On a conjecture of Seneta

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.

6 pages

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