Limiting crossing probabilities of random fields

Journal of Applied Probability, 43(3) (2006), 884-891
http://www.jstor.org/stable/27595781

Random fields on ${\Bbb Z}_{+}^{2}$, with long-range weak dependence for each coordinate individually, usually present clustering of high values. For each one of the eight directions in ${\Bbb Z}_{+}^{2}$, we formulate restriction conditions on local occurrence of two or more crossings of high levels. These smooth oscillation conditions enable computation of the extremal index as a clustering measure from the limiting mean number of crossings. In fact, only four directions must be inspected since for opposite directions we find the same local path crossing behaviour and the same limiting mean number of crossings. The general theory is illustrated with several 1-dependent nonstationary random fields.