On randomly spaced observations and continuous time random walks

TL;DR

This paper analyzes the asymptotic behavior of extreme observations in continuous time random walks with heavy-tailed steps, using point processes theory to generalize previous results and determine distributions of maximal excursions and sojourn times.

Contribution

It introduces a point processes approach to study the asymptotics of extremes in renewal-based observations with heavy tails, extending prior partial results.

Findings

Asymptotic distribution of maximal excursions derived

Distribution of sojourn times characterized

Generalization of earlier results achieved

Abstract

We consider random variables observed at arrival times of a renewal process, which possibly depends on those observations and has regularly varying steps with infinite mean. Due to the dependence and heavy tailed steps, the limiting behavior of extreme observations until a given time $t$ tends to be rather involved. We describe this asymptotics and generalize several partial results which appeared in this setting. In contrast to the earlier studies, our analysis is based in the point processes theory. The theory is applied to determine the asymptotic distribution of maximal excursions and sojourn times for continuous time random walks.