Consider a renewal process. The renewal events partition the process into i
.i.d. renewal cycles. Assume that on each cycle, a rare event called 'succe
ss' can occur. Such successes lend themselves naturally to approximation by
Poisson point processes. If each success occurs after a random delay, howe
ver. Poisson convergence may be relatively slow. because each success corre
sponds to a time interval, not a point. In 1996. Altschul and Gish proposed
a finite-size correction to a particular approximation by a Poisson point
process. Their correction is now used routinely (about once a second) when
computers compare biological sequences, although it lacks a mathematical fo
undation. This paper generalizes their correction. For a single renewal pro
cess or several renewal processes operating in parallel, this paper gives a
n asymptotic expansion that contains in successive terms a Poisson point ap
proximation, a generalization of the Aitschul-Gish correction. and a correc
tion term beyond that.