WAVELET TRANSFORMS OF SELF-SIMILAR PROCESSES

This paper deals with the wavelet transforms of self-similar random processes, of the kind assumed in the Kolmogorov (1941) theory of turbulence. It is shown that, after suitable rescaling, the wavelet transform at a given position becomes a stationary random function of the logarithm of the scale argument in the transform. The rescaling depends on the scaling exponent. The statistical properties of the resulting random fluctuations, such as its correlation function, depend on the choice of the analyzing wavelet. Some implications are: (i) the presence of fluctuations in log-log plots of the absolute value of wavelet transforms versus the scale; (ii) an estimate of the small scale fluctuations, generalizing the iterated logarithm law of Brownian motion; (iii) an ultraviolet ergodic formula giving the scaling exponents in terms of zoom-averages. Some observations are made on the issue of fluctuations observed in wavelet transforms of turbulence data.