Sharp large deviation estimates for heavy-tailed extrema

TL;DR

This paper derives precise large deviation estimates for the maximum of heavy-tailed i.i.d. variables, providing exact decay rates for tail probabilities and applications to ruin probabilities in insurance.

Contribution

It introduces sharp asymptotic formulas for heavy-tailed maxima applicable to all right tail Borel sets, extending existing large deviation results.

Findings

Exact decay rates for exceedance probabilities are established.

Polynomial decay rates for ruin probabilities are derived.

Results are valid for all Borel subsets of the right tail.

Abstract

We establish sharp large deviation asymptotics for the maximum order statistic of independent and identically distributed heavy-tailed random variables, valid for all Borel subsets of the right tail. This result yields exact decay rates for exceedance probabilities at thresholds that grow faster than the natural extreme-value scaling. As an application, we derive the polynomial rate of decay of ruin probabilities in insurance portfolios where insolvency is driven by a single extreme claim.