Detrending moving average algorithm: A closed-form approximation of the scaling law

The Hurst exponent H of long range correlated series can be estimated by means of the detrending moving average (DMA) method. The computational tool, on which the algorithm is based, is the generalized variance σDMA2=1/(N-n)∑i=nN[y(i)-y˜n(i)]2, with y˜n(i)=1/n∑k=0ny(i-k) being the average over the moving window n and N the dimension of the stochastic series y(i). The ability to yield H relies on the property of σDMA2 to vary as n2H over a wide range of scales [E. Alessio, A. Carbone, G. Castelli, V. Frappietro, Eur. J. Phys. B 27 (2002) 197]. Here, we give a closed form proof that σDMA2 is equivalent to CHn2H and provide an explicit expression for CH. We furthermore compare the values of CH with those obtained by applying the DMA algorithm to artificial self-similar signals.