Asymptotic analysis of rare events in high dimensions

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 $d^2\ll\lambda$, where $\lambda$ 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 $d^2/\lambda$. We also provide a nonasymptotic remainder bound which specifies the precise dependence of the remainder on $d$, $\lambda$, the density, and the boundary, and which shows that in certain cases, the condition $d^2\ll \lambda$ 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.