Asymptotic analysis of rare events in high dimensions
Anya Katsevich, Alexander Katsevich
TL;DR
This work develops the first asymptotic high-dimensional theory for rare events where the minimizing point of the exponent lies on the boundary of the rare-event set. It provides a structured Laplace-type expansion for integrals over such sets with explicit finite-sample (nonasymptotic) remainder bounds that quantify how the dimension $d$ interacts with the rarity parameter $\lambda$, and shows how these expansions extend to Gaussian settings and to densities conditional on the rare event. A canonical approximation $\hat{\pi}$ to the conditional density $\pi|_D$ is constructed, with a proven TV distance bound and a practical sampling scheme that uses independent Gaussian and exponential random variables. The framework yields explicit formulas for rare-event probabilities, conditional expectations, and the conditional density, enabling transparent sensitivity analyses and efficient rare-event sampling beyond traditional coordinate-transform methods like SORM, especially in high dimensions. Overall, the results provide both theoretical guarantees and practical tools for analyzing and sampling high-dimensional rare events with boundary minima.
Abstract
Understanding rare events is critical across domains ranging from signal processing to reliability and structural safety, extreme-weather forecasting, and insurance. The analysis of rare events is a computationally challenging problem, particularly in high dimensions $d$. In this work, we develop the first asymptotic high-dimensional theory of rare events. First, we exploit asymptotic integral methods recently developed by the first author to provide an asymptotic expansion of rare event probabilities. The expansion employs the geometry of the rare event boundary and the local behavior of the log probability density. Generically, the expansion is valid if $d^2\llλ$, where $λ$ characterizes the extremity of the event. We prove this condition is necessary by constructing an example in which the first-order remainder is bounded above and below by $d^2/λ$. We also provide a nonasymptotic remainder bound which specifies the precise dependence of the remainder on $d$, $λ$, the density, and the boundary, and which shows that in certain cases, the condition $d^2\ll λ$ can be relaxed. As an application of the theory, we derive asymptotic approximations to rare probabilities under the standard Gaussian density in high dimensions. In the second part of our work, we provide an asymptotic approximation to densities conditional on rare events. This gives rise to simple procedure for approximately sampling conditionally on the rare event using independent Gaussian and exponential random variables.