devised in statistical physics to describe systems at a critical point. The self similar system goes through a consecutive sequence of coarse grainings which generate a new signal with statistical properties similar to those of the original one.
Here you can find a description of the wavelet transform.
The wavelet transform technique makes something related to renormalization group transform. It can be used to investigate self similar processes, characterized by a 1/f^α power spectrum, also with α greater than 1. In this case, the correlation function gives no information on the involved scaling exponents, since it is a constant.