Renewal theory from the point of view of the theory of probability

Renewal theory is ordinarily reduced to the theory of certain types of integral equations. Since the basis for the integral equations is a simple probability process, however, it is to be expected that a treatment in terms of the theory of probability, which uses the modern developments of this theory, will shed new light on the subject. The purpose of this paper is to present such a treatment. Renewal theory, in the simplest case, deals with the following situation. A population of individuals is at hand; when any individual dies he is immediately replaced by a newly born individual. The problem is to investigate the development of the population, particularly the age distribution. Since the individuals are supposed to live and die independently of each other it is sufficient for most purposes to consider a population consisting at any time of only a single individual. At birth, the future lifetime of the individual is a chance variable with distribution function F(x) (the probability of death before age x). At any age x, the lifetime remaining then has distribution function