🤖 AI Summary
This paper addresses precise large-deviation estimation of tail probabilities for threshold exceedances in triangular arrays of conditionally dependent latent-factor models, focusing on how tail geometry and latent dependence shape the leading-order asymptotics. For three tail classes—Gaussian/exponential-power, regularly varying, and bounded support—we derive refined decay rates with prefactors of logarithmic polynomials, power laws, and $n^{-3/2}$, respectively. Methodologically, we develop a second-order characterization framework integrating localization, curvature analysis, and tilt identification. Combining Laplace–Olver asymptotics, conditional Bahadur–Rao approximations, and endpoint analysis, we establish a conditional Gibbs principle in total variation distance, revealing asymptotic independence and tilted loss-distribution structure under large-scale default events. The results yield second-order approximations for Value-at-Risk (VaR) and Expected Shortfall (ES), and explicitly characterize the entry criterion into the large-deviation regime.
📝 Abstract
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.