2025/11/17 by Anya Katsevich, Katsevich, Anya, Alexander Katsevich +1
Decision Sciences · Economics, Econometrics and Finance · Mathematics · #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Financial Risk and Volatility Modeling #Probability (math.PR) #Probability and Risk Models #Random Matrices and Applications #Statistics Theory (math.ST)
paper · pdf · doi:10.48550/arxiv.2511.13559
openalex publication_date 2025/11/17 · openalex created_date 2025/11/19 · openalex updated_date 2026/07/28
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 d2≪λ, 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 d2/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 d2≪ 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.