Well-behaved densities are typically log-convex with heavy tails and log-concave with light ones. We discuss a benchmark for distinguishing between the two cases, based on the observation that large values of a sum \(X_1+X_2\) occur as result of a single big jump with heavy tails whereas \(X_1,X_2\) are of equal order of magnitude in the light-tailed case. The method is based on the ratio \(|X_1-X_2|/(X_1+X_2)\), for which sharp asymptotic result are presented as well as a visual tool for distinguishing between the two cases. The study supplements modern non-parametric density estimation methods where log-concavity plays a main role, as well as heavy-tailed diagnostics such as the mean excess plot.

Keywords: Heavy-tailed; log-concave; Mean excess function; Principle of a single big jump Please add some keywords