Veraverbeke's theorem at large - On the maximum of some processes with negative drift and heavy tail innovations

TL;DR

This paper extends Veraverbeke's theorem to fractional integrated ARMA models with heavy-tailed innovations, providing new limit theorems for process trajectories conditioned on large maxima.

Contribution

It derives an analogue of Veraverbeke's theorem for fractional ARMA models with regularly varying tails and introduces a general proof scheme applicable to related problems.

Findings

Established an analogue of Veraverbeke's theorem for fractional ARMA models.

Proved limit theorems for process trajectories conditioned on large maxima.

Developed a general proof scheme for tail and maximum-related problems.

Abstract

Veraverbeke's (1977) theorem relates the tail of the distribution of the supremum of a random walk with negative drift to the tail of the distribution of its increments, or equivalently, the probability that a centered random walk with heavy-tail increments hits a moving linear boundary. We study similar problems for more general processes. In particular, we derive an analogue of Veraverbeke's theorem for fractional integrated ARMA models without prehistoric influence, when the innovations have regularly varying tails. Furthermore, we prove some limit theorems for the trajectory of the process, conditionally on a large maximum. Those results are obtained by using a general scheme of proof which we present in some detail and should be of value in other related problems.