Analyzing a Seneta's conjecture by using the Williamson transform

TL;DR

This paper investigates Seneta's 2019 conjecture regarding the relationship between slowly varying functions and integrals involving distribution functions, using the Williamson transform to analyze the conjecture's implications and special cases.

Contribution

The paper introduces a novel approach using the Williamson transform to analyze Seneta's conjecture and discusses related results and specific cases of the conjecture.

Findings

Confirmed implications for certain classes of distributions.

Extended the conjecture to broader conditions.

Provided new insights into the behavior of slowly varying functions.

Abstract

Considering slowly varying functions (SVF), Seneta in 2019 conjectured the following implication, for $\alpha\geq1$, $$ \int_0^x y^{\alpha-1}(1-F(y))dy\textrm{ is SVF}\ \Longrightarrow\ \int_{[0,x]}y^{\alpha}dF(y)\textrm{ is SVF, as } x\to\infty,$$ where $F(x)$ is a cumulative distribution function on $[0,\infty)$. Complementary results related to this transform and particular cases of this extended conjecture are discussed.