Correlation integrals for inhomogeneously distributed random data

In calculating the correlation dimension of chaotic time series, frequently the scaling behaviour is unsatisfactory. This is due to the influence of inhomogeneities in the probability distribution, such as boundary and lacunarity effects. To quantify these influences, the correlation integral for arbitrarily distributed random numbers is investigated. An analytical expression for the correlation integral is derived from a histogram of the probability. Different types of inhomogeneities in the probability distribution are then investigated. As an application, a normalization procedure is proposed that improves the scaling behaviour. It can also be extended to the analysis of data from chaotic systems.