Efficient Importance Sampling for the Left Tail of Positive Gaussian Quadratic Forms

Estimating the left tail of quadratic forms in Gaussian random vectors is major practical importance many applications. In this letter, we propose an efficient sampling estimator that endowed with bounded relative error property. This property significantly reduces number simulation runs required by proposed compared to naive Monte Carlo (MC), especially when probability interest very small. Selected results are presented illustrate efficiency our MC as well some well-known approximations.