Abstract: We prove a Tauberian theorem for the Laplace–Stieltjes transform, a Karamata-type theorem, and a monotone density theorem in the framework of regularly log-periodic functions. We provide several applications of these results: for example, we prove that the tail of a nonnegative random variable is regularly log-periodic if and only if the same holds for its Laplace transform at 0, and we determine the exact tail behavior of fixed points of certain smoothing transforms.

2010 AMS Mathematics Subject Classification: Primary 44A10; Secondary 60E99.

Keywords and phrases: regularly log-periodic functions, Tauberian theorem, Karamata theorem, monotone density theorem, smoothing transform, semistable laws, supercritical branching processes