Limit Theorems for the Symbolic Correlation Integral and the Renyi-2 Entropy under Short-range Dependence

The symbolic correlation integral provides a way to measure the complexity of time series and dynamical systems. In the present article we prove limit results for an estimator of this quantity which is based on U-statistics under the assumption of short-range dependence. To this end, we slightly generalize classical limit results in the framework of 1-approximating functionals. Furthermore, we carefully analyze the limit variance. A simulation study with ARMA and ARCH time series as well as a real world data example are also provided. In the latter we show how our method could be used to analyze EEG data in the context of epileptic seizures.