Fundamental limits for weighted empirical approximations of tilted distributions
This addresses a fundamental challenge in rare event simulation for fields like finance and climate science, offering theoretical insights into sample efficiency.
The paper tackles the problem of generating samples from tilted distributions when the underlying distribution is unknown, providing a sharp characterization of the asymptotic efficiency of a self-normalized importance sampler. It reveals that the number of samples needed increases polynomially for bounded random vectors but super-polynomially for unbounded ones, depending on the tilt amount.
Consider the task of generating samples from a tilted distribution of a random vector whose underlying distribution is unknown, but samples from it are available. This finds applications in fields such as finance and climate science, and in rare event simulation. In this article, we discuss the asymptotic efficiency of a self-normalized importance sampler of the tilted distribution. We provide a sharp characterization of its accuracy, given the number of samples and the degree of tilt. Our findings reveal a surprising dichotomy: while the number of samples needed to accurately tilt a bounded random vector increases polynomially in the tilt amount, it increases at a super polynomial rate for unbounded distributions.