On Series Expansions for the Renewal Moments

Abstract

Given a renewal situation specified by a `lifetime' distribution function $F(t)$, let $H(t)$ be the renewal function and $\phi_n(t)$ the $n$th $\phi$-moment. Then two (integral) recurrence equations are developed for $\phi_n$. The first expresses $\phi_n$ in terms of $\phi_{n - 1}$ and $H$, and the second is an integral equation involving $\phi_n,\phi_{n - 1}$ 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 $\phi_n(t)$ for any $n$. Further, the coefficients in the series expansion of $\phi_n(t)$ (in powers of $t^m$) may be calculated either from the coefficients in the series expansion for $F(t)$, or that for $H(t)$