On a Conjecture of Seneta on Regular Variation of Truncated Moments

Péter Kevei

We prove that $h_\beta(x)=\beta\int_0^x y^{\beta-1}\overline F(y)\,y$ is regularly varying with index $\rho\in[0,\beta)$ if and only if $V_\beta(x)=\int_{[0,x]} y^\beta dF(y)$ is regularly varying with the same index, where $\beta>0$, $F(x)$ is a distribution function of a nonnegative random variable, and $\overline F(x)=1-F(x)$. This contains at $\rho=0$, $\beta=1$ a result of Rogozin [8] on relative stability, and at $\rho=0$, $\beta=2$ a new, equivalent characterization of the domain of attraction of the normal law. For $\rho=0$ and $\beta>0$ our result implies a recent conjecture by Seneta [9].