On a conjecture of Seneta

TL;DR

This paper proves an equivalence in regular variation properties between two functions related to distribution tails, extending a recent conjecture by Seneta.

Contribution

It establishes the if and only if condition for regular variation of two related functions, extending Seneta's conjecture.

Findings

Proves the equivalence of regular variation between h_β and V_β functions.

Extends the validity of Seneta's conjecture to broader cases.

Provides a mathematical foundation for tail behavior analysis.

Abstract

In this short note we prove that $h_\beta(x) = \beta \int_0^x y^{\beta-1} \overline F(y) \mathrm{d} y$ is regularly varying with index $\rho \in [0,\beta)$ if and only if $V_\beta (x) = \int_{[0,x]} y^\beta \mathrm{d} F(y)$ is regularly varying with the same index. This implies an extended version of a recent conjecture by Seneta.