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 'success’ can occur. Such successes lend themselves naturally to approximation by Poisson point processes. If each success occurs after a random delay, however, Poisson convergence may be relatively slow, because each success corresponds 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 foundation. This paper generalizes their correction. For a single renewal process or several renewal processes operating in parallel, this paper gives an asymptotic expansion that contains in successive terms a Poisson point approximation, a generalization of the Altschul-Gish correction, and a correction term beyond that.

We investigate some preservation properties of two nonparametric classes of survival distributions and their duals, under appropriate reliability operations. The aging properties defining these nonparametric classes are based on comparing the mean life of a new unit to the mean residual life function of the asymptotic remaining survival time of the unit under repeated perfect repairs. They are motivated from a point of view that realistic notions of degradation, applicable to repairable systems, should be based on contrasting some aspect of the remaining life of a repairable unit (under a given repair strategy, such as renewals) to the life of a new unit.

A transient cumulative process based upon a sequence of possibly infinite lifetimes is defined, and examples of such a process are described. Given a mild condition on the improper lifetime distribution and given that all lifetimes observed by time t are finite, the expected value of this transient process at t is related to the expected value of a cumulative process based upon proper lifetimes. This relationship is exploited to show that the conditional behavior of the transient process is analogous to that of a proper process and, in particular, the transient process is asymptotically normal.

We study record times, mainly, and sizes in the following context. Let Xn denote the size of the nth event occurring in a point stochastic pacing process 𝒫. The Xn is i.i.d. and 𝒫 is, variously, Poisson, negative binomial, renewal and Furry. Explicit distributions of first-record times are found, domains of attraction studied and the asymptotic lognormality of the nth-record time is shown for Poisson 𝒫.