We analyse analyse the relation between the asymptotic behaviour of two sequences $\{a(n)\}$ and $\{b(n)\}$ related by the system of equations $nb(n) = a\ast b(n)$, where $\ast$ denotes convolution. This type of relation appears in studying discrete infinitely divisible laws and more recently in risk theory. In Hawkes and Jenkins (1978) the authors considered this relation and obtained the asymptotic behaviour of $b(n)$ in the cases where $a(n)\to\alpha$, or $\frac 1n\sum_{k=0}^na(k)\to \alpha$, where $\alpha>0$. We consider the case $\alpha = 0$ and consider O-analogues.