Clustering and Hitting Times of Threshold Exceedances and Applications

We investigate exceedances of the process over a sufficiently high threshold. The exceedances determine the risk of hazardous events like climate catastrophes, huge insurance claims, the loss and delay in telecommunication networks. Due to dependence such exceedances tend to occur in clusters. The cluster structure of social networks is caused by dependence (social relationships and interests) between nodes and possibly heavy-tailed distributions of the node degrees. A minimal time to reach a large node determines the first hitting time. We derive an asymptotically equivalent distribution and a limit expectation of the first hitting time to exceed the threshold $u_n$ as the sample size $n$ tends to infinity. The results can be extended to the second and, generally, to the $k$th ($k> 2$) hitting times. Applications in large-scale networks such as social, telecommunication and recommender systems are discussed.