On series expansions for the renewal moments
- 1 June 1963
- journal article
- Published by Oxford University Press (OUP) in Biometrika
- Vol. 50 (1-2) , 75-80
- https://doi.org/10.1093/biomet/50.1-2.75
Abstract
Given a renewal situation specified by a ‘lifetime’ distribution function F(t) let H(t) be the renewal function and Øn(t) the nth Ø-moment. Then two (integral) recurrence equations are developed for Øn. The first expresses Øn in terms of Øn-1 and H, and the second is an integral equation involving Øn-1 and F. It is then shown that if H(t) can be represented by an integral function of tm (for some m > 0), then so can Øn(t) for any n. Further, the coefficients in the series expansion of Øn(t) (in powers of tm) may be calculated either from the coefficients in the series expansion for F(t), or that for H(t).Keywords
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