On series expansions for the renewal moments

Abstract

SUMMARY Given a renewal situation specified by a ‘lifetime’ distribution function F( t ) let H( t ) be the renewal function and Ø n ( t ) the nth Ø-moment. Then two (integral) recurrence equations are developed for Ø n . The first expresses Ø n in terms of Ø n-1 and H , and the second is an integral equation involving Ø n-1 and F . It is then shown that if H ( t ) can be represented by an integral function of t m (for some m > 0), then so can Ø n ( t ) for any n . Further, the coefficients in the series expansion of Ø n ( t ) (in powers of t m ) may be calculated either from the coefficients in the series expansion for F ( t ), or that for H ( t ).