Correlation Dimension of Fractional Gaussian Noise: New Evidence from Wavelets

Abstract

In this paper, we study the behavior of the correlation dimension estimated using the Grassberger–Procaccia (GP) algorithm [Grassberger & Procaccia, 1983] in the wavelet domain for functions belonging to Hölder space. We prove that, as the wavelet scale level tends to infinity, the GP correlation dimension estimate tends to zero. Applying this result to the trajectories of the fractional Brownian motion process and using basic properties of the wavelet transform, we show that, for the fractional Gaussian noise process (fGn), the correlation dimension estimated by the GP procedure converges to the zero value. As the fractional Gaussian noise is a stochastic process with 1/fα spectrum, -1 < α < 1, our results confirm Osborne and Provenzale's assertion that colored random noise leads to the convergence of the GP-based correlation dimension estimator. However, our result holds for a different range of the spectrum exponent values. Moreover, for the fGn class of random processes, we found no correspondence between the value of the scaling exponent H and the value of the correlation dimension estimated by the GP algorithm as the latter is simply zero.