Scaling properties and entropy of long-range correlated time series

We calculate the Shannon entropy of a time series by using the probability density functions of the characteristic sizes of the long-range correlated clusters introduced in [A. Carbone, G. Castelli, H.E. Stanley, Phys. Rev. E 69 (2004) 026105]. We define three different measures of the entropy related, respectively, to the length, the duration and the area of the clusters. For all the three cases, the entropy increases as the logarithm of a power of the size with exponents equal to the fractal dimension of the cluster length, duration and area, respectively.