Fundamental limits for weighted empirical approximations of exponentially tilted distributions

Generating samples from exponentially tilting a given distribution of random vectors when samples from the given distribution are available finds applications in fields such as finance and climate science and in the broad area of rare event simulation. In this article, we discuss the asymptotic efficiency of an estimator obtained by exponentially tilting the empirical distribution. We provide a sharp characterization of how much one can accurately tilt distributions given a certain number of samples. Our findings reveal a surprising dichotomy: While twisting unbounded distributions is a fundamentally hard task, for bounded distributions, one can accurately tilt by a large amount using much fewer samples.