The univariate generalized extreme value (GEV) distribution is the most commonly used tool for analyzing the properties of rare events. The ever greater utilization of Bayesian methods for extreme value analysis warrants detailed theoretical investigation, which has thus far been underdeveloped. Even the most basic asymptotic results are difficult to obtain because the GEV fails to satisfy standard regularity conditions. Here, we prove that the posterior distribution of the GEV parameter vector, given $n$ independent and identically distributed samples, converges in distribution to a trivariate normal distribution. The proof necessitates analyzing integrals of the GEV likelihood function over the entire parameter space, which requires considerable care because the support of the GEV density depends on the parameters in complicated ways.

翻译：单变量广义极值（GEV）分布是分析罕见事件性质最常用的工具。贝叶斯方法在极值分析中的应用日益广泛，这需要深入的理论研究，但相关进展仍显不足。由于GEV不满足标准正则条件，即使最基本的渐近结果也难以获得。本文证明：给定$n$个独立同分布样本，GEV参数向量的后验分布依分布收敛于三元正态分布。该证明需对GEV似然函数在整个参数空间上的积分进行分析，由于GEV密度的支撑集与参数存在复杂依赖关系，这一过程需要格外审慎。