Analizing a Seneta's conjecture by using the Williamson transform

Abstract

Considering slowly varying functions (SVF), Seneta in 2019 conjectured the following implication, for α ≥ 1,where F (x) is a cumulative distribution function on [0, ∞). By applying the Williamson transform, an extension of this conjecture is proved. Complementary results related to this transform and particular cases of this extended conjecture are discussed.

Asymptotics for Kendall’s renewal function

“…Throughout we assume that the α−th moment is finite, and we write m(α) = H α (∞) = αW α (∞). The case of m(α) = ∞ has been treated in Jasiulis-Gołdyn et al (2020) and in Omey and Cadena (2022), Kevei (2021). Note that by dominated convergence we have lim x→∞ x −α H α (x) = 0.…”