A direct proof of a fundamental theorem of renewal theory
- 1 January 1953
- journal article
- research article
- Published by Taylor & Francis in Scandinavian Actuarial Journal
- Vol. 1953 (sup1), 139-150
- https://doi.org/10.1080/03461238.1953.10419467
Abstract
Let {X n } be a sequence of independent absolutely continuous random variables not necessarily non-negative. Let hn (χ) be the frequency function of X 1 + ··· + X n . A theorem is proved concerning the limiting behaviour as X 1 tends to infinity of Σa n h n (χ), for a general class of sequences {a n }. We first give a simple direct proof of a theorem due to Feller and to Täcklind that when the X n are identically distributed with mean μ, Σ h 2 (χ) = 1/ μ . Less is assumed than in Feller's proof and in certain respects our conditions are much weaker than Täcklind's. Applications are given to a problem in genetics and to Tauberian theory.Keywords
This publication has 6 references indexed in Scilit:
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- The theory of genetical recombination. I. Long-chromosome armsProceedings of the Royal Society of London. B. Biological Sciences, 1949
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- On the Integral Equation of Renewal TheoryThe Annals of Mathematical Statistics, 1941