A Seneta's Conjecture and the Williamson Transform

Abstract

Considering slowly varying functions (SVF), %Seneta (2019) Seneta in 2019 conjectured the following implication, for $\alpha\geq1$,$$\int_0^x y^{\alpha-1}(1-F(y))dy\textrm{\ is SVF}\ \Rightarrow\ \int_{0}^x y^{\alpha}dF(y)\textrm{\ is SVF, as $x\to\infty$,}$$where $F(x)$ is a cumulative distribution function on $[0,\infty)$. 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.

Keywords

• regular variation
• de haan class
• truncated moments
• williamson transform