  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. 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 asymptotically equivalent distribution and a limit expectation
of the first hitting time to exceed the threshold $u_n$ as sample size $n$
tends to infinity. The results can be extended to the second and, generally, to
$k$th ($k>2$) hitting times.
