Uniqueness of a self-decomposition remainder ($M_R(t)=M_X(t)/M_X(ct)$)

Suppose $M_X$ is finite on a neighborhood of zero, $0<c<1$, and $X$ is independent of each of $R,R^{\prime }$, with $cX+R$ and $cX+R^{\prime }$ 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 $R^{\prime }$ is required.