Ordinal Patterns in Long-Range Dependent Time Series

We analyze the ordinal structure of long-range dependent time series. To this end, we use so called ordinal patterns which describe the relative position of consecutive data points. We provide two estimators for the probabilities of ordinal patterns and prove limit theorems in different settings, namely stationarity and (less restrictive) stationary increments. In the second setting we encounter a Rosenblatt distribution in the limit. We derive the limit distribution for an estimation of the Hurst parameter H, if it is higher than 3/4. Thus, our theorems complement results for lower values of H which can be found in the literature. Finally, we provide some simulation studies.