Heavy-tailed distributions; extreme value theory; large deviations; ruin probabilities; solvency risk
José M. Zapata
TL;DR
Problem: quantify how ruin probabilities decay when the maximum claim in an $n$-policy portfolio exceeds a growing threshold under heavy-tailed losses. Approach: introduce a logarithmically scaled normalization $Z_n = \left(\frac{X_{(n)}}{a_n}\right)^{\frac{\alpha}{\log n}}$ with $a_n = F^{\leftarrow}\left(1-\frac{1}{n}\right)$ and prove a sharp large-deviation principle with speed $\log n$ and rate function $I(x)=\log x$ for $x\ge 1$, uniformly across Borel tail sets. Key contributions: (i) exact tail decay $\lim_{n\to\infty} \frac{1}{\log n} \log \mathbb{P}(Z_n \in A) = -\mathrm{ess.inf}_{x\in A} \log x$, (ii) explicit polynomial ruin decay $\mathrm{RP}_n \sim n^{-\alpha \beta}$ when $\pi_n = a_n n^{\beta-1}$, (iii) demonstration that the asymptotics depend only on the tail index $\alpha$, not on the slowly varying part $L$. Significance: provides precise solvency risk metrics under extreme capital constraints and improves over classical EVT in rare-event regimes.
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.