Correlation Integral for Stationary Gaussian Time Series

The correlation integral of a time series is a normalized coefficient that represents the number of close pairs of points of the series lying in phase space. It has been widely studied in a number of disciplines such as phisycs, mechanical engineering, bioengineering, among others, allowing the estimation of the dimension of an attractor in a chaotic regimen. The computation of the dimension of an attractor allows to distinguish deterministic behavior in stochastic processes with a weak structure on the noise. In this paper, we establish a power law for the limiting expected value of the correlation integral for Gaussian stationary time series. Examples with linear and nonlinear time series are used to illustrate the result.