Efficient estimation of density and probability of large deviations of sum of IID random variables

We consider the problem of efficient simulation estimation of the density function at the tails, and the probability of large deviations for an average of independent, identically distributed light-tailed random variables. The latter problem has been extensively studied in literature where state independent exponential twisting based importance sampling has been shown to be asymptotically efficient and a more nuanced state dependent exponential twisting has been shown to have a stronger bounded relative error property. We exploit the saddle-point based representations that exist for these rare quantities, which rely on inverting the characteristic functions of the underlying random variables. We note that these representations reduce the rare event estimation problem to evaluating certain integrals, which may via importance sampling be represented as expectations. Further, it is easy to identify and approximate the zero-variance importance sampling distribution to estimate these integrals. We identify such approximating importance sampling measures and argue that they possess the asymptotically vanishing relative error property.