Monte Carlo methods with infinite variances

Monte Carlo methods stand as a prominent computational technique extensively applied across a diverse array of disciplines in both natural and social sciences. Monte Carlo methods draw theoretical support from the strong law of large numbers and the central limit theorem, respectively, ensuring the convergence of estimates to the true mean and the weak convergence of the error to a normal distribution. While both theoretical foundations are commonly taken for granted, often without much skepticism, it is not very uncommon to encounter scenarios where the estimator possesses infinite variance, often without being readily apparent, thereby hindering the application of the central limit theorem and thus leading to unstable estimation convergence and the invalidity of confidence intervals, often in a subtle or obscured fashion. The present article aims to provide an accessible entry point into the exploration of diverse research avenues concerning Monte Carlo methods with infinite variances. It endeavors to elucidate those often overlooked yet critical instances, offering insightful discussions on feasible strategies and inherent limitations. Accompanied by a variety of relevant remarks and expositorial examples throughout, we aim to illuminate the potential, limitations and subtleties of Monte Carlo methods in applications when the estimator variance is infinite.