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On a class of renewal functions

Published online by Cambridge University Press:  24 October 2008

D. J. Daley
D. J. Daley
Affiliation:
Statistical Laboratory, University of Cambridge
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Extract

Renewal processes in discrete time (or as they are commonly termed, recurrent events) are appropriately described by renewal sequences {un} which are generated by 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 function f(x) of the lifetime distribution exists, so that a renewal density exists, the analogous property would be that for two renewal density functions h1(x) and h2(x), the function h(x) = h1(x) h2(x) is a renewal density function. A little intuitive reflexion shows that while h(x) dx has a probability density interpretation, this is not in general true of h1(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 densities h(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 which h1(x) and imply that .

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 61 , Issue 2 , April 1965 , pp. 519 - 526
Copyright
Copyright © Cambridge Philosophical Society 1965

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References

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