Hitting times of threshold exceedances and their distributions

TL;DR

This paper analyzes the distribution and expected value of the first time a stochastic process exceeds a high threshold, with applications to risk assessment in climate, insurance, and social networks, considering dependence and heavy tails.

Contribution

It derives asymptotic distributions and expectations for the first hitting time to high thresholds, extending to multiple hitting times, in dependent and heavy-tailed contexts.

Findings

Asymptotic distribution of first hitting time derived

Expected first hitting time asymptotically characterized

Extensions to multiple hitting times provided

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. 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.