On a class of renewal functions
Daryl J. Daley · Mathematical Proceedings of the Cambridge Philosophical Society · 1965
Renewal processes in discrete time (or as they are commonly termed, recurrent events) are appropriately described byrenewal sequences{un} which aregeneratedby discrete distributions , according to the equation Any two renewal sequences {u′n}, {u″n} define another renewal sequence {un} by means of their term-by-term product {un} = {u′nu″n}, for the joint occurrence of two independent recurrent events ℰ′ and ℰ″ is also a recurrent event. Considering a renewal process in continuous time for which we shall suppose a frequency functionf(x) of the lifetime distribution exists, so that a renewal density exists, the analogous property would be that for two renewal density functionsh1(x) andh2(x), the functionh(x) =h1(x)h2(x) is a renewal density function. A little intuitive reflexion shows that whileh(x)dxhas a probability density interpretation, this is not in general true ofh1(x)h2(x)dx. It is not surprising therefore to find in example 1 a case where the product of two renewal densities is not a renewal density. Example 2, on the other hand, shows that in some cases it is true, and taken together with example 1, there is suggested the problem of characterizing the class of renewal densitiesh(x) for which αh(x) is a renewal density for all finite positive α and not merely α in 0 < α ≤A< ∞. In turn this characterization enables us to define a class of renewal densities for whichh1(x) and imply that .