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Regularly log-periodic functions and some applications

Kevei Péter

Probability and Mathematical Statistics

2020

Vol. 40, Fasc. 1

159--182

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.

On an extension of min-semistable distributions

Ben Alaya M., Huillet T., Porzio A.

Probability and Mathematical Statistics

2007

Vol. 27, Fasc. 2

303--323

This work focuses on a functional equation which extends the notion of min-semistable distributions. Our main results are an existence theorem and a characterization theorem for its solutions. The first establishes the existence of a class of solutions of this equation under a condition on the first zero on the positive axis of the associated structure function. The second shows that solutions belonging to a subclass of complementary distribution functions can be identified by their behavior at the origin. Our constructed solutions are in this subclass. The uniqueness question is also discussed.