= Uniqueness of a self-decomposition remainder
{title2=$M_R(t)=M_X(t)/M_X(ct)$}
Suppose $M_X$ is finite on a neighborhood of zero, $0<c<1$, and $X$ is independent of each of $R,R'$, with $cX+R$ and $cX+R'$ both having the <law of a random variable> $X$. Independence and the <Tonelli theorem> first give $M_X(t)=M_X(ct)M_R(t)$ as an extended nonnegative <integral>. Since $0<M_X(ct)<\infty$ and $M_X(t)<\infty$, both remainder <moment-generating functions> are finite and equal to the displayed ratio on the same neighborhood. The <uniqueness theorem for moment-generating functions> gives equality of their <probability distributions>. No independence between $R$ and $R'$ is required.
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