%0 Journal Article
%T A Seneta's Conjecture and the Williamson Transform
%J Sahand Communications in Mathematical Analysis
%I University of Maragheh
%Z 2322-5807
%A Omey, Edward
%A Cadena, Meitner
%D 2023
%\ 09/01/2023
%V 20
%N 4
%P 227-241
%! A Seneta's Conjecture and the Williamson Transform
%K Regular variation
%K De Haan class
%K Truncated moments
%K Williamson transform
%R 10.22130/scma.2023.1983415.1223
%X 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.
%U https://scma.maragheh.ac.ir/article_706712_1874b9e9f26269a43add982e3eaf0042.pdf