Large deviations for point processes based on stationary sequences with heavy tails

TL;DR

This paper develops a framework for analyzing large deviations in point processes derived from stationary sequences with heavy tails, focusing on the order of extreme events and applying it to linear processes and ruin probabilities.

Contribution

It introduces a novel framework that captures both the magnitude and order of extreme values in heavy-tailed stationary sequences, especially for linear processes with random coefficients.

Findings

Describes joint asymptotic behavior of large values in stationary sequences.

Derives asymptotic decay rates for partial sum processes.

Analyzes ruin probabilities in the context of heavy-tailed sequences.

Abstract

In this paper we propose a framework that enables the study of large deviations for point processes based on stationary sequences with regularly varying tails. This framework allows us to keep track not of the magnitude of the extreme values of a process, but also of the order in which these extreme values appear. Particular emphasis is put on (infinite) linear processes with random coefficients. The proposed framework provide a rather complete description of the joint asymptotic behavior of the large values of the stationary sequence. We apply the general result on large deviations for point processes to derive the asymptotic decay of partial sum processes as well as ruin probabilities.