Renewal function asymptotics refined à la Feller

Abstract

RENEWAL FUNCTION ASYMPTOTICS REFINED À LA FELLERFeller’s volume 2 shows how to use the Key Renewal Theorem to prove that in the limit x!1, the renewal function Ux of a renewal process with nonarithmetic generic lifetime X with finite mean EX=1=and second moment differs from its linear asymptote x by the quantity 122EX2. His first edition 1966 but not the second in 1971 asserted that a similar approach would refine this asymptotic result when X has finite higher order moments. The paper shows how higher order moments may justify drawing conclusions from a recurrence relation that exploits a general renewal equation and further appeal to the Key Renewal Theorem.