# Eficient Monte Carlo Simulation of the Left Tail of Positive Gaussian Quadratic Forms

## TL;DR

This paper introduces an efficient importance sampling method for accurately estimating extremely small probabilities in the left tail of Gaussian quadratic forms, significantly reducing computational effort.

## Contribution

It presents a novel importance sampling estimator with bounded relative error for Gaussian quadratic forms, improving efficiency over naive Monte Carlo methods.

## Key findings

- Estimator achieves bounded relative error.
- Significantly fewer simulation runs needed.
- Effective in both real and complex, central and non-central cases.

## Abstract

Estimating the left tail of quadratic forms in Gaussian random vectors is of major practical importance in many applications. In this paper, we propose an efficient and robust importance sampling estimator that is endowed with the bounded relative error property. This property significantly reduces the number of simulation runs required by the proposed estimator compared to naive Monte Carlo. Thus, our importance sampling estimator is especially useful when the probability of interest is very small. Selected simulation results are presented to illustrate the efficiency of our estimator compared to naive Monte Carlo in both central and non-central cases, as well as both real and complex settings.