Finite-size corrections to Poisson approximations of rare events in renewal processes

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.