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II.—Asymptotic Renewal Theorems

Published online by Cambridge University Press:  14 February 2012

Walter L. Smith
Walter L. Smith
Affiliation:
Statistical Laboratory, University of Cambridge.
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Synopsis

A sequence of non-negative random variables {Xi} is called a renewal process, and if the Xi may only take values on some sequence it is termed a discrete renewal process. The greatest k such that X1 + X2 + … + Xk ≤ x(> o) is a random variable N(x) and theorems concerning N(x) are renewal theorems. This paper is concerned with the proofs of a number of renewal theorems, the main emphasis being on processes which are not discrete. It is assumed throughout that the {Xi} are independent and identically distributed.

If H(x) = Ɛ{N(x)} and K(x) is the distribution function of any non-negative random variable with mean K > o, then it is shown that for the non-discrete process

where Ɛ{Xi} need not be finite; a similar result is proved for the discrete process. This general renewal theorem leads to a number of new results concerning the non-discrete process, including a discussion of the stationary “age-distribution” of “renewals” and a discussion of the variance of N(x). Lastly, conditions are established under which

These new conditions are much weaker than those of previous theorems by Feller, Täcklind, and Cox and Smith.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 64 , Issue 1 , 1953 , pp. 9 - 48
Copyright
Copyright © Royal Society of Edinburgh 1953

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References

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