Title: Clustering and Hitting Times of Threshold Exceedances and Applications,
Abstract: 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.