The random sequence of inter-event times of a level-crossing is a statistical tool that can be used to investigate time series from complex phenomena. Typical features of observed series as the skewed distribution and long range correlations are modeled using non linear transformations applied to Gaussian ARMA processes. We investigate the distribution of the inter-event times of the level-crossing events in ARMA processes in function of the probability corresponding to the level. For Gaussian ARMA processes we establish a representation of this indicator, prove its symmetry and that it is invariant with respect to the application of a non linear monotonic transformation. Using simulated series we provide evidence that the symmetry disappears if a non monotonic transformation is applied to an ARMA process. We estimate this indicator in wind speed time series obtained from three different databases. Data analysis provides evidence that the indicator is non symmetric, suggesting that only highly non linear transformations of ARMA processes can be used in modeling. We discuss the possible use of the inter-event times in the prediction task.
Robust State Estimations in Controlled ARMA Processes with the Non-Gaussian Noises: Applications to the Delayed Dynamics
