We obtain sharp large deviation estimates for exceedance probabilities in dependent triangular array threshold models with a diverging number of latent factors. The prefactors quantify how latent-factor dependence and tail geometry enter at leading order, yielding three regimes: Gaussian or exponential-power tails produce polylogarithmic refinements of the Bahadur-Rao $n^{-1/2}$ law; regularly varying tails yield index-driven polynomial scaling; and bounded-support (endpoint) cases lead to an $n^{-3/2}$ prefactor. We derive these results through Laplace-Olver asymptotics for exponential integrals and conditional Bahadur-Rao estimates for the triangular arrays. Using these estimates, we establish a Gibbs conditioning principle in total variation: conditioned on a large exceedance event, the default indicators become asymptotically i.i.d., and the loss-given-default distribution is exponentially tilted (with the boundary case handled by an endpoint analysis). As illustrations, we obtain second-order approximations for Value-at-Risk and Expected Shortfall, clarifying when portfolios operate in the genuine large-deviation regime. The results provide a transferable set of techniques-localization, curvature, and tilt identification-for sharp rare-event analysis in dependent threshold systems.

翻译：针对具有发散潜在因子数量的相依三角阵列阈值模型，我们获得了超出概率的尖锐大偏差估计。前因子量化了潜在因子相依性与尾部几何结构如何主导一阶项，从而产生三种机制：高斯或指数幂尾部产生Bahadur-Rao $n^{-1/2}$律的多对数修正；正则变化尾部产生指数驱动的多项式标度；有界支撑（端点）情形导致$n^{-3/2}$前因子。我们通过指数积分的Laplace-Olver渐近分析及三角阵列的条件Bahadur-Rao估计推导了这些结果。基于这些估计，我们在全变差范数下建立了吉布斯条件化原理：在给定大超出事件的条件下，违约指示变量渐近独立同分布，违约损失分布呈指数倾斜（边界情形通过端点分析处理）。作为示例，我们获得了风险价值与期望短缺的二阶近似，明确了投资组合何时处于真实大偏差机制。该研究为相依阈值系统中的尖锐罕见事件分析提供了一套可迁移的技术体系——局部化、曲率与倾斜识别。