Heavy tailed random variables (rvs) have proven to be an essential element in modeling a wide variety of natural and human induced processes, and the sums of heavy tailed rvs represent a particularly important construct in such models. Oriented toward both geophysical and statistical audiences, this paper discusses the appearance of the Pareto law in seismology and addresses the problem of the statistical approximation for the sums of independent rvs with common Pareto distribution F(x)=1 – xα for 1/2 < α < 2. Such variables have infinite second moment which prevents one from using the Central Limit Theorem to solve the problem. This paper presents five approximation techniques for the Pareto sums and discusses their respective accuracy. The main focus is on the median and the upper and lower quantiles of the sum