On a conjecture of Seneta

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

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