Clustering of extreme events in time series generated by the fractional Ornstein-Uhlenbeck equation.

Abstract

We analyze the time clustering phenomenon in sequences of extremes of time series generated by the fractional Ornstein-Uhlenbeck (fO-U) equation as the source of long-term correlation. We used the percentile-based definition of extremes based on the crossing theory or run theory, where a run is a sequence of L contiguous values above a given percentile. Thus, a sequence of extremes becomes a point process in time, being the time of occurrence of the extreme the starting time of the run. We investigate the relationship between the Hurst exponent related to the time series generated by the fO-U equation and three measures of time clustering of the corresponding extremes defined on the base of the 95th percentile. Our results suggest that for persistent pure fractional Gaussian noise, the sequence of the extremes is clusterized, while extremes obtained by antipersistent or Markovian pure fractional Gaussian noise seem to behave more regularly or Poissonianly. However, for the fractional Ornstein-Uhlenbeck equation, the clustering of extremes is evident even for antipersistent and Markovian cases. This is a result of short range correlations caused by differential and drift terms. The drift parameter influences the extremes clustering effect-it drops with increasing value of the parameter.